International Mathematics Research Notices

Hai, Phùng Hô; dos Santos, João Pedro

doi: 10.1093/imrn/rnaa247pmid: N/A

Abstract In the first part of this work [ 12], we studied affine group schemes over a discrete valuation ring (DVR) by means of Neron blowups. We also showed how to apply these findings to throw light on the group schemes coming from Tannakian categories of |$\mathcal{D}$|-modules. In the present work, we follow up this theme. We show that a certain class of affine group schemes of “infinite type,” Neron blowups of formal subgroups, are quite typical. We also explain how these group schemes appear naturally in Tannakian categories of |$\mathcal{D}$|-modules. To conclude, we isolate a Tannakian property of affine group schemes, named prudence, which allows one to verify if the underlying ring of functions is a free module over the base ring. This is then successfully applied to obtain a general result on the structure of differential Galois groups over complete DVRs. 1 Introduction In this paper, we continue the analysis of affine flat group schemes over a discrete valuation ring (DVR) |$R$| started in [12] and use it to derive results in differential Galois theory. Recall that the main idea behind the latter theory is to attach to a linear differential systems, or |$\mathcal{D}$|-module, a linear algebraic group and to study these concurrently. Now, when dealing with objects having an “arithmetic” input—like differential systems with coefficients in |$R$|—the group theoretical side of the picture is better grasped with the help of Tannakian categories and flat group schemes that are not necessarily of finite type over |$R$|⁠. (A very simple example is below and details are in [12, Section 7]. See also Section 2.3.) Indeed, for a smooth |$R$|-scheme |$X$| with geometrically connected fibres and a point |$x_0\in X(R)$|⁠, and a |$\mathcal{D}$|–module |$\mathcal{M}$| on |$X$|⁠, the category of representations of the (full) differential Galois group is “generated” by |$\mathcal{M}$|⁠, but, contrary to the case of a base field, this property does not assure the existence of an immersion into |$\textbf{GL}(x_0^*\mathcal{M})$|⁠. It then becomes relevant to throw light on general properties of arbitrary affine flat group schemes. Some of these properties were spotted in [9] and [10] and further meditation conducted to answers which we write down here. To focus our attention, we shall raise three questions—named (UB), (SA), and (PR) below—and structure the paper around them. Let |$R=k [\![ \pi ]\!] $|⁠, |$k$| of characteristic zero, and consider the |$\mathcal{D}$|–module on |$\textrm{Spec}\, R[x]$| associated to the differential equation |$y^{\prime}(x)=\pi y(x)$|⁠. Example 7.11 in [11], coming from [2], shows that the associated differential Galois group fails to be of finite type. In addition, and more importantly, this failure is related to a simple construction—the Neron blowup of a formal subgroup—performed on |$\textbf{G}_{m,R}$|⁠. Assume now only that |$R$| is complete with uniformizer |$\pi $|⁠. The Neron blowup produces, starting from an affine flat group scheme of finite type |$G$| and a flat closed formal subgroup |$\mathfrak{H}$| of the completion |$\widehat G$|⁠, a new group |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$| having the same generic fibre as |$G$|⁠. More precisely, yet still loosely speaking, the ring of |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$| is obtained from the ring of |$G$| by adjoining rational functions of the form $$\begin{align*} & \frac{\textrm{element of the ideal of}\ \mathfrak{H}\otimes R/(\pi^{n})}{\pi^n} \end{align*}$$ for varying |$n$|⁠. See Section 2.2 and [12, Section 5]. It is then natural to raise the following question. (UB) “Ubiquity of blowups”: Assume that |$R$| is complete and let |$G$| be an affine and flat group scheme over |$R$| whose generic fibre |$G\otimes K$| is of finite type. Is it the case that |$G$| is the blowup of a formal subgroup of some group scheme of finite type over |$R$|? This is addressed in Section 3, where we show that, if |$R$| in (UB) has equicharacteristic zero, then the answer is affirmative; see Corollary 3.3. In fact, the particularity that gives these blowups such a prominent role is not exclusive of characteristic |$(0,0)$| and has to do, as visible in Theorem 3.1, with the “constancy” of the centres in the standard sequence (recalled in Section 2.1). This property had already been detected in [12, Section 6] and used to arrive at a less clear result than Theorem 3.1. Now, if Theorem 3.1 and Corollary 3.3 say that Neron blowups of formal subgroups are typical among affine and flat group schemes, nothing assures their relevance in all Tannakian theories over |$R$|⁠. The task of detecting their “presence” in simple instances—we have in mind categories of representations of abstract groups and |$\mathcal{D}$|-modules on proper schemes—comes thus to the front. (Here we understand “presence,” at least in the case of |$\mathcal{D}$|–modules, as “being a full differential Galois group.” Note also that the equation |$y^{\prime}=\pi y$| above assures that Neron blowups of formal subgroups are present in the category of |$\mathcal{D}$|-modules on affine schemes.) As argued below, this task is closely linked to a problem raised in [11, Question 4.1.4] and a fundamental result concerning affine group schemes over a field [25, 3.3 and 14.1]. We therefore prefer to highlight this point: (SA) “Strict pro-algebraicity”: Let |$G$| be an affine and flat group scheme over |$R$|⁠; is it possible to write |$G$| as |$\varprojlim _iG_i$|⁠, where each |$G_j\to G_i$| is a faithfully flat morphism of affine and flat group schemes of finite type? As hinted above, Neron blowups of formal subgroups provide negative answers to (SA): if |$G=\varprojlim _iG_i$| is as in (SA), then each faithfully flat quotient |$G\to H$| having a generic fibre of finite type must be noetherian, a property that these blowups can easily violate. (The proof of Corollary 4.16 contains details of the last argument.) With Corollary 4.16 and Corollary 5.11, we show that, in case |$R=\textbf{C} [\![ t ]\!] $|⁠, a certain |$\mathcal{N}_{\mathfrak{H}}^\infty (\textbf{G}_{a,R}\times \textbf{G}_{m,R})$| appears both in the theory of representations of abstract groups (⁠|$\textbf{Z}$| actually!) and in the theory of |$\mathcal{D}$|-modules on proper schemes. This has the double effect of extending the number of negative answers to (SA) and increasing the relevance of blowing up formal subgroups. Finally, we address a basic property of affine group schemes. (PR) “Projectivity”: Let |$G$| be an affine and flat group scheme over |$R$|⁠. What conditions should the category of representations |$\textrm{Rep}_{R}(G)$| enjoy so that |$R[G]$| is projective as an |$R$|-module? That projectivity as a module is indeed a basic property of group schemes and not simply a commutative algebraic one should be made clearer by pointing out its relation to the existence of arbitrary intersections in representations (Proposition 6.19). The analysis of (PR) is the theme of Section 6 and the answer we offer, for a complete |$R$|⁠, is named prudence, see Definition 6.7 and Theorem 6.11. When the category of representations in question is related to a category of modules on a scheme, the property of being prudent is linked to Grothendieck’s existence theorem in formal geometry. This permits us to apply our findings to verify that differential Galois groups on proper|$R$|-schemes have rings of functions that are free |$R$|-modules, see Theorem 6.20. We end this introduction by summarizing the remaining sections of the paper while throwing light on results that serve to complement or support the ones described above. Section 2 exists mainly to establish notations and explain colloquially the operation of blowing up a formal subgroup scheme plus the concept of differential Galois group. Section 3 addresses Question (UB) and proves Theorem 3.1 and Corollary 3.3. In it, the reader shall also find applications of Theorem 3.1 to unipotent group schemes in equicharacteristic |$(0,0)$|⁠; this particular situation allows to show that formal subgroups are algebraizable (Proposition 3.5), something that will be put to good use once in possession of Theorem 6.20, see Corollary 6.22. Section 4 develops carefully the theory of the Tannakian envelope over |$R$| of an abstract group (Definition 4.1) and demonstrates Corollary 4.16. Section 5 serves to show that on a smooth and proper |$\textbf{C} [\![ t ]\!] $|-scheme |$X$|⁠, the theory of |$\mathcal{D}$|-modules is tantamount to that of the representations of |$\pi _1((X\otimes \textbf{C})^{\textrm{an}})$|⁠, see Theorem 5.10. This result is then applied to establish Corollary 5.11, highlighted above. Section 6 develops the concept of prudence (Definition 6.7) to deal with (PR), as already mentioned before. Other noteworthy points elaborated in Section 6 are a link between prudence and a property of representations appearing in [29, Exposé |$\textrm{VI}_B$|]—see Proposition 6.19—, and, building up on material from Section 3, a finiteness result for differential Galois groups with unipotent generic fibre, see Corollary 6.22. Notations, conventions and standard terminology (1) The ring |$R$| is a DVR with quotient field |$K$| and residue field |$k$|⁠. A uniformizer is denoted by |$\pi $|⁠, except in Section 5, where we call it |$t$| to avoid confusion. (2) The spectrum of |$R$| is denoted by |$S$|⁠, and the quotient ring |$R/(\pi ^{n+1})$| is denoted by |$R_n$|⁠. (3) The characteristic of |$R$| is the ordered pair |$(\textrm{char}.\,K,\textrm{char.}\,k)$|⁠. (4) If |$M$| is an |$R$|-submodule of |$N$|⁠, we say that |$M$| is saturated in |$N$| if |$N/M$| has no |$\pi $|-torsion. (5) Given an object |$X$| over |$R$| (a scheme, a module, etc), we sometimes find useful to write |$X_k$| instead of |$X\otimes _Rk$|⁠, |$X_K$| instead of |$X\otimes _RK$|⁠, etc. If context prevents any misunderstanding, we also employ |$X_n$| instead of |$X\otimes _RR_n$|⁠. (6) To avoid repetitions, by a group scheme over some ring |$A$|⁠, we understand an affine group scheme over |$A$|⁠. (7) The category of flat group schemes over a ring |$A$| will be denoted by |$(\textbf{FGSch}/A)$|⁠. (8) If |$V$| is a free |$R$|-module of finite rank, we write |$\textbf{GL}(V)$| for the general linear group scheme representing |$A\mapsto \textrm{Aut}_A(V\otimes A)$|⁠. If |$V=R^n$|⁠, then |$\textbf{GL}(V)=\textbf{GL}_{n}$|⁠. (9) If |$G$| is a group scheme over |$R$|⁠, we let |$\textrm{Rep}_R(G)$| stand for the category of representations of |$G$| which are, as |$R$|-modules, of finite type over |$R$|⁠. (We adopt Jantzen’s and Waterhouse’s definition of representation. See [15, Part I, 2.7 and 2.8, 29ff] and [25, 3.1-2, 21ff].) The full subcategory of |$\textrm{Rep}_{R}(G)$| having as objects those underlying a free |$R$|-module is denoted by |$\textrm{Rep}^\circ _{R}(G)$|⁠. (In [12] we employed the more confusing notation |$\textrm{Rep}_{R}(G)^o$|⁠.) (10) Given an arrow |$\rho :G\to H$| of group schemes over |$R$|⁠, we let |$\rho ^\#:\textrm{Rep}_{R}(H)\to \textrm{Rep}_{R}(G)$| stand for the associated functor. (11) If |$\Gamma $| is an abstract group and |$A$| is a commutative ring, we write |$A\Gamma $| for the group ring of |$\Gamma $| with coefficients in |$A$|⁠. We let |$\textrm{Rep}_{A}(\Gamma )$| stand for the category of left |$A\Gamma $|-modules whose underlying |$A$|-module is of finite type. The full subcategory of |$\textrm{Rep}_{A}(\Gamma )$| whose objects are moreover projective |$A$|-modules shall be denoted by |$\textrm{Rep}^\circ _{A}(\Gamma )$|⁠. (12) For an affine scheme |$X$| over |$R$|⁠, we call |$\mathcal{O}(X)$| the ring of functions of |$X$|⁠, and denote it by |$R[X]$|⁠. More generally, if |$A$| is any |$R$|-algebra, we write |$A[X]$| to denote |$\mathcal O(X\otimes _R A)$|⁠. (13) If |$G\in (\textbf{FGSch}/R)$|⁠, its right-regular module is obtained by letting |$G$| act on |$R[G]$| via right translations. It is denoted by |$R[G]_{\textrm{right}}$| below. (14) For an affine |$R$|-adic formal scheme |$\mathfrak{X}$|⁠, we write |$R\langle \mathfrak{X}\rangle $| for the ring |$\mathcal O(\mathfrak{X})$|⁠. If |$\mathfrak{X}$| is the |$\pi $|-adic completion (or completion along the closed fibre) of some affine |$R$|-scheme |$X$|⁠, we write |$R\langle X\rangle $| or |$R[X]^\wedge $| to denote |$R\langle \mathfrak{X}\rangle $|⁠. (15) A formal group scheme over |$R$| is a group object in the category of affine formal |$R$|-adic schemes. (16) If |$X$| is a ringed space, we let |$|X|$| stand for the its underlying topological space. (17) If |$G$| is a group scheme over a ring |$A$|⁠, and |$\mathfrak{a}\subset R[G]$| is the augmentation ideal, we define the conormal module |$\omega (G/A)$| as being the |$A$|-module |$\mathfrak{a}/\mathfrak{a}^2$| [6, II.4.3.4]. Its dual |$A$|-module is the Lie algebra |$\textrm{Lie}(G)$| of |$G$| (see II.4.3.6 and II.4.4.8 of [6]). Similar conventions are in force when dealing with formal affine group schemes. (18) By a |$\otimes $| category or tensor category, we mean a |$\otimes $|-category ACU in the sense of [21, I.2.4, 38ff], or a tensor category in the sense of [8, Definition 1.1, page 105]. Unless mentioned otherwise, all |$\otimes $|-functors are ACU [21, I.4.2] and all |$\otimes $|-natural transformation are unital (which are the conventions of [8, Section 1]). 2 Preliminary Material For the convenience of the reader, we briefly recall some thoughts from [12] employed here. More detailed explanations are to be found in [12] and its references. 2.1 The standard sequences [12, Section 2], [26, Section 1] Let $$\begin{align*} &\rho:G^{\prime}\longrightarrow G\end{align*}$$ be an arrow of |$(\textbf{FGSch}/R)$| inducing an isomorphism on generic fibres. We then associate to |$\rho $| its standard sequence: as follows. The arrow |$\varphi _0:G_1\to G_0$| is the Neron blowup of |$B_0:=\textrm{Im}(\rho _0\otimes k)$|⁠, and |$\rho _1$| is the morphism obtained by the universal property. Now, if |$\rho _n:G^{\prime}\to G_n$| is defined, then |$\varphi _n:G_{n+1}\to G_n$| is the Neron blowup of $$\begin{align*} & B_n:=\textrm{Im}(\rho_n\otimes k) \end{align*}$$ and |$\rho _{n+1}$| is once more derived from the universal property. In particular, for each |$n$|⁠, the morphisms |$\rho _n\otimes k:G^{\prime}\otimes k\to B_n$| and |$\varphi _n:B_{n+1}\to B_{n}$| are faithfully flat because of [25, Theorem 14.1]. In [12, Theorem 2.11], it is proved that the morphism $$\begin{align*} &G^{\prime}\longrightarrow\varprojlim_n G_n\end{align*}$$ obtained from the above commutative diagram is an isomorphism of group schemes. 2.2 Blowing up formal subgroup schemes [12, Section 5] We assume that |$R$| is complete. Let |$G$| be a group scheme over |$R$|⁠, which is flat and of finite type. Denote by |$\widehat G$| its |$\pi $|-adic completion (or completion along the closed fibre): it is a group object in the category of |$R$|-adic affine formal schemes. Let |$\mathfrak{H}\subset \widehat G$| be a closed, flat formal subgroup scheme and let |$I_n\subset R[G]$| be the ideal of |$H_n:=\mathfrak{H}\otimes R_n$|⁠; note that |$\pi ^{n+1}\in I_n$| and that |$R\langle \mathfrak{H}\rangle =\varprojlim _nR_n[G]/I_nR_n[G]$| (since all ideals in |$R\langle G\rangle $| are closed). We then define the Neron blowup of |$\mathfrak{H}$|⁠, |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$|⁠, as being the group scheme whose Hopf algebra, viewed as a subring of |$K[G]$|⁠, is $$\begin{align*} & \varinjlim_n R[G][\pi^{-n-1}I_n]. \end{align*}$$ By definition, |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$| lies in |$(\textbf{FGSch}/R)$|⁠; it usually fails to be of finite type over |$R$| and is the source of a morphism of group schemes |$\mathcal{N}_{\mathfrak{H}}^\infty (G)\to G$| inducing an isomorphism on generic fibres. One fundamental feature of |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$| that is worth mentioning here is that |$\mathcal{N}_{\mathfrak{H}}^\infty (G)\otimes R_n\to G\otimes R_n$| induces an isomorphism between the source and |$H_n$| [12, Corollary 5.11]. Let us profit to observe that |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$| represents the sub-functor of |$\textrm{Hom}_R(R[G],-)$| given by $$\begin{align*} & \left\{R\textrm{-flat algebras}\right\}\longrightarrow (\textrm{Groups}),\end{align*}$$ $$\begin{align*} &A\longmapsto\left\{g\in\textrm{Hom}_R(R[G],A)\,:\,\begin{array}{c} \hat g:R\langle G\rangle\to \hat A \\ \textrm{factors through}\ R\langle\mathfrak{H}\rangle\end{array}\right\}. \end{align*}$$ If |$H\subset G$| is a closed, flat subgroup scheme, we let |$\mathcal{N}_H^\infty (G)$| stand for the Neron blowup of the (necessarily |$R$|-flat) closed subgroup |$\widehat H\subset \widehat G$|⁠. It is a simple matter to verify that |$R[\mathcal{N}_H^\infty (G)]=\varinjlim _n R[G][\pi ^{-n-1}I]$|⁠, where |$I$| is the ideal of |$H$|⁠. Example 2.1 (See [12, Example 1.1]). Consider the pro-system of affine group schemes $$\begin{align}& \ldots\longrightarrow \textbf{G}_{a,R}\xrightarrow{\times \pi}\textbf{G}_{a,R}\xrightarrow{\times \pi}\textbf{G}_{a,R}, \end{align}$$(1) which corresponds, on the level of rings, to the inductive system $$\begin{align}& R[x_0]\longrightarrow R[x_1]\longrightarrow\ldots,\qquad x_i\longmapsto \pi x_{i+1}. \end{align}$$(2) The limit of diagram (1) is simply |$\mathcal{N}_{\{e\}}^\infty (\textbf{G}_a)$| and the associated ring, the colimit of (2), is |$\left \{ P\in K[x_0]\, | \,P(0)\in R\right \}$|⁠. Note that the |$R$|-module |$R[G]$|⁠, being isomorphic to |$R\oplus K\oplus K\oplus \cdots $|⁠, is not projective and that |$G\otimes R_n$| is always the trivial group scheme over |$R_n$|⁠. 2.3 Differential Galois groups [12, Section 7] Let |$X$| be a smooth |$R$|-scheme and |$\mathcal{D}(X/R)$| be the ring of |$R$|–linear differential operators on |$X$|⁠, see [13, |$\textrm{IV}_{\textrm{4}}$|⁠, 16.8] or [4, §2]. The existence of a morphism of rings |$\mathcal{O}_X\to \mathcal{D}(X/R)$| allows us to see any left |$\mathcal{D}(X/R)$|-module as an |$\mathcal{O}_X$|-module and we let |$\mathcal{D}(X/R)\text{-}\textbf{mod}$| stand for the category of left |$\mathcal{D}(X/R)$|-modules, which, when regarded as |$\mathcal{O}_X$|-modules, are coherent. As is well known, in case |$R$| is a |$\textbf{Q}$|-algebra, the category |$\mathcal{D}(X/R)\text{-}\textbf{mod}$| is equivalent to the category of integrable connections [4, Theorem 2.15]. The category |$\mathcal{D}(X/R)\text{-}\textbf{mod}$| possess a natural tensor product, which, on the level of coherent sheaves, is simply the standard tensor product. In addition, an object |$\mathcal{M}\in \mathcal{D}(X/R)\text{-}\textbf{mod}$| is locally free (as an |$\mathcal{O}_X$|-module) if and only if it is free of |$\pi $|–torsion. As in [11, Section 7], the full subcategory of |$\mathcal{D}(X/R)\text{-}\textbf{mod}$| formed by those objects that are locally free as |$\mathcal{O}_X$|-modules will be denoted by |$\mathcal{D}(X/R)\text{-}\textbf{mod}^\circ $|⁠. These facts allow us to apply techniques from Tannakian categories. We now suppose that |$X$| has geometrically connected fibres over |$R$| and is endowed with an |$R$|-point |$x_0$|⁠. Let |$\mathcal{M}\in \mathcal{D}(X/R)\text{-}\textbf{mod}^\circ $| be given; for |$a,b\in \textbf{N}$|⁠, we put |$\textbf{T}^{a,b}\mathcal{M}=\mathcal{M}^{\otimes a}\otimes \mathcal{M}^{\vee \otimes b}$| (here |$\mathcal{M}^\vee $| is the dual of |$\mathcal{M}$|⁠). The full subcategory of |$\mathcal{D}(X/R)\text{-}\textbf{mod}$| whose objects are subquotients (= quotients of subobjects) of direct sums of the form |$\bigoplus _i\textbf{T}^{a_i,b_i}\mathcal{M}$| is denoted |$\langle \mathcal{M}\rangle _\otimes $|⁠. The functor “fibre at |$x_0$|⁠,” $$\begin{align*} &x_0^*:\langle\mathcal{M}\rangle_\otimes\longrightarrow R\text{-}\textbf{mod},\end{align*}$$ produces, by Tannkian duality (cf. [21, II.4.1, 152ff] or [11, Theorem 1.2.6]), a flat group scheme |$\textrm{Gal}^{\prime}(\mathcal{M})$| and an equivalence $$\begin{align*} &\overline{x}_0^*: \langle\mathcal{M}\rangle_\otimes\ \stackrel{\sim}{\longrightarrow}\textrm{Rep}_{R}(\textrm{Gal}^{\prime}(\mathcal{M})) \end{align*}$$ of |$R$|-linear tensor categories. The group |$\textrm{Gal}^{\prime}(\mathcal{M})$| is called the full differential Galois group of |$\mathcal{M}$|⁠. Contrary to the case of a field, the natural representation $$\begin{align*} & \rho:\textrm{Gal}^{\prime}(\mathcal{M})\longrightarrow \textbf{GL}(x_0^*\mathcal{M}) \end{align*}$$ obtained from |$\overline{x}^*_0$| is not necessarily a closed immersion. On the other hand, it is possible to factor |$\rho $| as $$\begin{align*} & \textrm{Gal}^{\prime}(\mathcal{M})\stackrel{\sigma}{\longrightarrow} \textrm{Gal}(\mathcal{M})\stackrel{\tau}{\longrightarrow} \textbf{GL}(x_0^*\mathcal{M}), \end{align*}$$ where |$ \textrm{Gal}(\mathcal{M})$| is a flat group scheme called the restricted differential Galois group. The arrow |$\sigma $| induces an isomorphism of generic fibres and |$\tau $| is a closed immersion. 3 The Ubiquity of Blowups of Formal Subgroups In this section, we assume that |$R$| is complete. Let |$G^{\prime}\in (\textbf{FGSch}/R)$| have a generic fibre |$G^{\prime}\otimes K$| of finite type (over |$K$|⁠). This being so, there exists some |$G\in (\textbf{FGSch}/R)$|⁠, now of finite type over |$R$|⁠, which is the target of a morphism $$\begin{align*} &\rho:G^{\prime}\longrightarrow G\end{align*}$$ inducing an isomorphism on generic fibres. Then, as explained on Section 2.1, we have the standard sequence of |$\rho $|⁠: where each |$G_n$| is of finite type over |$R$| (by construction). To state Theorem 3.1 below, we let $$\begin{align*} &\begin{split} B_n&= \textrm{image of}\ \rho_n\otimes k:G^{\prime}\otimes k\to G_n\otimes k \\&=\textrm{centre of}\ \varphi_n:G_{n+1}\to G_n. \end{split}\end{align*}$$ Theorem 3.1. (1) Assume that for all |$n$|⁠, the morphisms |$\varphi _n:B_{n+1}\to B_n$| are isomorphisms. Then (a) the morphism induced between |$\pi $|-adic completions |$R\langle G\rangle \to R\langle G^{\prime}\rangle $| is surjective. (b) Let |$\mathfrak{H}$| stand for the closed formal subgroup scheme of |$\widehat G$| cut out by the kernel of |$R\langle G\rangle \to R\langle G^{\prime}\rangle $|⁠. (Clearly, |$\widehat G^{\prime}\simeq \mathfrak{H}$|⁠.) Then there exists an isomorphism of group schemes $$\begin{align*} &\sigma:G^{\prime}\stackrel{\sim}{\longrightarrow} \mathcal{N}_{\mathfrak{H}}^\infty(G)\end{align*}$$ rendering commutative. (2) Assume that the characteristic of |$k$| is zero. Then, there exists |$n_0\in \textbf{N}$| such that, for all |$n\ge n_0$|⁠, the arrows |$\varphi _{n}:B_{n+1}\to B_n$| are isomorphisms. The proof shall need the following: Lemma 3.2. The obvious morphisms |$G^{\prime}\otimes k\to B_n$| induce an isomorphism $$\begin{align*} & G^{\prime}\otimes k\stackrel{\sim}{\longrightarrow}\varprojlim_nB_n. \end{align*}$$ Proof. One considers the commutative diagram of |$k$|-algebras obtained from the construction of the standard sequence (As usual, |$\hookrightarrow $| means injection, and |$\twoheadrightarrow $| surjection.) It is then clear that $$\begin{align*} &\varinjlim_nk[G_n]\longrightarrow \varinjlim_nk[B_n]\end{align*}$$ is an isomorphism. As explained in Section 2.1, the canonical arrow |$G^{\prime}\to \varprojlim G_n$| induces an isomorphism. Since taking tensor products commutes with direct limits [17, Theorem A.1], the result follows. Proof. of Theorem 3.1 (1a) The morphism |$G^{\prime}\otimes k\to G\otimes k$| induces an isomorphism between |$G^{\prime}\otimes k$| and |$B_0$| because of the assumption and of Lemma 3.2. Hence, the arrow between |$k$|-algebras |$k[G]\to k[G^{\prime}]$| is surjective. As is well known (see e.g., [3, 10.23(ii), page 112]), the induced arrow between completions |$R\langle G\rangle \to R\langle G^{\prime}\rangle $| is then surjective. (1b) Let |$I_n\subset R[G]$| be the ideal of the closed immersion |$G^{\prime}\otimes R_n\to G\otimes R_n\to G$| (in particular |$\pi ^{n+1}\in I_n$|⁠). Writing |$\mathcal{N}$| instead of |$\mathcal{N}_{\mathfrak{H}}^\infty (G)$|⁠, the definition says that $$\begin{align*} &R[\mathcal{N}]=\bigcup_nR[G][\pi^{-n-1}I_n].\end{align*}$$ Since, by construction, the morphism of |$R$|-algebras |$R[G]\to R[G^{\prime}]$| sends |$I_n$| to the ideal |$\pi ^{n+1}R[G^{\prime}]$|⁠, we conclude that |$R[G]\to R[G^{\prime}]$| factors thought |$R[\mathcal{N}]$|⁠, that is, |$\rho :G^{\prime}\to G$| factors through |$\mathcal{N}\to G$|⁠. Using that |$\mathcal{N} \otimes k\to G\otimes k$| induces an isomorphism between |$\mathcal{N} \otimes k$| and |$\mathfrak{H}\otimes k$| [12, Corollary 5.11], we conclude that |$G^{\prime}\otimes k\to \mathcal{N}\otimes k$| is also an isomorphism. Together with the fact that |$G^{\prime}\otimes K\to \mathcal{N}\otimes K$| is an isomorphism, this easily implies that |$G^{\prime}\to \mathcal{N}$| is an isomorphism [26, Lemma 1.3]. (2) As already remarked in Section 2.1, the morphisms $$\begin{align*} &\varphi_n|_{B_{n+1}}:B_{n+1}\longrightarrow B_n\end{align*}$$ are all faithfully flat. It then follows from [27, |$ \textrm{VI}_B$|⁠, Proposition 1.2, page 335] that $$\begin{align*} & \dim B_{n+1}= \dim B_n+\dim \textrm{Ker}(\varphi_n|_{B_{n+1}}). \end{align*}$$( * ) Using the equality |$\dim G_n\otimes k=\dim G_n\otimes K$| [27, |$ \textrm{VI}_B$|⁠, Corollary 4.3, p.358], we conclude that $$\begin{align*} \dim B_n&\le \dim G_n\otimes k \\ &=\dim G_n\otimes K \\ &=\dim G\otimes K. \end{align*}$$ Let |$n_0\in \textbf{N}$| be such that |$\dim B_{n_0}$| is maximal. We then derive from equation (*) above that for all |$n\ge n_0$|⁠, $$\begin{align*} & \dim \textrm{Ker}(\varphi_n|_{B_{n+1}})=0. \end{align*}$$( * *) Because of the assumption on the characteristic, we know that |$B_n$| and |$G_n\otimes k$| are smooth|$k$|-schemes, so that |$B_n\otimes k\to G_n\otimes k$| is a regular immersion [13, |$\textrm{IV}_4$|⁠, 17.12.1, page 85]. Now, as explained in [12, Section 2.2], the group scheme |$\textrm{Ker}\,(\varphi _n\otimes k)$| is isomorphic to a vector group |$\textbf{G}_{a,k}^{c_n}$| for some integer |$c_n\ge 0$|⁠. Consequently, |$\textrm{Ker}(\varphi _n|_{B_{n+1}})$| is isomorphic to a closed subgroup scheme of |$\textbf{G}_{a,k}^{c_n}$|⁠; the hypothesis on the characteristic then implies that |$\textrm{Ker}(\varphi _n|_{B_{n+1}})$| is either trivial or positive dimensional (by [6, Proposition IV.2.4.1], say). Equation (**) then shows that |$\textrm{Ker}(\varphi _n|_{B_{n+1}})=\{e\}$| if |$n\ge n_0$|⁠. We have therefore proved that |$\varphi _n|_{B_{n+1}}$| is a closed immersion [27, |$ \textrm{VI}_B$|⁠, Corollary 1.4.2, page 341], and consequently an isomorphism if |$n\ge n_0$|⁠. Corollary 3.3. If |$k$| is of characteristic zero, there exists |$n_0\in \textbf{N}$| and a formal closed flat subgroup |$\mathfrak{H}\subset \widehat G_{n_0}$| such that |$G^{\prime}\simeq \mathcal{N}_{\mathfrak{H}}^\infty (G_{n_0})$|⁠. Remark 3.4. Statement (2) in Theorem 3.1 fails if |$k$| is allowed to have characteristic |$p>0$| as the following construction shows. Let |$k$| and |$K$| be of characteristic |$p$| (the case of mixed characteristic will be considered elsewhere). Let |$A_0= R[x_0]$| be a polynomial algebra in one variable. As in Example 2.1, we introduce |$\tilde A=\{P\in K[x_0]\,:\,P(0)\in R\}$| and define $$\begin{align*} & \Delta:\tilde A\longrightarrow\tilde A\otimes_R\tilde A,\quad \varepsilon:\tilde A\to R,\quad\textrm{and}\quad \sigma:\tilde A\longrightarrow\tilde A \end{align*}$$ by requiring that |$\Delta x_0=x_0\otimes 1+1\otimes x_0$|⁠, |$\varepsilon x_0=0$|⁠, and |$\sigma (x_0)=-x_0$|⁠. These morphisms give |$A_0$| and |$\tilde A$| the structure of a commutative Hopf algebra over |$R$|⁠. (Clearly, the group scheme |$\textrm{Spec}\,A_0$| is just |$\textbf{G}_{a,R}$|⁠.) We construct inductively elements |$x_n$| of |$\tilde A$| by putting |$x_{n+1}=\pi ^{-1}(x_n^p-x_n)$| and then write |$A_n:=R[x_0,\ldots ,x_n]$|⁠, which is an |$R$|-subalgebra of |$\tilde A$|⁠. A simple argument shows that $$\begin{align*} & \Delta x_n=x_n\otimes1+1\otimes x_n,\quad\varepsilon(x_n)=0,\quad\textrm{and}\quad \sigma(x_n)=-x_n. \end{align*}$$ In particular, |$\Delta (A_n)\subset A_n\otimes A_n$| and |$\sigma (A_n)\subset A_n$|⁠. Let |$G_n$| be |$\textrm{Spec}\, A_n$| and give it the structure of a group scheme arising from |$\Delta $|⁠, |$\varepsilon $| and |$\sigma $|⁠. It follows that |$\{x_n^p-x_n=0\}$| cuts out a closed subgroup scheme |$B_n$| of |$G_{n}\otimes k$|⁠, and that the natural arrow |$G_{n+1}\to G_n$| is simply the blowup of |$B_n$|⁠. Proceeding by induction, one shows that |$A_{n+1}=A_n[T]/(\pi T-(x_n^p-x_n))$| and that the natural arrow |$k[B_n]\to k[G_{n+1}]$| identifies |$k[G_{n+1}]$| with a polynomial algebra in one variable over |$k[B_n]$|⁠. (Here we apply the description of the blowup of an ideal generated by a regular sequence given in [5, Theorem 1, §9, no. 7].) Consequently, |$B_{n+1}\to B_n$| is faithfully flat and has a non-trivial kernel, which is isomorphic to |$\textbf{Z}/p\textbf{Z}$|⁠. We now wish to derive from Theorem 3.1 some consequences concerning unipotent group schemes over |$R$| of characteristic |$(0,0)$|⁠; all hinges on the well-known principle that, in this case, Lie subalgebras are “algebraic.” Recall that |$U\in (\textbf{FGSch}/R)$| is called unipotent if |$U\otimes K$|⁠, respectively |$U\otimes k$|⁠, is a unipotent group scheme over |$K$|⁠, respectively |$k$|⁠. (It should be noted that in certain cases it is sufficient to require this condition only for |$K$|⁠. See e.g., Theorem 2.11 of [24].) Let us prepare the terrain: assume that |$R$| has characteristic |$(0,0)$| and that |$U\in (\textbf{FGSch}/R)$| is unipotent and of finite type; we also abbreviate |$\omega _U=\omega (U/R)$|⁠. The most fundamental tool for studying |$U$| is the exponential morphism $$\begin{align*} & \exp:\textrm{Lie}(U)_a\longrightarrow U ,\end{align*}$$ which is explained in terms of functors in [6, II.6.3.1, 264ff]. (In loc.cit. the authors work over a base field, but this assumption is unnecessary for the construction.) Endowing |$\textrm{Lie}(U)_a$| with its Baker-Campbell-Hausdoff (BCH) multiplication, |$\exp $| becomes an isomorphism of group schemes [24, Section 1.3]. Since we wish to deal with formal group schemes as well, it is convenient to elaborate on a purely algebraic expression for |$\exp ^\#:R[U]\to \textbf{S}(\omega _U)$|⁠. (Recall that |$\textrm{Lie}(U)_a=\textrm{Spec}\,\textbf{S}(\omega _U)$|⁠.) Write |$D:R[U]\to \omega _U$| for the canonical |$\varepsilon $|-derivation; this gives rise to an |$\varepsilon $|-derivation, denoted likewise, |$D:R[U]\to \textbf{S}(\omega _U)$| into the symmetric |$R$|-algebra. For each |$n>0$|⁠, we write |$D^n:R[U]\to \textbf{S}(\omega _U)$| for the convolution product of |$D$| with itself |$n$| times. In terms of Sweedler’s notation, this reads $$\begin{align}& D^n(\,f)=\sum_{(\,f)}D(\,f_{(1)})\cdots D(\,f_{(n)}),\qquad f\in R[U]. \end{align}$$(3) Then, agreeing to fix |$D^0(\,f)=\varepsilon (\,f)\cdot 1$|⁠, we define $$\begin{align*} & \eta_t(\,f):=\sum_{n=0}^\infty\frac{D^n}{n!}(\,f)t^n,\quad f\in R[U]. \end{align*}$$ (This is an element of |$\textbf{S}(\omega ) [\![ t ]\!] $|⁠.) Leibniz’ formula |$D^n(\varphi \psi )=\sum _{m=0}^n {{n}\choose{m}} D^m(\varphi )D^{n-m}(\psi )$| proves that |$\eta _t$| is a morphism of |$R$|-algebras, while the expression $$\begin{align*} & D^{\ell+m}(\,f)=\sum_{(\,f)}D^\ell(\,f_{(1)})D^m(\,f_{(2)}) \end{align*}$$ allows one to verify that |$\eta _t$| satisfies the uniqueness statement concerning the exponential given in [6, II.3.6.1, page 264]. In conclusion, $$\begin{align*} &\exp^\#=\sum_{n=0}^\infty\frac{D^n}{n!}.\end{align*}$$ We are now in a position to comfortably prove the Proposition 3.5. We keep the notations and assumptions made just above. Write |$\mathfrak{U}$| for the |$\pi $|-adic completion of |$U$| and let |$\mathfrak{H}\subset \mathfrak{U}$| be a formal flat closed subgroup scheme. Then, there exists a closed flat subgroup |$H\subset U$| inducing |$\mathfrak{H}$|⁠. Said differently, these formal subgroup schemes are algebraizable. Proof. We start by remarks on the structure of |$\mathfrak{H}$|⁠. We note that |$\mathfrak{H}$| is connected because the unipotent group scheme |$\mathfrak{H}\otimes k$| is, as a scheme, an affine space [6, Proposition IV.2.4.1, page 497]. Also, since |$\mathfrak{H}\otimes k$| is smooth over |$k$|⁠, the fibre-by-fibre smoothness criterion [13, |$\textrm{IV}_4$|⁠, 17.8.2, page 79] proves that |$\mathfrak{H}\otimes R_n$| is smooth over |$R_n$|⁠. We conclude that the conormal module |$\omega (\mathfrak{H}\otimes R_n/R_n)$| is free over |$R_n$|⁠; as the canonical arrow |$\omega (\mathfrak{H}/R)\otimes R_n\to \omega (\mathfrak{H}\otimes R_n/R_n)$| is bijective [6, I.4.1.6, page 99], |$\omega (\mathfrak{H}/R)$| is free over |$R$|⁠. The canonical morphism |$R[U]\to R\langle U\rangle $| induces a bijection |$\omega (U/R)\to \omega (\mathfrak{U}/R)$| so that we can associate to |$\omega (\mathfrak{H}/R)$|⁠, which is a quotient Lie coalgebra of |$\omega (\mathfrak{U}/R)$|⁠, a quotient Lie coalgebra |$\omega (\mathfrak{H}/R)$| of |$\omega (U/R)$|⁠. Let |$H\subset U$| be a closed subgroup scheme that corresponds, via |$\exp $|⁠, to |$\omega (\mathfrak{H}/R)$|⁠. (In this case, |$H\simeq \textrm{Spec}\,\textbf{S}( \omega (\mathfrak{H}/R))$| and the kernel of |$\omega (U/R)\to \omega (H/R)$| is the kernel of |$\omega (U/R)\to \omega (\mathfrak{H}/R)$|⁠.) Claim. The closed immersion |$\widehat H\to \mathfrak{U}$| factors as |$\widehat H\to \mathfrak{H}\to \mathfrak{U}$|⁠, where |$\widehat H\to \mathfrak{H}$| is a closed immersion inducing an isomorphism |$\omega (\mathfrak{H}/R)\stackrel{\sim }\to \omega (\widehat H/R)$|⁠. We ease notation by putting |$\omega _{\mathfrak{H}}=\omega (\mathfrak{H}/R)$|⁠, etc. Write |$\nu :=\textrm{Ker}(\omega _U\to \omega _{\mathfrak{H}})$| and let |$I\subset R\langle U\rangle $| stand for the ideal cutting out |$\mathfrak{H}$|⁠. Let |$\widehat D^n$|⁠, respectively |$\widehat \exp ^\#$|⁠, stand for the prolongation of |$D^n$|⁠, respectively |$\exp ^\#$|⁠, to |$\pi $|-adic completions. Then |$\widehat D(I)\subset \nu $|⁠, and using equation (3), we conclude that |$\widehat \exp ^\#$| takes |$I$| to the ideal |$(\nu )\cdot \textbf{S}(\omega _U)^\wedge $|⁠, which means that |$\mathfrak{H}\to \mathfrak{U}$| factors |$\widehat H\to \mathfrak{U}$|⁠. Since |$\widehat H\to \mathfrak{U}$| is a closed immersion so is |$\widehat H\to \mathfrak{H}$|⁠; since |$\omega _{\mathfrak{U}}\to \omega _{\widehat H}$| and |$\omega _{\mathfrak{U}}\to \omega _{\mathfrak{H}}$| have the same kernel, the claim is proved. Using again that |$\omega (\mathfrak{H}/R)\otimes R_n\stackrel \sim \to \omega (\mathfrak{H}_n/R_n)$| and |$\omega (H/R)\otimes R_n\stackrel \sim \to \omega (H_n/R_n)$| [5, I.4.1.6, page 99], we conclude that the closed immersion |$H_n\to \mathfrak{H}_n$| induces an isomorphism |$\omega (\mathfrak{H}_n/R_n)\to \omega (H_n/R_n)$|⁠. Now, [6, I.4.4.2, page 109] shows that |$\widehat{\mathcal{O}}_{H_n,e}$|⁠, resp. |$\widehat{\mathcal{O}}_{\mathfrak{H}_n,e}$|⁠, is isomorphic to the completion of |$\textbf{S}(\omega (H_n/R_n))$|⁠, resp. |$\textbf{S}(\omega (\mathfrak{H}_n/R_n))$|⁠, with respect to the augmentation ideal, so that we derive an isomorphism |$\widehat{\mathcal{O}}_{\mathfrak{H}_n,e}\stackrel{\sim }{\to } \widehat{\mathcal{O}}_{H_n,e}$|⁠. This, in turn, implies |$\mathcal{O}_{\mathfrak{H}_n,e}\stackrel{\sim }{\to } \mathcal{O}_{H_n,e}$|⁠. Hence, |$H_n\to \mathfrak{H}_n$| is flat at the point |$e$|⁠, and consequently flat all over [29, |$ \textrm{VI}_B$|⁠, Proposition 1.3]. We can therefore say that the closed immersion |$H_n\to \mathfrak{H}_n$| is an open immersion. Because |$H_n$| and |$\mathfrak{H}_n$| are connected, we conclude that |$H_n\to \mathfrak{H}_n$| is an isomorphism, which, in turn, implies that |$\widehat H\to \mathfrak{H}$| is an isomorphism. The following corollary shall prove useful in Section 6. Corollary 3.6. Assume that |$R$| is of characteristic |$(0,0)$| and let |$U\in (\textbf{FGSch}/R)$| be unipotent and of finite type. Given an arrow $$\begin{align*} &\rho:G\longrightarrow U\end{align*}$$ of |$(\textbf{FGSch}/R)$|⁠, which induces an isomorphism on generic fibres, either |$G$| is of finite type or |$R[G]$| contains an |$R$|-submodule isomorphic to |$K$|⁠. Proof. We assume that |$G$| is not of finite type over |$R$|⁠. Let $$\begin{align*} & \cdots\longrightarrow U_{n}\longrightarrow \cdots\longrightarrow U \end{align*}$$ be the standard sequence of |$\rho $|⁠. It follows from Corollary 3.3 that for some |$n_0\in \textbf{N}$|⁠, |$G\simeq \mathcal{N}^\infty _{\mathfrak{H}}(U_{n_0})$|⁠, where |$\mathfrak{H}$| is a closed and flat subgroup of |$\widehat U_{n_0}$|⁠. Since the kernel of |$U_{n+1}\otimes k\to U_n\otimes k$| is unipotent [26, Theorem 1.5], basic theory [6, IV.2.2.3, page 485] tells us that |$U_{n_0}$| is unipotent. So Proposition 3.5 can be applied: there exists some flat and closed |$H\subset U_{n_0}$| such that |$\mathfrak{H}=\widehat H$|⁠. In addition, as |$G$| is not of finite type over |$R$|⁠, the closed immersion |$H\subset U_{n_0}$| is not an isomorphism. Now, if |$f\in R[U_{n_0}]\setminus \{0\}$| belongs to the ideal cutting out |$H$|⁠, then, in |$R[G]\simeq \mathcal{N}_{H}^\infty (U_{n_0})$|⁠, |$f$| can be uniquely divided by any power of |$\pi $|⁠, so that |$\cup _mR\pi ^{-m}f$| is isomorphic to |$K$|⁠. 4 Neron Blowups of Formal Subgroup Schemes in the Tannakian Theory of “Abstract” Groups In Section 3, we saw that blowing up a formal subgroup scheme is a very pertinent operation in the theory of flat group schemes over |$R$|⁠. We now investigate if such group schemes play a role in the Tannakian theory of abstract groups. Apart from the results of Section 5, our motivation to carry the present study comes from the fact that Tannakian categories of abstract groups produce easily many interesting examples of group schemes. Section 4.1 contains preparatory material on the Tannakian categories in question. Section 4.2 explains how to mimic, in our setting, the folkloric computation of differential Galois groups from the monodromy representation. Section 4.3 shows that Neron blowups of formal subgroup schemes appear naturally in the Tannakian theory of abstract groups, already in characteristic |$(0,0)$|⁠. The results here shall be employed in Section 5 to study categories of |$\mathcal{D}$|-modules. 4.1 Representations of abstract groups (In this section we make no supplementary assumption on |$R$|⁠.) Let |$\Gamma $| be an abstract group. In what follows, we convey some thoughts on the existence and basic properties of a “Tannakian envelope” for |$\Gamma $| over |$R$|⁠, that is, a flat affine group scheme |$\Pi $| such that |$\textrm{Rep}_{R}(\Pi )\simeq \textrm{Rep}_{R}(\Gamma )$|⁠. Of course, contrary to the case of a ground field, flatness is not gratuitous, so that the existence of such an envelope imposes one extra property on |$\textrm{Rep}_{R}(\Gamma )$| which we briefly explain. Recall from [11, Definition 1.2.5] that |$\textrm{Rep}_{R}(\Gamma )$| is Tannakian if every |$V\in \textrm{Rep}_{R}(\Gamma )$| is the target of an epimorphism |$\widetilde V\to V$| from an object |$\widetilde V\in \textrm{Rep}^\circ _{R}(\Gamma )$|⁠. If |$\textrm{Rep}_{R}(\Gamma )$| is not Tannakian, it still contains the full Tannakian subcategory |$\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}$| consisting of those objects satisfying the aforementioned condition. Definition 4.1. (1) We say that |$\Gamma $| is Tannakian over |$R$| if |$\textrm{Rep}_{R}(\Gamma )$| is Tannakian in the sense explained above. (2) The Tannakian envelope of |$\Gamma $| over |$R$| is the flat group scheme constructed from |$\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}$| and the forgetful functor |$\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}} \to R\text{-}\textbf{mod}$| by means of the main theorem of Tannakian duality (see [21, II.4.1, 152ff] or [11, Theorem 1.2.6]). Note that, whether |$\Gamma $| is Tannakian or not, its envelope does exist. Example 4.2. Any finite abstract group |$\Gamma $| is Tannakian. (One uses that the multiplication morphism |$R\Gamma \otimes _RV\to V$| is equivariant.) In order to explore the defining property of a Tannakian group, we use the following terminology. Definition 4.3. Let |$V\in \textrm{Rep}_R(\Gamma )$|⁠. (1) Assume that |$\pi V=0$|⁠, that is, |$V$| is a |$k$|-module. An object |$\widetilde V\in \textrm{Rep}^\circ _{R}(\Gamma )$| such that |$\widetilde V/\pi \widetilde V\simeq V$| is called a lift of |$V$| from |$k$| to |$R$|⁠. (2) An objet |$\widetilde V\in \textrm{Rep}^\circ _{R}(\Gamma )$|⁠, which is the source of an epimorphism |$\widetilde V\to V$|⁠, is called a weak lift of |$V$|⁠. Proposition 4.4 (cf. [11, Proposition 5.2.2]). Assume that each |$V\in \textrm{Rep}_R(\Gamma )$|⁠, which is annihilated by |$\pi $|⁠, has a weak lift. Then every |$E\in \textrm{Rep}_R(\Gamma )$| has a weak lift. Proof. Given |$M\in \textrm{Rep}_{R}(\Gamma )$|⁠, we define $$\begin{align*} &r(M)=\min\{s\in\textbf{N}\,:\,\pi^sM_{\textrm{tors}}=0\}. \end{align*}$$ We shall proceed by induction on |$r(E)$|⁠, the case |$r(E)=0$| being trivial. Assume |$r(E)=1$|⁠, so that |$\pi E$| is torsion free. Let |$q:E\to C$| be the quotient by |$\pi E$|⁠; since |$C$| is annihilated by |$\pi $|⁠, the hypothesis gives us |$\widetilde C\in \textrm{Rep}^\circ _{R}(\Gamma )$| and an epimorphism |$\sigma :\widetilde C\to C$|⁠. We then have a commutative diagram with exact rows where the rightmost square is cartesian and |$\tau $| is surjective. Since |$\pi E$| and |$\widetilde C$| are torsion free, so is |$\widetilde E$|⁠, and we have found a weak lift of |$E$|⁠. Let us now assume that |$r(E)>1$|⁠. Let |$N=\{e\in E\,:\,\pi e=0\}$| and denote by |$q:E\to C$| the quotient by |$N$|⁠. It then follows that |$\pi ^{r(E)-1}C_{\textrm{tors}}=0$|⁠, so that |$r(C)\le r(E)-1$|⁠. By induction, there exists |$\widetilde C\in \textrm{Rep}^\circ _{R}(\Gamma )$| and a surjection |$\sigma :\widetilde C\to C$|⁠. We arrive at commutative diagram with exact rows where the rightmost square is cartesian and |$\tau $| is surjective. Since |$\widetilde C$| is torsion free as an |$R$|-module, we conclude that |$\widetilde E_{\textrm{tors}}= N$|⁠, so that |$r(\widetilde E)\le 1$|⁠. We can therefore find |$\widetilde E_1$| and a surjection |$\widetilde E_1\to \widetilde E$| and consequently a surjection |$\widetilde E_1\to E$|⁠. The simplest case where any |$V\in \textrm{Rep}_{R}(\Gamma )$| annihilated by |$\pi $| has a lift occurs when |$R$| has a coefficient field [17, page 215] (in this case, the reduction morphism |$\textrm{GL}_r(R)\to \textrm{GL}_r(k)$| has a section) or when the group in question is free. Hence: Corollary 4.5. A free group is Tannakian. If |$R$| has a coefficient field, then any abstract group is Tannakian. Corollary 4.6. Let |$\Pi $| denote the Tannakian envelope of |$\Gamma $| over |$R$|⁠. Let |$\Theta $| denote the Tannakian envelope of |$\Gamma $| over |$k$|⁠. The following claims hold true. (1) There exists a canonical faithfully flat arrow of group schemes |$h:\Theta \to \Pi \otimes k$|⁠. (2) The morphism |$h$| is an isomorphism if and only if |$\Gamma $| is Tannakian. Proof. (1) According to [15, Part I, 10.1], |$\textrm{Rep}_{k}(\Pi \otimes k)$| can be identified with the full subcategory of |$\textrm{Rep}_{R}(\Pi )$| consisting of those representations annihilated by |$\pi $|⁠. Hence, |$\textrm{Rep}_{k}(\Pi \otimes k)$| can be identified with the full tensor subcategory of |$\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}$| of objects that are annihilated by |$\pi $|⁠. We then derive a fully faithful |$\otimes $|-functor |$\eta :\textrm{Rep}_{k}(\Pi \otimes k)\to \textrm{Rep}_{k}(\Gamma )$| which, on the level of vector spaces, is just the identity. (Note that if |$V\in \textrm{Rep}_{k}(\Pi \otimes k)$|⁠, then |$\eta (V)$| always admits a weak lift.) Moreover, |$\eta $| is also closed under taking subobjects. Tannakian duality then produces a morphism |$h:\Theta \to \Pi \otimes k$|⁠, which is, in addition, faithfully flat [7, Proposition 2.21, page 139]. (2) Assume now that |$h$| is an isomorphism, so that |$\eta $| is an equivalence. Then, any |$V\in \textrm{Rep}_{k}(\Gamma )$| admits a weak lift, so that, by Proposition 4.4, any |$E\in \textrm{Rep}_{R}(\Gamma )$| admits a weak lift. This means that |$\textrm{Rep}_{R}(\Gamma )$| is Tannakian. The converse is also simple. Example 4.7 (Non-Tannakian groups). We assume that |$R$| is of mixed characteristic |$(0,p)$|⁠. Let |$\Gamma $| be a periodic group of finite exponent (all orders are divisors of a fixed integer). A theorem of Burnside [10, Theorem 2.9, page 40] says that every morphism |$\Gamma \to \textrm{GL}_r(R)$| must then have a finite image, so that a morphism |$\Gamma \to \textrm{GL}_r(k)$| that admits a lift must have a finite image. This fact puts heavy restrictions on a Tannakian periodic group of finite exponent. For, if |$\Gamma $| is moreover Tannakian, then all representations |$\rho :\Gamma \to \textrm{GL}_r(k)$| must have a finite image. (By the Tannakian property some |$\sigma :\Gamma \to \textrm{GL}_{r+h}(k)$| of the form $$\begin{align*} &\begin{bmatrix}*&*\\ *&\rho\end{bmatrix} \end{align*}$$ admits a lift.) Hence, such a periodic group cannot be “linear over |$k$|⁠.” A specific counter-example is then given by the additive group |$(k,+)$|⁠, if |$k$| is infinite. 4.2 Computing faithfully flat quotients of the Tannakian envelope (In this section, we make no supplementary assumption on |$R$|⁠.) We fix |$G\in (\textbf{FGSch}/R)$| and an abstract group |$\Gamma $|⁠, whose Tannakian envelope over |$R$| is denoted by |$\Pi $| (see Definition 4.1). In this section, we shall first explain how to associate to each arrow of abstract groups |$\varphi :\Gamma \to G(R)$| a morphism |$u_\varphi :\Pi \to G$|⁠. Then, we set out to determine under which conditions on |$\varphi $|⁠, the map |$u_\varphi $| is faithfully flat. We inform the reader that the analogous situation over a base field is folkloric (meaning that some cursory discussions can be found in the literature, for example, [7, 10.24] and [1, Appendix]). Before starting, let us fix some notations. We write |$\omega _\Gamma :\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}\to R\text{-}\textbf{mod}$| and |$\omega _G:\textrm{Rep}_{R}(G)\to R\text{-}\textbf{mod}$| for the forgetful functors. The set |$\textbf{Fun}^\otimes _*(\textrm{Rep}_{R}(G),\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}})$| is formed by all |$\otimes $|-functors |$\eta $| satisfying the equality |$\omega _G\circ \eta =\omega _\Gamma $|⁠. Given |$H\in (\textbf{FGSch}/R)$|⁠, we define |$\textbf{Fun}^\otimes _*(\textrm{Rep}_{R}(G),\textrm{Rep}_{R}(H))$| analogously. Elements in these sets shall be called pointed|$\otimes $|-functors. Finally, if |$\gamma \in \Gamma $| and |$M\in \textrm{Rep}_{R}(\Gamma )$|⁠, respectively |$g\in G(R)$| and |$N\in \textrm{Rep}_{R}(G)$|⁠, we let |$\gamma _M:M\to M$|⁠, respectively |$g_N$|⁠, stand for the action of |$\gamma $| on |$M$|⁠, respectively |$g$| on |$N$|⁠. Let |$\varphi :\Gamma \to G(R)$| be a morphism of abstract groups. For each |$M\in \textrm{Rep}_{R}(G)$| and each |$\gamma \in \Gamma $|⁠, we have an |$R$|-linear automorphism |$\varphi (\gamma )_M$|⁠. In this way, we obtain a pointed |$\otimes $|-functor $$\begin{align*} & \varphi^\natural\colon\textrm{Rep}_{R}(G)\longrightarrow\textrm{Rep}_{R}(\Gamma)^{\textrm{tan}} \end{align*}$$ verifying the equations $$\begin{align*} & \gamma_{\varphi^\natural(M)}=\varphi(\gamma)_M\ \textrm{as elements of}\ \textrm{Aut}_R(M), \end{align*}$$ for all |$\gamma \in \Gamma $| and |$M\in \textrm{Rep}_{R}(G)$|⁠. Conversely, let |$T:\textrm{Rep}_{R}(G)\to \textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}$| be a pointed |$\otimes $|-functor. If |$\gamma \in \Gamma $| is fixed, and |$M\in \textrm{Rep}_{R}(G)$| is given, then |$\gamma _{TM}$| is an |$R$|-linear automorphism of |$M$|⁠. It is a simple matter to show that the family |$\{\gamma _{TM}\,:\,M\in \textrm{Rep}_{R}(G)\}$| defines a |$\otimes $|-automorphism of |$\omega _G$|⁠, that is, an element of |$\textrm{Aut}^\otimes (\omega _G)$|⁠. (It is equally simple to note that |$\gamma \mapsto \{\gamma _{TM}\,:\,M\in \textrm{Rep}_{R}(G)\}$| is in fact an arrow of groups.) Since the canonical arrow |$G(R)\to \textrm{Aut}^\otimes (\omega _G)$| is bijective [21, II.2.5.4, page 133], we arrive at a morphism of groups $$\begin{align*} &T^\flat:\Gamma\longrightarrow G(R)\end{align*}$$ verifying the equations $$\begin{align*} & [T^\flat(\gamma)]_M=\gamma_{TM}\ \textrm{as elements of}\ \textrm{Aut}_R(M), \end{align*}$$ for all |$\gamma \in \Gamma $| and |$M\in \textrm{Rep}_{R}(G)$|⁠. Proposition 4.8. The arrows |$\varphi \mapsto \varphi ^\natural $| and |$T\mapsto T^\flat $| are inverse bijections between |$\textrm{Hom}_{}(\Gamma ,G(R))$| and |$\textbf{Fun}_*^\otimes (\textrm{Rep}_{R}(G),\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}})$|⁠. In addition, the construction |$(-)^\natural $| is functorial in the sense that, for each |$\rho :G\to H$| in |$(\textbf{FGSch}/R)$|⁠, the diagram commutes. Proof. One immediately sees that |$[(\varphi ^\natural )^\flat (\gamma )]_M=\varphi (\gamma )_M$| for any |$M\in \textrm{Rep}_{R}(G)$|⁠. Since the natural arrow |$G(R)\to \textrm{Aut}^\otimes (\omega _G)$| is bijective, we conclude that |$(\varphi ^\natural )^\flat (\gamma )=\varphi (\gamma )$|⁠. To prove that |$(T^\flat )^\natural :\textrm{Rep}_{R}(G)\to \textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}$| coincides with |$T$|⁠, we need to show that for each |$\gamma \in \Gamma $| and each |$M\in \textrm{Rep}_{R}(G)$|⁠, the |$R$|-linear automorphisms |$\gamma _{(T^\flat )^\natural (M)}$| and |$\gamma _{TM}$| are one and the same. Now, |$\gamma _{(T^\flat )^\natural (M)}$| is just |$[T^\flat (\gamma )]_M$|⁠, which is the only element of |$G(R)$| inducing the automorphism |$\gamma _{TM}$|⁠. The verification of the last property is also very simple and we omit it. Given |$\gamma \in \Gamma $|⁠, let |$u(\gamma )\in \Pi (R)=\textrm{Aut}^\otimes (\omega _\Gamma )$| be defined by $$\begin{align*} & u(\gamma)_M=\gamma_M \ \textrm{as elements of}\ \textrm{Aut}_R(M). \end{align*}$$ In this way, we construct a morphism of groups $$\begin{align*} &u:\Gamma\longrightarrow \Pi(R)\end{align*}$$ and then a pointed |$\otimes $|-functor |$u^\natural :\textrm{Rep}_{R}(\Pi )\to \textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}$|⁠. Composing the canonical equivalence |$\textrm{Rep}_{R}(\Gamma )^{\textrm{tan}}\to \textrm{Rep}_{R}(\Pi )$| with |$u^\natural $|⁠, we obtain precisely the identity, so that |$u^\natural $| is also a |$\otimes $|-equivalence. Using Proposition 4.8 and the bijection $$\begin{align*} &\textrm{Hom}_{}(G,H)\longrightarrow\textbf{Fun}^\otimes_*(\textrm{Rep}_{R}(H),\textrm{Rep}_{R}(G)),\quad\rho\longmapsto\rho^\#,\end{align*}$$ explained in [21, II.3.3.1, page 148], we arrive at the following: Corollary 4.9. The function $$\begin{align*} & \textrm{Hom}_{}(\Pi,G)\longrightarrow \textrm{Hom}_{}(\Gamma,G(R)),\quad \rho\longmapsto \rho(R)\circ u \end{align*}$$ is a bijection. Remark 4.10. Corollary 4.9 gives a different way to think about the Tannakian envelope; needless to say, this is a common viewpoint in algebraic Topology, see [1, Appendix A]. Using Corollary 4.9, we can easily detect when a morphism |$\Gamma \to G(R)$| presents |$G$| as a faithfully flat quotient of |$\Pi $|⁠. Definition 4.11. Let |$G$| be a flat group scheme over |$R$|⁠, and |$\varphi :\Gamma \to G(R)$| a homomorphism of abstract groups. Write |$\varphi (\gamma )_k:\textrm{Spec}\,k\to G_k$|⁠, respectively |$\varphi (\gamma )_K:\textrm{Spec}\,K\to G_K$|⁠, for the closed immersions associated to |$\varphi (\gamma ):\textrm{Spec}\,R\to G$| by base change. We say that |$\varphi $| is dense if |$\{\varphi (\gamma )_k\}_{\gamma \in \Gamma }$|⁠, respectively |$\{\varphi (\gamma )_K\}_{\gamma \in \Gamma }$|⁠, is schematically dense in |$G_k$| [13, |$ \textrm{IV}_3$|⁠, 11.10.2, page 171], respectively in |$G_K$|⁠. Remark 4.12. The somewhat complicated definition given above is due to the usual discrepancy between topological and schematic properties. Needless to say, if in the above definition |$G_k$| and |$G_K$| are reduced, then the condition simply means that the image of |$\Gamma $| in |$|G_k|$| and in |$|G_K|$| is dense [13, |$ \textrm{IV}_3$|⁠, 11.10.4]. But in the case of non-reduced schemes one has to proceed with caution as a simple example shows (take |$\Gamma =\{\pm 1\}$|⁠, |$R=\textbf{Z}_2$| and |$G=\mu _{2,R}$|⁠). Also note that requiring the family |$\{\varphi (\gamma )\}_{\gamma \in \Gamma }$| to be schematically dense does not imply, in general, the above condition (same example as before). Proposition 4.13. The following claims about the Tannakian envelope |$\Pi $| of |$\Gamma $| are true. (1) The morphism |$u:\Gamma \to \Pi (R)$| is dense. (2) Both fibres of |$\Pi $| are reduced. (3) Let |$G\in (\textbf{FGSch}/R)$| and let |$\varphi :\Gamma \to G(R)$| be a dense morphism. Then, the morphism |$u_\varphi :\Pi \to G$| associated to |$\varphi $| by means of Corollary 4.9 is faithfully flat. Proof. (1) and (2). Let |$B$| be the smallest |$K$|-subgroup scheme of |$\Pi _K$| bounding (“majorant” in French) the family |$\{u(\gamma )_K\}_{\gamma \in \Gamma }$| [29,|$\textrm{ VI}_B$|⁠, Proposition 7.1, page 384]. Let |$i:\overline{B}\to \Pi $| be the schematic closure of |$B$| in |$\Pi $| [13, |$ \textrm{IV}_2$|⁠, 2.8, 33ff]. Note that, for each |$\gamma \in \Gamma $|⁠, the arrow |$u(\gamma ):\textrm{Spec}\,R\to \Pi $| factors as $$\begin{align*} & u(\gamma)=i(R)\circ \psi(\gamma), \end{align*}$$( * ) where |$\psi (\gamma ):\textrm{Spec}\,R\to \overline{B}$| is an |$R$|-point of |$\overline{B}$|⁠. In addition, since |$i(R):\overline{B}(R)\to \Pi (R)$| is a monomorphism, |$\psi :\Gamma \to \overline{B}(R)$| is a morphism of groups. By Corollary 4.9, there exists a unique morphism |$u_\psi :\Pi \to \overline{B}$| such that $$\begin{align*} & \psi(\gamma)=u_\psi(R)\circ u(\gamma). \end{align*}$$( * *) From (*) and (**) we deduce that |$u(\gamma )=(i\circ u_\psi )(R)\circ u(\gamma )$|⁠, and hence |$\textrm{id}_\Pi = i\circ u_\psi $| (due to Corollary 4.9). Therefore, |$i$| must be an isomorphism. Using [29,|$\textrm{ VI}_B$|⁠, Proposition 7.1(ii)], we conclude that |$\Pi _K$| is reduced, and that the set of points associated to |$\{u(\gamma )_K\}_{\gamma \in \Gamma }$| is dense in it. In this case, it is obvious that |$\{u(\gamma )_K\}_{\gamma \in \Gamma }$| is also schematically dense [13, |$ \textrm{IV}_3$|⁠, 11.10.4]. In the same spirit, let |$j:H\to \Pi _k$| be the smallest |$k$|-subgroup bounding |$\{u(\gamma )_k\}_{\gamma \in \Gamma }$| and let |$\zeta :\Pi ^{\prime}\to \Pi $| be the Neron blowup of |$\Pi $| at |$H$|⁠. By the definition of a Neron blowup, for each |$\gamma \in \Gamma $|⁠, there exists |$u^{\prime}(\gamma ):\textrm{Spec}\,R\to \Pi ^{\prime}$| such that $$\begin{align*} & u(\gamma)=\zeta(R)\circ u^{\prime}(\gamma). \end{align*}$$($$\dagger $$) Since |$\zeta (R):\Pi ^{\prime}(R)\to \Pi (R)$| is a monomorphism of groups, we conclude that |$u^{\prime}:\Gamma \to \Pi ^{\prime}(R)$| is a homomorphism. Corollary 4.9 provides |$v:\Pi \to \Pi ^{\prime}$| such that $$\begin{align*} & u^{\prime}(\gamma)=v(R)\circ u(\gamma). \end{align*}$$($$\ddagger $$) As a consequence of (⁠|$\dagger $|⁠) and (⁠|$\ddagger $|⁠), we have |$u(\gamma )=(\zeta v)(R)\circ u(\gamma )$|⁠. Another application of Corollary 4.9 leads to |$\zeta v=\textrm{id}_\Pi $|⁠. Now, for each |$a^{\prime}\in R[\Pi ^{\prime}]$|⁠, there exist |$m\in \textbf{N}$| and |$a\in R[\Pi ]$| such that |$\pi ^ma^{\prime}=\zeta ^\#(a)$|⁠, so that |$\pi ^mv^\#(a^{\prime})=a$|⁠, which shows that |$\pi ^ma^{\prime}=\pi ^m\zeta ^\#(v^\#(a))$| thereby proving that |$a^{\prime}=\zeta ^\#(v^\#(a))$|⁠. (It is equally possible to note that |$\zeta (A):\Pi ^{\prime}(A)\to \Pi (A)$| is injective for each flat |$R$|-algebra |$A$|⁠, and that the equation |$\zeta v=\textrm{id}_\Pi $| further implies that |$\zeta (A)$| is surjective.) Consequently, |$\zeta $| is an isomorphism and |$H=\Pi _k$|⁠. As in the previous paragraph, these findings show that |$\Pi _k$| is reduced and that |$\{u(\gamma )_k\}_{\gamma \in \Gamma }$| is schematically dense. (3) Because |$\varphi (\gamma )=u_\varphi \circ u(\gamma )$| for each |$\gamma $|⁠, it follows that the arrows |$K[G]\to K[\Pi ]$| and |$k[G]\to k[\Pi ]$| derived from |$u_\varphi $| are injective. The result then follows by applying [11, Theorem 4.1.1]. Remark 4.14. Let |$\varphi :\Gamma \to \Gamma ^{\prime}$| be a homomorphism of abstract groups. Then there exists, by the universal property, a unique group scheme homomorphism |$\Phi :\Pi \to \Pi ^{\prime}$| rendering the following diagram commutative 4.3 The Tannakian envelope of an infinite cyclic group In this section, we assume |$R=k [\![ \pi ]\!] $|⁠, with |$k$| of characteristic zero, and we give ourselves an infinite cyclic group |$\Gamma $| generated by |$\gamma $|⁠. In the terminology of Definition 4.1, this is certainly a Tannakian group (apply Corollary 4.5); let |$\Pi $| stand for its Tannakian envelope over |$R$|⁠. We wish to show that, despite the very simple nature of |$\Gamma $|⁠, its envelope does allow rather “large” faithfully flat quotients. A precise statement is Proposition 4.15 below. Let |$G=\textbf{G}_{a,R}\times \textbf{G}_{m,R}$| have ring of functions |$R[G]=R[x,y^\pm ]$|⁠. In the |$\pi $|-adic completion |$R\langle G\rangle $| of |$R[G]$|⁠, we single out the element $$\begin{align*} & \Phi=y-\exp(\pi x)\in R\langle G\rangle. \end{align*}$$ Obviously, the closed formal subscheme |$\mathfrak{H}$| cut out by the ideal |$(\Phi )\subset R\langle G\rangle $| is a formal subgroup, isomorphic to the |$\pi $|-adic completion of |$\textbf{G}_{a,R}$|⁠; let |$\mathcal{N}$| stand for the Neron blowup of |$\mathfrak{H}$| (see Section 2.2). By definition, $$\begin{align*} &R[\mathcal{N}]=\varinjlim_n R[G][\pi^{-{n-1}}\Phi_n],\end{align*}$$ where $$\begin{align*} &\Phi_n=y-\left(1+\pi x+\frac{\pi^2}{2!}x^2+\cdots+\frac{\pi^n}{n!}x^n\right).\end{align*}$$ We now introduce the “evaluation” at the point |$P=(1,e^\pi )\in G(R)$|⁠: $$\begin{align*} & \textrm{ev}_P:R[G]\longrightarrow R,\quad \begin{array}{cc} x&\longmapsto 1\\ y&\longmapsto e^\pi. \end{array} \end{align*}$$ Since |$\textrm{ev}_P(\Phi _n) \equiv 0\mod \pi ^{n+1}$|⁠, |$P$| is a point of |$\mathcal{N}$|⁠. Let |$\varphi :\Gamma \to \mathcal{N}(R)$| send |$\gamma $| to |$P$|⁠. According to Corollary 4.9, there exists a morphism |$u_\varphi :\Pi \to \mathcal{N}$| and a commutative diagram Proposition 4.15. The morphism |$u_\varphi :\Pi \to \mathcal{N}$| is faithfully flat. Proof. According to [12, Corollary 5.11], |$\mathcal{N}\otimes k$| is just |$\mathfrak{H}\otimes k\simeq \textbf{G}_{a,k}$|⁠; as the image of |$P$| in |$\mathcal{N}(k)$| is not the neutral element, the assumption on the characteristic proves that the subgroup it generates is schematically dense. On the generic fibre, the Zariski closure |$C$| of the subgroup generated by |$P$| must be of dimension at least two since both restrictions |$C\to \textbf{G}_{m,K}$| and |$C\to \textbf{G}_{a,K}$| are faithfully flat (the standard facts employed here are in [25], see the Corollary on page 65 and the Theorem on the bottom of page 114). This implies that |$C=G_K$| and Proposition 4.13 then finishes the proof. Corollary 4.16. The group scheme |$\Pi $| is not strictly pro-algebraic (in the sense of Question (SA) from the Introduction). In fact, the Tannakian envelope |$\Pi ^\star $| of any abstract group |$\Gamma ^\star $| admitting |$\Gamma $| as a quotient fails to be strictly pro-algebraic. Proof. We assume that |$\Pi $| is strictly pro-algebraic and write |$R[\Pi ]=\varinjlim _iR[\mathcal{G}_i]$|⁠, where each |$\mathcal{G}_i\in (\textbf{FGSch}/R)$| is of finite type, and each arrow |$R[\mathcal{G}_i]\to R[\mathcal{G}_j]$| is faithfully flat. In particular, each arrow |$R[\mathcal{G}_i]\to R[\Pi ]$| is faithfully flat (since tensor products commute with direct limits), and |$R[\mathcal{G}_i]$| is saturated in both |$R[\mathcal{G}_j]$| and |$R[\Pi ]$|⁠. Let |$i_0$| be such that |$x,y,y^{-1}\in R[\mathcal{G}_{i_0}]$| (we identify |$R[\mathcal{N}]$| with its image in |$R[\Pi ]$|⁠). As |$K[\mathcal{N}]$| is generated by |$\{x,y,y^{-1}\}$| and |$R[\mathcal{G}_{i_0}]$| is saturated in |$R[\Pi ]$|⁠, we arrive at a commutative diagram factoring |$u_\varphi $|⁠: It then follows, since |$u_\varphi $| and |$q$| are faithfully flat, that |$\chi $| is faithfully flat [17, page 46]. Together with Exercise 7.9 on page 53 of [17], this implies that |$R[\mathcal{N}]$| is a noetherian ring, something that is certainly false. (To verify this last claim, one proceeds as follows. Let |$\mathfrak{a}$| be the augmentation ideal of |$R[\mathcal{N}]$|⁠, and let |$\Psi _n=\pi ^{-n-1}\Phi _n$|⁠. Then |$\pi \Psi _{n+1}\equiv \Psi _n\mod \mathfrak{a}^2$| for all |$n\ge 1$|⁠, so that |$\mathfrak{a}/\mathfrak{a}^2$| is either not finitely generated over |$R=R[\mathcal{N}]/\mathfrak{a}$|⁠, or |$y-1-\pi x\in \mathfrak{a}^2$|⁠. But this latter condition is impossible.) To verify the 2nd statement, we observe that if |$\psi :\Gamma ^\star \to \Gamma $| is an epimorphism, then the induced arrow |$\Psi :\Pi ^\star \to \Pi $| is faithfully flat due to [11, Proposition 3.2.1(ii)]. The exact same proof as above now proves the 2nd statement. 5 Neron Blowups of Formal Subgroup Schemes in Differential Galois Theory In this section, we set out to investigate whether blowups of closed formal subgroup schemes, as in Section 2.2, appear in differential Galois theory over |$R$|⁠. (This theory is to be understood as the one explained in Section 2.3.) Our strategy is to restrict attention to the discrete valuation ring |$\textbf{C} [\![ t ]\!] $|⁠. We shall see that through a “Riemann–Hilbert correspondence,” which is Theorem 5.10 below, we are able to transplant the results of Section 4.3 concerning the category of representations of an abstract group to swiftly arrive at a conclusion: see Corollary 5.11. The proof of Theorem 5.10 is routine and employs Serre’s GAGA, Grothendieck’s Existence Theorem in formal Geometry (GFGA), and Deligne’s dictionary [9, I.2]. 5.1 Preliminaries on |$\mathcal{D}$|-modules We now review some useful notions concerning |$\mathcal{D}$|-modules. These are employed to establish an algebraization result, Proposition 5.3, further ahead. Let |$T$| be a noetherian scheme and |$Y$| a smooth |$T$|-scheme; we write |$\mathcal{D}(Y/T)$| for the ring of differential operators described in [13, |$ \textrm{IV}_4$|⁠, 16.8] and [4, §2]. Following the notation of [12, §7], we let |$\mathcal{D}(Y/T)\text{-}\textbf{mod}$| stand for the category of (sheaves of) |$\mathcal{D}(Y/T)$|-modules on |$Y$| whose underlying |$\mathcal{O}_Y$|-module is coherent. If |$T$| is not a |$\textbf{Q}$|-scheme, then the ring |$\mathcal{D}(Y/T)$| is usually not finitely generated over |$\mathcal{O}_Y$|⁠, so that a fundamental tool in dealing with |$\mathcal{D}$|-modules is Grothendieck’s notion of stratification. Let |$\mathcal{P}_{Y/T}^\nu $| stand for the sheaf of principal parts of order |$\nu $| of |$Y\to T$| [13, |$\textrm{IV}_4$|⁠, 16.3.1, page 14]. The scheme |$(|Y|,\mathcal{P}_{Y/T}^\nu )$| [13, |$ \textrm{IV}_4$|⁠, 16.1], usually named the scheme of principal parts of order |$\nu $|, is denoted by |$P_{Y/T}^\nu $|⁠. It comes with two projections |$p_0,p_1:P_{Y/T}^\nu \to Y$| and a closed immersion |$\Delta :Y\to P_{Y/T}^\nu $|⁠, and we follow the convention of EGA fixing the first projection as the structural one, while the other is supplementary. Given |$\mathcal{E}\in \mathcal{D}(Y/T)\text{-}\textbf{mod}$|⁠, for any |$\nu \in \textbf{N}$|⁠, we can associate an isomorphism of |$\mathcal{O}_{P^\nu _{Y/T}}$|-modules $$\begin{align*} &\theta(\nu):p_1^*(\mathcal{E})\stackrel{\sim}{\longrightarrow} p_0^*(\mathcal{E}),\end{align*}$$ which, when pulled-back along |$\Delta $|⁠, gives back the identity. These are called Taylor isomorphisms of order |$\nu $|⁠. Clearly, using the obvious closed immersions |$P_{Y/T}^{\nu }\to P_{Y/T}^{\nu +1}$|⁠, we obtain the notion of a compatible family of Taylor isomorphisms. These considerations prompt the definition of a pseudo-stratification of a coherent module |$\mathcal{F}$|⁠: It is a compatible family of isomorphisms $$\begin{align*} & \sigma(\nu):p_1^*(\mathcal{F})\stackrel{\sim}{\longrightarrow} p_0^*(\mathcal{F}),\quad\nu=0,\ldots, \end{align*}$$ which induce the identity when pulled back along the diagonal. The pseudo-stratification |$\{\sigma (\nu )\}_{\nu \in \textbf{N}}$| is a stratification if the isomorphisms satisfy, in addition, a cocycle condition [4, Definition 2.10]. Coherent modules endowed with stratifications constitute a category, denoted by |$\textbf{str}(Y/T)$|⁠. The main point of these definitions is that the Taylor isomorphisms define an equivalence of categories |$\mathcal{D}(Y/T)\text{-}\textbf{mod}\to \textbf{str}(Y/T)$|⁠, which preserves all the underlying coherent modules and morphisms [4, 2.11]. Although stratifications are a more intricate version of the natural concept of |$\mathcal{D}$|-module, their functoriality properties are most welcome. From a commutative diagram of schemes we obtain another such diagram where |$P^\nu (\varphi )$| is the morphism defined by |$|\varphi |:|Z|\to |Y|$| and the arrow of |$\mathcal{O}_Z$|-algebras (abusively denoted by the same symbol) $$\begin{align*} & P^\nu(\varphi):\mathcal{O}_Z\underset{\mathcal{O}_Y}{\otimes}\mathcal{P}^\nu_{Y/T}\longrightarrow \mathcal{P}^\nu_{Z/U}, \end{align*}$$ which is explained on [13, |$\textrm{IV}_4$|⁠, 16.4.3, 17ff]. Consequently, we have an isomorphism of functors $$\begin{align*} & P^\nu(\varphi)^*\circ p_{i,Y}^*\simeq p_{i,Z}^*\circ\varphi^*, \end{align*}$$ which allows us to prolong |$\varphi ^*:\textbf{Coh}(Y)\to \textbf{Coh}(Z)$| into $$\begin{align*} & \varphi^\#:\textbf{str}(Y/T)\longrightarrow\textbf{str}(Z/U),\quad(\mathcal{E},\{\theta(\nu)\})\longmapsto(\varphi^*\mathcal{E},\{\varphi^\#\theta(\nu)\} ); \end{align*}$$ just define |$\varphi ^\#\theta (\nu )$| by decreeing that commutes. Using the equivalence |$\mathcal{D}(Y/T)\text{-}\textbf{mod}\simeq \textbf{str}(Y/T)$| mentioned above, we can also prolong |$\varphi ^*$| to a functor |$\varphi ^\#:\mathcal{D}(Y/T)\text{-}\textbf{mod}\to \mathcal{D}(Z/U)\text{-}\textbf{mod}$|⁠. Remark 5.1. Similar notions hold for the case where |$g:Y\to T$| is replaced by a smooth morphism of complex analytic spaces [9, I.2.22]. This being so, we adopt similar notations. 5.2 An algebraization result In this section, we assume that |$R$| is complete. Let $$\begin{align*} &f:X\longrightarrow S\end{align*}$$ be a smooth morphism (recall that |$S=\textrm{Spec}\,R$|⁠). For every |$n\in \textbf{N}$|⁠, we obtain cartesian commutative diagrams and where the vertical arrows are the evident closed embeddings. In this case, the arrows $$\begin{align}& P^\nu(v_n): \mathcal{O}_{X_n}\underset{\mathcal{O}_X}{\otimes}\mathcal{P}_{X/S}^\nu\longrightarrow\mathcal{P}^\nu_{X_n/S_n} \end{align}$$(4) are isomorphisms [13, |$ \textrm{IV}_4$|⁠, 16.4.5]. Hence, the dual arrows $$\begin{align*} & \mathcal{D}(X_n/S_n)\longrightarrow \mathcal{O}_{X_n}\underset{\mathcal{O}_X}{\otimes}\mathcal{D}(X/S) \end{align*}$$ are isomorphisms of |$\mathcal{O}_{X_n}$|-modules (here we used that the modules of principal parts are locally free [4, Proposition 2.6]). This particular situation allows us to construct an arrow of |$\mathcal{O}_X$|-modules $$\begin{align}& \rho_n:\mathcal{D}(X/S)\longrightarrow\mathcal{D}(X_n/S_n). \end{align}$$(5) A moment’s thought proves that for any differential operator (on some open of |$X$|⁠) |$\partial :\mathcal{O}_V\to \mathcal{O}_V$|⁠, the operator |$\rho _n(\partial ):\mathcal{O}_{V_n}\to \mathcal{O}_{V_n}$| is just the reduction of |$\partial $| modulo |$\pi ^{n+1}$|⁠, so that |$\rho _n$| is a morphisms of |$v_n^{-1}\mathcal{O}_X$|-algebras. Moreover, |$\rho _n$| allows us to identify |$\mathcal{D}(X/S)/(\pi ^{n+1})$| and |$\mathcal{D}(X_n/S_n)$| in such a way that the functor |$v_n^\#:\mathcal{D}(X/S)\text{-}\textbf{mod}\to \mathcal{D}(X_n/S_n)\text{-}\textbf{mod}$| is the obvious one. Definition 5.2. Denote by |$\textbf{Coh}^\wedge (X)$| the category constructed as follows. Objects are sequences |$(\mathcal{E}_n,\alpha _n)$|⁠, where |$\mathcal{E}_n\in \textbf{Coh}(X_n)$|⁠, and |$\alpha _n:u_n^*\mathcal{E}_{n+1}\to \mathcal{E}_n$| an isomorphism. Morphisms between |$(\mathcal{E}_n,\alpha _n)$| and |$(\mathcal{F}_n,\beta _n)$| are families |$\varphi _n:\mathcal{E}_n\to \mathcal{F}_n$| fulfilling |$\varphi _n\circ \alpha _n=\beta _n\circ u_n^*(\varphi _n)$|⁠, see [14, 8.1.4]. (Needless to say, this is the category of coherent modules on the formal completion.) We let |$\mathcal{D}(X/S)\text{-}\textbf{mod}^\wedge $| be constructed analogously from |$\mathcal{D}(X/S)\text{-}\textbf{mod}$| and the functors |$u_n^\#$|⁠. Let $$\begin{align*} & \textbf{Coh}(X)\stackrel{\Phi}{\longrightarrow}\textbf{Coh}(X)^\wedge\quad \textrm{and} \quad\mathcal{D}(X/S)\text{-}\textbf{mod}\stackrel{\textrm{D}\Phi}{\longrightarrow}\mathcal{D}(X/S)\text{-}\textbf{mod}^\wedge\end{align*}$$ be the obvious functors. Grothendieck’s existence theorem [14, Theorem 8.4.2], henceforth called GFGA, says that if |$f$| is proper, then |$\Phi $| is an equivalence of |$R$|-linear |$\otimes $|-categories. Using the notion of stratification, we have the following: Proposition 5.3. Assume that |$f:X\to S$| is proper. Then, |$\textrm{D}\Phi $| is an equivalence. Proof. One obvious proof is to apply GFGA to the isomorphisms of coherent sheaves on the proper |$S$|-schemes |$P^\nu _{X/S}$| (the Taylor isomorphisms) and then prove that these define a stratification. For the sake of clarity, we choose to avoid the 2nd step, so that the number of verifications relying on canonical identifications is reduced. In taking this approach, we shall require Lemma 5.4 below; its proof is simple and we leave it to the reader. Essential surjectivity: due to GFGA, we need to prove the following: Claim. Let |$\mathcal{E}\in \textbf{Coh}(X)$|⁠, and assume that (i) each |$\mathcal{E}_n:=\mathcal{O}_{X_n}\otimes \mathcal{E}$| has the structure of a |$\mathcal{D}(X_n/S_n)$|-module, and (ii) the canonical isomorphisms |$\alpha _n:u_n^*\mathcal{E}_{n+1}\to \mathcal{E}_n$| preserve these structures. Then |$\mathcal{E}$| carries a |$\mathcal{D}(X/S)$|-module structure, and |$\textrm{D}\Phi (\mathcal{E})=(\mathcal{E}_n,\alpha _n)$|⁠. Let |$\nu \in \textbf{N}$| be fixed, and for each |$n\in \textbf{N}$|⁠, write $$\begin{align*} & \theta(\nu)_{n}:p_{1,X_n}^*(\mathcal{E}_n)\longrightarrow p_{0,X_n}^*(\mathcal{E}_n) \end{align*}$$ for the Taylor isomorphism of order |$\nu $| associated to the |$\mathcal{D}(X_n/S_n)$|-module |$\mathcal{E}_n$|⁠. The fact that |$\alpha _n$| is an isomorphism of stratified modules implies, after the necessary identifications, that $$\begin{align}& P^\nu(u_n)^*(\theta(\nu)_{n+1})=\theta(\nu)_n,\quad\textrm{for all}\ n. \end{align}$$(6) By the isomorphism (4), we can apply GFGA to the proper |$S$|-scheme |$P_X^\nu $| to obtain an isomorphism of |$\mathcal{O}_{P^\nu _X}$|-modules $$\begin{align*} & \theta(\nu):p_{1,X}^*(\mathcal{E})\longrightarrow p_{0,X}^*(\mathcal{E})\end{align*}$$ such that $$\begin{align*} & P^\nu(v_n)^*(\theta(\nu))=\theta(\nu)_n\quad\textrm{for all}\ n. \end{align*}$$ Let |$V\subset X$| be an open subset constructed from |$\mathcal{E}$| as in Lemma 5.4 below, and let |$\partial \in \mathcal{D}(X/S)(V)$| be of order |$\le \nu $|⁠. Let |$e\in \mathcal{E}(V)$|⁠. We define $$\begin{align*} &\nabla_{\nu}(\partial)(e):= (\textrm{id}\otimes\partial)\left[\theta(\nu)(1\otimes e)\right],\end{align*}$$ where we remind the reader that |$p_{0,X}^*(\mathcal{E})=\mathcal{E}\otimes \mathcal{P}_{X}^\nu $| and |$p_{1,X}^*(\mathcal{E})= \mathcal{P}_{X}^\nu \otimes \mathcal{E}$| as sheaves on |$|X|$|⁠. At this point, |$\nabla _{\nu }(\partial )(e)$| is just a name for a section of |$\mathcal{E}(V)$|⁠. We have |$\mathcal{O}(V)$|-linear maps |$q_n:\mathcal{E}(V)\to \mathcal{E}_n(V\cap X_0)$| and |$\rho _n$| (as in eq. (5)) and it is clear from the definition and eq. (6) that $$\begin{align}& q_n[\nabla_{\nu}(\partial)(e)]=\rho_n(\partial)\cdot q_n(e),\qquad\textrm{for each}\ n. \end{align}$$(7) As |$(q_n):\mathcal{E}(V)\to \varprojlim _n\mathcal{E}_n(V\cap X_0)$| is injective (Lemma 5.4-(b)), we can say (1) if |$\nu \le \nu ^{\prime}$|⁠, then |$\nabla _{\nu }(\partial )(e)=\nabla _{\nu ^{\prime}}(\partial )(e)$|⁠. (2) If |$\partial ^{\prime}$| is such that |$\partial \partial ^{\prime}$| has order |$\le \nu $|⁠, then |$\nabla _{\nu }(\partial \partial ^{\prime})(e)=\nabla _{\nu }(\partial )[\nabla _{\nu }(\partial ^{\prime})(e)]$|⁠. This means that |$\mathcal{E}(V)$| now carries the structure of a |$\mathcal{D}(X/S)(V)$|-module. Because |$V$| is affine, this endows |$\mathcal{E}|_V$| with a structure of |$\mathcal{D}(V/S)$|-module. Furthermore, by definition, the Taylor isomorphism of level |$\nu $| associated to this structure is simply the restriction of |$\theta (\nu )$| to |$V$|⁠. Hence, covering |$X$| by open subsets as these, |$\mathcal{E}$| becomes a |$\mathcal{D}$|-module for which the Taylor isomorphism of order |$\nu $| is simply |$\theta (\nu )$|⁠. By construction (see eq. (6)), we then conclude that |$\textrm{D}\Phi (\mathcal{E})=(\mathcal{E}_n,\alpha _n)$|⁠. This proves the claim. Fully faithfulness: As before, by GFGA, it is sufficient to show the following: Claim. Let |$\mathcal{E}$| and |$\mathcal{F}$| be |$\mathcal{D}(X/S)$|-modules, and let |$\phi :\mathcal{E}\to \mathcal{F}$| be an |$\mathcal{O}_X$|-linear arrow such that |$\phi _n:v_n^*\mathcal{E}\to v_n^*\mathcal{F}$| is |$\mathcal{D}(X_n/S_n)$|-linear for all |$n$|⁠. Then |$\phi $| is a morphism of |$\mathcal{D}(X/S)$|-modules. Let |$V$| be constructed from |$\mathcal{F}$| as in Lemma 5.4(a); we show that |$\phi |_V$| is |$\mathcal{D}(V/S)$|-linear. Let |$e\in \mathcal{E}(V)$|⁠, |$\partial \in \mathcal{D}(X/S)(V)$|⁠, and write |$r_n$| for the natural |$\mathcal{O}(V)$|-linear morphisms |$\mathcal{F}(V)\to v_n^* \mathcal{F}(V\cap X_0)$|⁠. It follows easily that |$r_n(\partial \cdot \phi (e))=r_n(\phi (\partial \cdot e))$|⁠, so that injectivity of |$(r_n):\mathcal{F}(V)\to \varprojlim v_n^*(\mathcal{F})(V\cap X_0)$| assures that |$\phi |_V$| is |$\mathcal{D}(V/S)$|-linear. Since |$X$| can be covered by open subsets like |$V$| (Lemma 5.4), we are done. The following result was employed in the proof of Proposition 5.3. Lemma 5.4. Let |$\mathcal{M}$| be a coherent |$\mathcal{O}_X$|-module. (a) Denote by |$\xi _1,\ldots ,\xi _r$| the associated points of |$\mathcal{M}$| [13, |$ \textrm{IV}_2$|⁠, 3.1] lying on the generic fibre |$X_K$|⁠. Then, every point |$x\in X_k$| has an open affine coordinate neighbourhood |$V$| such that if |$\xi _j\in V$|⁠, then |$\overline{\{\xi _j\}}\cap V_k\not =\varnothing $|⁠. (b) If |$V$| is as before, then |$\mathcal{M}(V)$| is |$\pi $|-adically separated. (c) The scheme |$X$| can be covered by finitely many open subsets as |$V$| above. 5.3 Local systems and representations of the topological fundamental group Let |$\Lambda $| be a local |$\textbf{C}$|-algebra, which, as a complex vector space, is finite dimensional. (In this case, the residue field of |$\Lambda $| is |$\textbf{C}$|⁠.) Writing |$T$| for the analytic space associated to |$\Lambda $| [27, Exposé 9, 2.8], we give ourselves a smooth morphism |$g:Y\to T$| of connected complex analytic spaces [27, Exposé 13, §3]. Let us now bring to stage the category |$\textbf{LS}(Y/T)$| of relative local systems [9, I.2.22]. These are |$g^{-1}(\mathcal{O}_T)$|-modules |$\mathcal{F}$| on |$Y$| enjoying the ensuing property: for every |$y\in Y$|⁠, there exists an open |$V\ni y$| such that the |$g^{-1}(\mathcal{O}_T)$|-module |$\mathcal{F}|V$| belongs to |$g^{-1}\textbf{Coh}(T)$|⁠. Under these conditions, any |$\mathcal{F}\in \textbf{LS}(Y/T)$| is a locally constant sheaf of |$\Lambda $|-modules whose values on connected open sets are of finite type. Conversely, any such locally constant sheaf of |$\Lambda $|-modules satisfies the defining property of |$\textbf{LS}(Y/T)$|⁠. Hence, the following arguments constitute a well-known exercise, see Exercise F of Ch. 1 and Exercise F of Ch. 6 in [23]. (Here one should recall that |$|Y|$| is the topological space of a manifold.) Let |$y_0\in |Y|$| be a point and |$\Gamma $| the fundamental group of |$|Y|$| based at it. Given a locally constant sheaf of |$\Lambda $|-modules |$\mathcal{F}$| on |$Y$|⁠, we can endow its stalk |$\mathcal{F}_{y_0}$| with a left action of |$\Gamma $|⁠. This construction then allows the definition of a functor $$\begin{align*} & \textsf{IM}_{y_0}: \left\{\begin{array}{c}\textrm{locally constant sheaves}\\ \textrm{of}\ \Lambda\textrm{-modules whose}\\ \textrm{stalks are of finite type}\end{array}\right\}\longrightarrow \textrm{Rep}_\Lambda(\Gamma). \end{align*}$$ Proposition 5.5. The functor |$\textsf{IM}_{y_0}$| defines a |$\Lambda $|-linear |$\otimes $|-equivalence between |$\textbf{LS}(Y/T)$| and |$\textrm{Rep}_{\Lambda }(\Gamma )$|⁠. 5.4 |$\mathcal{D}$|-modules on the analytification of an algebraic |$\textbf{C}$|-scheme Let |$\Lambda $| be a local |$\textbf{C}$|-algebra, which, as a complex vector space, is finite dimensional and write |$T=\textrm{Spec}\,\Lambda $|⁠. We give ourselves a smooth morphism of algebraic |$\textbf{C}$|-schemes|$g:Y\to T$| and set out to explain briefly the equivalence of |$\mathcal{D}$|-modules on |$Y$| and on its analytification, see Proposition 5.6 below. (We observe that these constructions are routine and often left aside, see pages 97–98 in [9], for example.) Let stand for the diagram of locally ringed spaces obtained from the analytification functor [28, XII, 1.2]. Proceeding as in [13, |$\textrm{IV}_{\textrm{4}}$|⁠, 16.4], while suppressing reference to |$\alpha ^{-1}$|⁠, |$T$| and |$T^{\textrm{an}}$|⁠, we arrive at a morphism of |$\mathcal{O}_{Y}$|-modules $$\begin{align*} &\{\alpha\}_\nu:\mathcal{P}^\nu_Y\longrightarrow\mathcal{P}^\nu_{Y^{\textrm{an}}} \end{align*}$$ such that, if |$d^\nu $| is as in [13, |$\textrm{IV}_{\textrm{4}}$|⁠, 16.3.6, page 15], then |$\{\alpha \}_\nu ( d^\nu (\,f))=d^\nu (\alpha ^\#(\,f))$|⁠. (The justification given on page 17 of [13, |$\textrm{IV}_{\textrm{4}}$|⁠, 16.4.3] for the existence of |$\{\alpha \}_\nu $| relies on the case of affine schemes, but a proof based only on formal properties of normal invariants is possible.) We then have a morphism of |$\mathcal{O}_{Y^{\textrm{an}}}$|-modules $$\begin{align}& P^\nu(\alpha):\mathcal{O}_{Y^{\textrm{an}}}\underset{\mathcal{O}_Y}{\otimes}\mathcal{P}_{Y}^\nu \longrightarrow \mathcal{P}_{Y^{\textrm{an}}}^\nu \end{align}$$(8) such that $$\begin{align}& P^\nu(\alpha)(1\otimes d^\nu(\,f))=d^\nu(\alpha^\#(\,f)). \end{align}$$(9) If |$V\subset Y$| is an open subset having etale coordinates |$\{y_j\}$| over |$T$|⁠, then |$\{d^\nu ( y^q)\,:\,\sum q_j\le \nu \}$| is a basis of |$\mathcal{P}^\nu _Y|_V$| [4, Proposition 2.6]. Similarly, as |$V^{\textrm{an}}$| is locally isomorphic to an open set of an affine space over |$T^{\textrm{an}}$| [27, Exposé 13, 3.1], the set |$\{d^\nu ( \alpha ^\#(y)^q)\,:\,\sum q_j\le \nu \}$| is a basis of |$\mathcal{P}^\nu _{Y^{\textrm{an}}}|_{V^{\textrm{an}}}$|⁠; we conclude with the help of equation (9) that |$P^\nu (\alpha )$| is an isomorphism. In particular, |$(\mathcal{P}^\nu _Y)^{\textrm{an}}\stackrel \sim \to \mathcal{P}^\nu _{Y^{\textrm{an}}}$| via |$P^\nu (\alpha )$| so that |$(|Y^{\textrm{an}}|,(\mathcal{P}^\nu _Y)^{\textrm{an}})\simeq P_{Y^{\textrm{an}}}^\nu $|⁠. Now we note that |$(|Y^{\textrm{an}}|,(\mathcal{P}_Y^\nu )^{\textrm{an}})$| is just |$(P^{\nu }_Y)^{\textrm{an}}$|⁠. Taking the dual of arrow (8) for varying |$\nu $|⁠, we obtain an isomorphism of |$\mathcal{O}_{Y^{\textrm{an}}}$|-modules |$\mathcal{D}(Y^{\textrm{an}})\to \mathcal{O}_{Y^{\textrm{an}}}\otimes _{\mathcal{O}_Y}\mathcal{D}(Y)$|⁠, which allows us to construct an arrow of |$\mathcal{O}_Y$|-modules $$\begin{align*} & \rho:\mathcal{D}(Y)\longrightarrow\mathcal{D}(Y^{\textrm{an}}). \end{align*}$$ Using |$V$| and |$\{y_j\}$| as above and equation (9), the verification that |$\rho $| is a morphism of |$\mathcal{O}_Y$|-algebras is straightforward. If |$\mathcal{E}\in \mathcal{D}_Y\text{-}\textbf{mod}$| and |$\{\theta (\nu )\}$| are its Taylor isomorphisms (see Section 5.1), we arrive at a commutative diagram of |$\mathcal{P}^\nu _{Y^{\textrm{an}}}$|-modules |$ $|(10) allowing us to define an analytification functor: $$\begin{align*} &(-)^{\textrm{an}}:\mathcal{D}(Y/T)\text{-}\textbf{mod} \longrightarrow\mathcal{D}(Y^{\textrm{an}}/T^{\textrm{an}})\text{-}\textbf{mod}. \end{align*}$$ An argument similar to the one used in proving Proposition 5.3 gives the following: Proposition 5.6. Suppose that |$g$| is proper. Then the functor |$(-)^{\textrm{an}}:\mathcal{D}(Y/T)\text{-}\textbf{mod}\to \mathcal{D}(Y^{\textrm{an}}/T^{\textrm{an}})\text{-}\textbf{mod}$| is an equivalence of |$\Lambda $|-linear |$\otimes $|-categories. Proof. We only offer a sketch. We deal only with essential surjectivity. By GAGA [28, XII, Theorem 4.4], we only need to show that if |$\mathcal{E}$| is a coherent |$\mathcal{O}_{Y}$|-module such that |$\mathcal{E}^{\textrm{an}}$| carries a structure of |$\mathcal{D}(Y^{\textrm{an}})$|-module, then |$\mathcal{E}$| carries a structure of |$\mathcal{D}(Y)$|-module inducing the preceding one. So let |$\nu \in \textbf{N}$| be fixed and write |$\sigma (\nu ):p^{\textrm{an}*}_{1}\mathcal{E}^{\textrm{an}}\to p_0^{\textrm{an}*}\mathcal{E}^{\textrm{an}}$| for the Taylor isomorphism of order |$\nu $|⁠. Applying GAGA to the proper |$\textbf{C}$|-scheme |$P_{Y}^\nu =(|Y|,\mathcal{P}_Y^\nu )$| we see that |$\sigma (\nu )$| comes from an isomorphism |$\theta (\nu ):p_1^*\mathcal{E}\to p_0^*\mathcal{E}$| of |$\mathcal{P}_{Y}^\nu $|-modules. Let |$V$| be an affine and open subscheme of |$Y$| having etale coordinates |$\{y_j\}$| as above, |$e$| a section of |$\mathcal{E}$| over |$V$|⁠, and |$\partial \in \mathcal{D}(Y)(V)$| of order |$\le \nu $|⁠. Define |$\nabla _\nu (\partial )(e)=(\textrm{id}_{\mathcal{E}}\otimes \partial )[\theta (\nu )(1\otimes e)]$|⁠, which is a section of |$\mathcal{E}$|⁠. If |$(-)^{\textrm{an}}:\Gamma (V,\mathcal{E})\to \Gamma (V^{\textrm{an}},\mathcal{E}^{\textrm{an}})$| denotes the canonical map, then equation (9) and diagram (10) show that |$\rho (\partial )(e^{\textrm{an}})=(\nabla _\nu (\partial )(e))^{\textrm{an}}$|⁠. This allows us to define on |$\mathcal{E}$| the structure of a |$\mathcal{D}(Y)$|-module having |$\{\theta (\nu )\}$| as Taylor isomorphisms; details are simple and similar to those in the proof of Proposition 5.3. To complement this Proposition, we note the following. Let |$y$| be a |$T$|-point of |$Y$| sending |$|T|$| to |$y_0$|⁠. If |$y^{\textrm{an}}$| is the associated |$T^{\textrm{an}}$|-point of |$Y^{\textrm{an}}$|⁠, then it sends |$|T^{\textrm{an}}|$| to |$y_0$|⁠, and, for an object |$\mathcal{E}$| of |$\mathcal{D}(Y/T)\text{-}\textbf{mod}$|⁠, the natural morphism of stalks |$\mathcal{E}_{y_0}\to (\mathcal{E}^{\textrm{an}})_{y_0}$| defines an isomorphism |$y^*\mathcal{E}\to y^{\textrm{an}*}\mathcal{E}^{\textrm{an}}$|⁠. This produces a natural isomorphism |$y^*\stackrel{\sim }{\to } y^{\textrm{an}*}\circ (-)^{\textrm{an}}$|⁠, where we abuse notation and identify |$\textbf{Coh}(T)$| and |$\textbf{Coh}(T^{\textrm{an}})$|⁠. 5.5 |$\mathcal{D}$|-modules and relative local systems Notations now are those of Section 5.3, so that we are concerned only with complex analytic spaces. For each |$\mathcal{E}\in \mathcal{D}(Y/T)\text{-}\textbf{mod}$|⁠, we consider the sheaf of |$\Lambda $|-modules (“solutions,” “horizontal sections”) $$\begin{align*} & \textbf{Sol}(\mathcal{E})(U):=\left\{s\in\mathcal{E}(U)\,:\,\begin{array}{c} \textrm{for}\ V\subset U\ \textrm{open, the section}\ s|_V\ \textrm{is annihilated}\\ \textrm{by the augmentation ideal of}\ \mathcal{D}(Y/T)(V) \end{array}\right\}. \end{align*}$$ Now, recalling that |$\mathcal{D}(Y/T)$|-modules are exactly the same thing as integrable connections [4, Theorem 2.15], Deligne shows in [9, I.2.23] that: Theorem 5.7. For each |$\mathcal{E}\in \mathcal{D}(Y/T)\text{-}\textbf{mod}$|⁠, the sheaf of |$\Lambda $|-modules |$\textbf{Sol}(\mathcal{E})$| is a relative local system and the morphism of |$\mathcal{O}_Y$|-modules $$\begin{align*} &\mathcal{O}_Y\otimes_\Lambda \textbf{Sol}(\mathcal{E})\longrightarrow\mathcal{E}\end{align*}$$ is an isomorphism. In addition, the functor $$\begin{align*} & \textbf{Sol}:\mathcal{D}(Y/T)\text{-}\textbf{mod}\longrightarrow \textbf{LS}(Y/T) \end{align*}$$ is a |$\otimes $|-equivalence of |$\Lambda $|-linear categories. Putting together Proposition 5.5 and Theorem 5.7, we have the following: Corollary 5.8. The composite functor $$\begin{align*} &\mathcal{D}(Y/T)\text{-}\textbf{mod}\stackrel{\textbf{Sol}}{\longrightarrow} \textbf{LS}(Y/T) \stackrel{\textsf{IM}_{y_0}}{\longrightarrow} \textrm{Rep}_{\Lambda}(\Gamma) \end{align*}$$ defines a |$\otimes $|-equivalence of |$\Lambda $|-linear categories. To complement this corollary, we make the following observations. Let |$y:T\to Y$| be a section to the structural morphism, which takes |$|T|$| to |$y_0$|⁠, and let |$\mathcal{E}$| be an object of |$\mathcal{D}(Y/T)\text{-}\textbf{mod}$|⁠. On the level of stalks at |$y_0$|⁠, Theorem 5.7 implies that the natural morphism |$\mathcal{O}_{y_0}\otimes _\Lambda \textbf{Sol}(\mathcal{E})_{y_0}\to \mathcal{E}_{y_0}$| is an isomorphism, so that the composite arrow |$\textbf{Sol}(\mathcal{E})_{y_0}\to \mathcal{E}_{y_0}\to y^*(\mathcal{E})$| is an isomorphism. Hence, if |$\omega :\textrm{Rep}_{\Lambda }(\Gamma )\to \Lambda \text{-}\textbf{mod}$| stands for the forgetful functor, it follows that |$\omega \circ \textsf{IM}_{y_0}\circ \textbf{Sol}\stackrel \sim \to y^*$| as |$\otimes $|-functors. 5.6 The category |$\textrm{Rep}_R(\Gamma )^\wedge $| In this section, we shall assume that |$R$| is complete and denote by |$t$| a uniformizer. Given an abstract group |$\Gamma $|⁠, we introduce the category |$\textrm{Rep}_R(\Gamma )^\wedge $| following the pattern of Definition 5.2. Its objects are sequences |$(E_n,\sigma _n)_{n\in \textbf{N}}$| where each |$E_n$| is a left |${R_n}\Gamma $|-module whose underlying |$R_n$|-module is finite, and $$\begin{align*} & \sigma_n:R_{n}\otimes_{R_{n+1}}E_{n+1}\longrightarrow E_n \end{align*}$$ is an isomorphism of |$R_n\Gamma $|-modules. Term-by-term tensor product (of |$R_n$|-modules) defines on |$\textrm{Rep}_{R}(\Gamma )^\wedge $| the structure of a |$R$|-linear |$\otimes $|-category. The categories |$\textrm{Rep}_{R}(\Gamma )$| and |$\textrm{Rep}_{R}(\Gamma )^\wedge $| are related by two functors $$\begin{align*} &\mathfrak{F}:\textrm{Rep}_{R}(\Gamma)\longrightarrow \textrm{Rep}_{R}(\Gamma)^\wedge\end{align*}$$ and $$\begin{align*} & \mathfrak{L}:\textrm{Rep}_R(\Gamma)^\wedge\longrightarrow \textrm{Rep}_R(\Gamma). \end{align*}$$ On the level of modules these are defined by $$\begin{align*} &\mathfrak{F}(E)=(R_n\otimes_R E,\textrm{canonic})\quad\textrm{and}\quad\mathfrak{L}(E_n,\sigma_n)=\varprojlim_nE_n.\end{align*}$$ (That the projective limit above is of finite type over |$R$| is proved in [13, |$\textrm{0}_{\textrm{I}}$|⁠, 7.2.9].) Needless to say, |$\mathfrak{L}$| and |$\mathfrak{F}$| are inverse |$\otimes $|-equivalences of |$R$|-linear categories, see 7.2.9-11 and 7.7.1 in [13, |$0_{\textrm{I}}$|]. Remark 5.9. The additive category |$\textrm{Rep}_R(\Gamma )^\wedge $| is abelian and |$\mathfrak{F}$| respects that structure, but kernels are not defined effortlessly because for an arrow |$(\varphi _n):(E_n)\to (F_n)$|⁠, the projective system of kernels |$(\textrm{Ker}\,\varphi _n)$| is not necessarily an object of |$\textrm{Rep}_R(\Gamma )^\wedge $|⁠. For example, one can consider the obvious object |$(R_n)$| of |$\textrm{Rep}_{R}(\Gamma )^\wedge $| and the arrow |$(\varphi _n):(R_n)\to (R_n)$| of multiplication by |$t$|⁠; then the kernel is certainly trivial, while |$\textrm{Ker}(\varphi _0)=R_0$|⁠. 5.7 The main result We suppose here that |$R=\textbf{C} [\![ t ]\!] $| and return to the setting of Section 5.2, where, among others, we considered a smooth morphism of schemes $$\begin{align*} & f:X\longrightarrow S. \end{align*}$$ We assume in addition that |$f$| is proper and that |$X_0$| is connected. Note that, the latter condition implies that each |$|X_n^{\textrm{an}}|$| is also connected [28, XII Proposition 2.4]. Following the model of Definition 5.2, we use the |$\textbf{N}$|-indexed family of categories |$\mathcal{D}(X_n^{\textrm{an}}/S_n^{\textrm{an}})\text{-}\textbf{mod}$| to produce $$\begin{align*} & \mathcal{D}(X^{\textrm{an}}/S^{\textrm{an}})\text{-}\textbf{mod}^\wedge. \end{align*}$$ (Here |$X^{\textrm{an}}$| and |$S^{\textrm{an}}$| are just graphical signs carrying no mathematical meaning.) Since |$X_n^{\textrm{an}}$| is proper over |$S_n^{\textrm{an}}$| [28, XII, Proposition 3.2, page 245], Proposition 5.6 produces |$\otimes $|-equivalences of |$R_n$|-linear categories $$\begin{align*} & {(-)^{\textrm{an}}}:\mathcal{D}(X_n/S_n)\text{-}\textbf{mod}\stackrel{\sim}{\longrightarrow} \mathcal{D}(X_n^{\textrm{an}}/S_n^{\textrm{an}})\text{-}\textbf{mod}, \end{align*}$$ which in turn give rise to a |$\otimes $|-equivalence of |$R$|-linear categories $$\begin{align*} & \mathfrak{A}:\mathcal{D}(X/S)\text{-}\textbf{mod}^\wedge \stackrel{\sim}{\longrightarrow} \mathcal{D}(X^{\textrm{an}}/S^{\textrm{an}})\text{-}\textbf{mod}^\wedge. \end{align*}$$ Let |$x$| be an |$R$|-point of |$X$| and write |$x_n$| for the corresponding |$R_n$|-point of |$X_n$|⁠. Clearly, each |$x_n$| induces the same point, call it |$x_0$|⁠, on the topological space |$|X_0^{\textrm{an}}|=|X_n^{\textrm{an}}|$|⁠. Let |$\Gamma $| be the fundamental group of |$|X_0^{\textrm{an}}|$| based at |$x_0$|⁠. From Corollary 5.8, we have |$\otimes $|-equivalences of |$R_n$|-linear categories: $$\begin{align*} & \textsf{IM}_{x_0}\circ \textbf{Sol}:\mathcal{D}(X_n^{\textrm{an}}/S_n^{\textrm{an}})\text{-}\textbf{mod}\stackrel{\sim}{\longrightarrow} \textrm{Rep}_{R_n}(\Gamma), \end{align*}$$ which in turn induce an equivalence $$\begin{align*} & \mathfrak{M}_{x_0}: \mathcal{D}(X^{\textrm{an}}/S^{\textrm{an}})\text{-}\textbf{mod}^\wedge\stackrel{\sim}{\longrightarrow} \textrm{Rep}_R(\Gamma)^\wedge. \end{align*}$$ We have therefore a diagram |$ $|(11) where each arrow is a |$\otimes $|-equivalence of |$R$|-linear categories. We can therefore say: Theorem 5.10. The |$R$|-linear |$\otimes $|-categories |$\mathcal{D}(X/S)\text{-}\textbf{mod}$| and |$\textrm{Rep}_{R}(\Gamma )$| are equivalent by means of the composition of (11) above. Since we are also interested in the differential Galois groups associated to objects in |$\mathcal{D}(X/S)\text{-}\textbf{mod}$| (see [12, Section 7] and Section 2.3), we complement Theorem 5.10 by explaining how the aforementioned equivalence relates fibre functors. As we observed after Proposition 5.6 and Corollary 5.8, |$x_n^*\simeq x_n^{\textrm{an}*}\circ (-)^{\textrm{an}}$| and |$x_n^{\textrm{an}*}\simeq \omega \textsf{IM}_{x_0}\textbf{Sol}$|⁠. This implies that |$\omega \mathfrak{M}_{x_0}\mathfrak{A}$| is |$\otimes $|-isomorphic to |$(\mathcal{E}_n)\mapsto (x_n^*\mathcal{E}_n)$| so under the above equivalence the forgetful functors |$\textrm{Rep}_{R}(\Gamma )\to R\text{-}\textbf{mod}$| and |$x^*:\mathcal{D}(X/S)\text{-}\textbf{mod}\to R\text{-}\textbf{mod}$| are |$\otimes $|-isomorphic. Corollary 5.11. If |$\Gamma $| has an infinite cyclic quotient, then the group scheme |$\mathcal{N}$| of Section 4.3 appears as the full differential Galois group of some |$(\mathcal{E},\nabla )\in \mathcal{D}(X/S)\text{-}\textbf{mod}^\circ $|⁠. Proof. Let |$\Pi $| be the Tannakian envelope of |$\Gamma $| (Definition 4.1). Using [11, 3.2.1(ii)] and then Proposition 4.15 we produce a faithfully flat morphism |$u:\Pi \to \mathcal{N}$|⁠. Let now |$E$| be a free |$R$|-module of finite rank and |$\rho :\textbf{G}_a\times \textbf{G}_m\to \textbf{GL}(E)$| a faithful representation. We have a commutative diagram in |$(\textbf{FGSch}/R)$| from [12, Proposition 4.10] we know that the full subcategory |$\langle E\rangle _\otimes $| of |$\textrm{Rep}_{R}(\Pi )$| is equivalent to |$\textrm{Rep}_{R}(\mathcal{N})$| under |$u$|⁠. Now we translate these observations to the category |$\mathcal{D}(X/S)\text{-}\textbf{mod}$|⁠. 6 Prudence of Flat Group Schemes and Projectivity of the Underlying Hopf Algebra In this section, |$R$| is assumed to be a complete DVR. In what follows, |$G$| is a flat group scheme over |$R$|⁠, which is not assumed to be of finite type. We shall investigate under which conditions |$R[G]$| is a projective |$R$|-module. Our method is based on the following fundamental result. Theorem 6.1 (cf. [16, page 338]). Let |$M$| be a flat |$R$|-module whose associated |$K$|-module |$M\otimes K$| has countable rank. Then |$M$| is free if and only if |$\cap \pi ^nM=(0)$|⁠. Therefore, all efforts should be concentrated on understanding the divisible ideal, which we now introduce. Definition 6.2. Let |$M$| be an |$R$|-module. We denote by |$\mathfrak{d}(M)$| the submodule |$\cap \pi ^nM$|⁠, and call it the divisible submodule of |$M$|⁠. If |$A$| is an |$R$|-algebra, |$\mathfrak{d}(A)$| is an ideal, which we call the divisible ideal of |$A$|⁠. If |$X=\textrm{Spec}\,A$|⁠, we write alternatively |$\mathfrak{d}(X)$| to mean |$\mathfrak{d}(A)$| and let |$\textrm{Haus}(X)$| stand for the closed subscheme of |$X$| defined by |$\mathfrak{d}(X)$|⁠. It is quite easy to see that |$\mathfrak{d}(G)$| cuts out a closed subgroup scheme (see Proposition 6.12). This being so, in order to study how far |$\mathfrak{d}(G)$| is from being |$(0)$|⁠, the natural idea is, following Chevalley, to search semi-invariants. This is the crucial idea behind our notion of prudence, see Definition 6.7. Section 6.1 explains what we understand by semi-invariants in the context of group schemes over |$R$|⁠. In it we also derive basic results on this concept, which prove useful further ahead. In Section 6.2, we introduce the notion of a prudent group scheme; this property is compared to the vanishing of the divisible ideal in Section 6.3. The function of Section 6.4 is to relate prudence to other existing properties (appearing in [29], [20], and [11]). Finally, Section 6.5 links prudence of differential Galois groups to Grothendieck’s existence theorem of projective geometry. 6.1 Generalities on semi-invariants Let |$V$| be an object of |$\textrm{Rep}_{R}(G)$|⁠; comultiplication |$V\to V\otimes _RR[G]$| is denoted by |$\rho $|⁠. Definition 6.3. We say that |$\textbf{v}\in V$| is a semi-invariant if |$R \textbf{v}$| is a subrepresentation [15, Part I, 2.9]. The set of all semi-invariants in |$V$| is denoted by |$\textrm{SI}_G(V)$|⁠. We need the following basic results on semi-invariants. Lemma 6.4. The element |$\textbf{v}\in V$| is a semi-invariant if and only if |$\rho (\textbf{v})=\textbf{v}\otimes \theta $| for some |$\theta \in R[G]$|⁠. Proof. If |$R\textbf{v}$| is a subrepresentation, then |$\rho (\textbf{v})$| belongs to the |$R$|-submodule |$R\textbf{v}\otimes _RR[G]$| of |$V\otimes R[G]$|⁠, and hence |$\rho (\textbf{v})=\textbf{v}\otimes \theta $| for some |$\theta \in R[G]$|⁠. On the other hand, if |$\rho (\textbf{v})=\textbf{v}\otimes \theta $| for some |$\theta \in R[G]$|⁠, then |$\rho |_{R\textbf{v}}$| factors through |$R\textbf{v}\otimes R[G]$|⁠, which amounts to being a subrepresentation [15, Part I, 2.9]. Remarks 6.5. We refrain from calling |$\theta $| a group-like element because we do not assume that |$R\textbf{v}$| is isomorphic to |$R$|⁠. Note that [29, Exposé |$\textrm{VI}_B$|] only uses the notion of semi-invariant in the case of a base field, see Definition 11.15 on page 420 in there. The set of semi-invariants is not usually stable under addition, so that it cannot be an |$R$|-submodule. On the other hand, it is stable under scalar multiplication (apply Lemma 6.4). Lemma 6.6. The following claims are true. (1) Let |$\varphi :V\to W$| be a morphism of representations of |$G$|⁠. The image of |$\textrm{SI}_G(V)$| lies in |$\textrm{SI}_G(W)$|⁠. (2) Let |$H$| be another flat affine group scheme over |$R$| and |$f:G\to H$| a morphism. Then, for any |$V\in \textrm{Rep}_{R}(H)$|⁠, we have |$\textrm{SI}_H(V)\subset \textrm{SI}_G(V)$|⁠. (3) Under the notations of the previous item, assume that |$f^\#:R[H]\to R[G]$| is injective and that |$V$| belongs to |$\textrm{Rep}^\circ _{R}(H)$|⁠. Then |$\textrm{SI}_H(V)=\textrm{SI}_G(V)$|⁠. Proof. The verification of (1) and (2) follows the obvious method and we omit it. (3) Let |$\textbf{v}\in V$| be |$G$|-semi-invariant. We assume, to begin with, that |$\textbf{v}$| is the 1st element of an ordered basis |$\{\textbf{v}_1,\ldots ,\textbf{v}_r\}$| of |$V$|⁠. Write |$\sigma :V\to V\otimes R[H]$| for the coaction and let |$\sigma (\textbf{v}_j)=\sum _i\textbf{v}_i\otimes \sigma _{ij}$| with |$\sigma _{ij}\in R[H]$|⁠. Our hypothesis jointly with Lemma 6.4 then implies that $$\begin{align*} &\sum_i\textbf{v}_i\otimes f^\#(\sigma_{i1}) =\textbf{v}_1\otimes \theta\end{align*}$$ for some |$\theta \in R[G]$|⁠. This shows that |$f^\#(\sigma _{i1})=0$| if |$i>1$| and, consequently, that |$\sigma _{i1}=0$| if |$i>1$|⁠. This implies that |$\textbf{v}$| is |$H$|-semi-invariant (again by Lemma 6.4). Now, for a general |$\textbf{v}\in V\setminus \{0\}$|⁠, there exists an ordered basis |$\{\textbf{v}_1,\ldots ,\textbf{v}_r\}$| and some non-negative integer |$m$| such that |$\pi ^m\textbf{v}_1=\textbf{v}$|⁠. We then see that |$\textbf{v}_1$| is also |$G$|-semi-invariant: there exists |$\theta \in R[G]$| such that |$(\textrm{id}_V\otimes f^\#)\circ \sigma (\pi ^m\textbf{v}_1)=\pi ^m\textbf{v}_1\otimes \theta $|⁠, and hence |$(\textrm{id}_V\otimes f^\#)\circ \sigma (\textbf{v}_1)=\textbf{v}_1\otimes \theta $| as |$\pi ^m$| is not a zero divisor in |$V\otimes R[G]$|⁠; Lemma 6.4 proves that |$\textbf{v}_1$| is semi-invariant for |$G$|⁠. From the previous considerations, |$\textbf{v}_1$| is |$H$|-semi-invariant, so that |$\textbf{v}=\pi ^m\textbf{v}_1$| is also |$H$|-semi-invariant. 6.2 The concept of prudence We are now able to isolate the property of |$\textrm{Rep}_{R}(G)$| which forces the ideal |$\mathfrak{d}(G)$| to vanish. Definition 6.7. Let |$V\in \textrm{Rep}_R^{\circ }(G)$|⁠. We say that |$V$| is prudent if an element |$\textbf{v}$| whose reductions modulo |$\pi ^{n+1}V$| are semi-invariant for all |$n$| must necessarily be semi-invariant. We say that |$G$| is prudent if each |$V\in \textrm{Rep}_R^{\circ }(G)$| is prudent. The definition of prudence admits the following reformulation (the proof is similar to the proof of Lemma 6.6(3), and we omit it). Lemma 6.8. Let |$V\in \textrm{Rep}^\circ _{R}(G)$|⁠. Assume that each |$\textbf{v}\in V$| that |$\bullet $| belongs to a basis, and |$\bullet $| has a reduction modulo |$\pi ^{n+1}V$|⁠, which is semi-invariant for all |$n$|⁠, is a semi-invariant. Then |$V$| is prudent. The following result shall be employed in deriving deeper consequences in Section 6.3. Lemma 6.9. Let |$f:G\to H$| be a morphism in |$(\textbf{FGSch}/R)$| such that the associated morphism of Hopf algebras |$f^\#:R[H]\to R[G]$| is injective. Then, if |$G$| is prudent, so is |$H$|⁠. Proof. Let |$V\in \textrm{Rep}^\circ _{R}(H)$| and consider an element |$\textbf{v}\in V$| whose reduction modulo |$\pi ^{n+1}V$| is |$H$|-semi-invariant for any given |$n$|⁠. From Lemma 6.6-(2), we have $$\begin{align*} & \textrm{SI}_H(V/\pi^{n+1})\subset \textrm{SI}_G(V/\pi^{n+1}). \end{align*}$$ This forces, as |$G$| is prudent, |$\textbf{v}$| to be |$G$|-semi-invariant. But by Lemma 6.6-(3), we know that |$\textbf{v}$| is then |$H$|-semi-invariant. Example 6.10. Let |$\Gamma $| be an abstract group with Tannakian envelope |$\Pi $| (see Section 4.1 for definitions). We contend that |$\Pi $| is prudent. Let |$V$| be a free |$R$|-module of rank |$r$| affording a linear representation of |$\Gamma $|⁠. Let |$\textbf{v}\in V$| be such that, for any given |$n\in \textbf{N}$|⁠, the element |$\textbf{v}+\pi ^{n+1}V$| is semi-invariant. This implies that |$R\textbf{v}+\pi ^{n+1}V$|⁠, regarded as an |$R$|-submodule of |$V$|⁠, is invariant under |$\Gamma $|⁠. Hence, |$\cap _n(R\textbf{v}+\pi ^{n+1}V)$| is invariant under |$\Gamma $|⁠. But this intersection is the closure of |$R\textbf{v}$| for the |$\pi $|-adic topology [17, pages 55–56], which is simply |$R\textbf{v}$| [17, Theorem 8.10]. 6.3 Prudence and the divisible ideal Our objective here is to establish the following: Theorem 6.11. If |$G$| is prudent, then |$\mathfrak{d}( G)=0$|⁠. Before embarking on a proof, we note a fundamental fact. Proposition 6.12. The closed subscheme of |$G$| defined by |$\mathfrak{d}(G)$|⁠, |$\textrm{Haus}(G)$|⁠, is in fact a closed and flat subgroup scheme. The proof of this Proposition follows easily from item (3) of Lemma 6.13. Let |$M$| and |$N$| be flat |$R$|-modules. (1) Assume that |$\mathfrak{d}(M)=(0)$| and let |$E\subset M$| be finitely generated. Then |$E^{\textrm{sat}}$| is free. (2) If |$\mathfrak{d}(M)=(0)$| and |$\mathfrak{d}(N)=(0)$|⁠, then |$\mathfrak{d}(M\otimes _R N)=(0)$|⁠. (3) The divisible submodule of |$M\otimes N$| is simply |$\mathfrak{d}(M)\otimes N+M\otimes \mathfrak{d}(N)$|⁠. Proof. (1) Clearly, |$E\otimes K\stackrel{\sim }{\to } E^{\textrm{sat}}\otimes K$|⁠, and we can apply Theorem 6.1 to conclude. (2) Let |$t=\sum u_i\otimes v_i\in M\otimes N$| belong to |$\mathfrak{d}(M\otimes N)$|⁠. By saturating |$\sum Ru_i$| and |$\sum Rv_i$|⁠, and applying (1), we see that there exist a free and saturated submodule |$U$| of |$M$| and a free and saturated submodule |$V$| of |$N$| such that |$t\in U\otimes V$|⁠. Now |$U\otimes V$| is saturated in |$M\otimes N$|⁠, and hence |$\mathfrak{d}(M\otimes N)\cap (U\otimes V)=\mathfrak{d}(U\otimes V)$|⁠. Therefore, |$t=0$| since |$\mathfrak{d}(U\otimes V)=(0)$|⁠. (3) This is a simple consequence of (2). We now head towards the proof of Theorem 6.11. In fact, this theorem is a consequence of the ensuing result, which therefore becomes the object of our efforts. To state it, we employ the notion of the diptych [12, Section 4], briefly recalled here for the convenience of the reader. Given an arrow |$u:\mathcal{H}\to \mathcal{G}$| of |$(\textbf{FGSch}/R)$|⁠, there are two reasonable ways to define its image: the closed subgroup scheme of |$\mathcal{G}$| cut out by the kernel of |$u^\#:R[\mathcal{G}]\to R[\mathcal{H}]$| (the schematic image of |$u$|⁠) and the spectrum of the saturation of |$u^\#(R[\mathcal{G}])$| in |$R[\mathcal{H}]$|⁠. If |$\Psi $| stands for the former and |$\Psi ^{\prime}$| for the latter, we then have a commutative diagram called the diptych. Theorem 6.14. Assume that |$G$| is prudent and let |$V\in \textrm{Rep}^\circ _{R}(G)$| be arbitrary. Consider the diptych of |$\rho $| so that |$\Psi ^{\prime}\otimes K=\Psi \otimes K$| is of finite type over |$K$|⁠. Then |$\mathfrak{d}(\Psi ^{\prime})=0$|⁠. Proof that Theorem 6.14|$\Rightarrow $|Theorem 6.11. Let |$a\in \mathfrak{d}(G)$|⁠. Endowing |$R[G]$| with its right-regular action and using “local finiteness” [15, Part I, 2.13], we can find a subcomodule |$V\subset R[G]$| that contains |$a$| and is finitely generated as an |$R$|-module. Since |$R[\Psi ]$| is constructed as the image of the obvious morphism |$R[\textbf{GL}(V)]\to R[G]$|⁠, and |$R[\Psi ^{\prime}]$| is the saturation of |$R[\Psi ]$| in |$R[G]$| (see [12, Section 4]), we conclude that |$a\in R[\Psi ^{\prime}]$|⁠. Now, because |$R[\Psi ^{\prime}]$| is saturated in |$R[G]$|⁠, we have |$\mathfrak{d}(\Psi ^{\prime})=\mathfrak{d}(G)\cap R[\Psi ^{\prime}]$|⁠. If Theorem 6.14 is true, then |$a=0$|⁠. We need some lemmas to establish Theorem 6.14. In these results, we employ the notion of closure of a closed subscheme of the generic fibre [13, |$\textrm{IV}_{\textrm{2}}$|⁠, 2.8, 33ff] as well as the constructions of Neron blowups appearing in Section 2.2. Lemma 6.15. Let |$f:\mathcal{G}^{\prime}\to \mathcal{G}$| be a morphism of |$(\textbf{FGSch}/R)$|⁠. We suppose that |$\mathcal{G}$| is of finite type and that |$f\otimes K$| is an isomorphism. Let |$H\subset \mathcal{G}$| stand for the closure of $$\begin{align*} &\textrm{Haus}(\mathcal{G}^{\prime})\otimes K\subset \mathcal{G}^{\prime}\otimes K=\mathcal{G}\otimes K\end{align*}$$ in |$\mathcal{G}$|⁠. Then |$f$| factors through $$\begin{align*} & \mathcal{N}^\infty_H(\mathcal{G})\longrightarrow \mathcal{G}. \end{align*}$$ Proof. Algebraically, |$H$| is cut out by the ideal $$\begin{align*} & I=\{ a\in R[\mathcal{G}]\,:\, 1\otimes a\in K\otimes \mathfrak{d}(\mathcal{G}^{\prime}) \}. \end{align*}$$ By definition, |$R[\mathcal{N}_H^\infty (\mathcal{G})]$| is the |$R$|-subalgebra of |$K[\mathcal{G}]$| obtained by adjoining to |$R[\mathcal{G}]$| all elements of the form |$\pi ^{-n}\otimes a$|⁠, where |$a\in I$|⁠. We then need to prove that for each |$a\in I$| and each |$n\in \textbf{N}$|⁠, there exists |$\delta _n\in R[\mathcal{G}^{\prime}]$| such that |$a=\pi ^n\delta _n$|⁠. So let |$a\in I$|⁠. By construction, |$1\otimes a$| belongs to the ideal |$K\otimes \mathfrak{d}(\mathcal{G}^{\prime})$|⁠. Hence, there exists some |$m\in \textbf{N}$| and some |$\delta \in \mathfrak{d}(\mathcal{G}^{\prime})$| such that |$1\otimes a=\pi ^{-m}\otimes \delta $|⁠. This shows that |$\pi ^m a\in \mathfrak{d}(\mathcal{G}^{\prime})$|⁠. Now, for every |$n\in \textbf{N}$|⁠, we pick |$\delta _n$| such that |$\pi ^{n+m}\delta _n=\pi ^ma$|⁠, so that |$\pi ^n\delta _n=a$|⁠. Lemma 6.16. Let |$\mathcal{H}\to \mathcal{G}$| be a closed embedding of flat group schemes of finite type over |$R$|⁠. If |$\mathcal{N}_{\mathcal{H}}^\infty (\mathcal{G})$| is prudent, then |$\mathcal{H}=\mathcal{G}$|⁠. Proof. From Lemma 6.17 below, we can find |$V\in \textrm{Rep}^\circ _{R}(\mathcal{G})$| and a line |$R\textbf{v}\subset V$| such that $$\begin{align*} & \textbf{Stab}_{\mathcal{G}}(R\textbf{v})=\mathcal{H}. \end{align*}$$ The |$\mathcal{H}$|-semi-invariance of |$\textbf{v}$| forces |$\textbf{v}+\pi ^{n+1}V\in V/\pi ^{n+1}$| to be |$\mathcal{H}$|-semi-invariant for all |$n$|⁠. (Use Lemma 6.6-(1).) But since for each |$n$| the base change $$\begin{align*} &\mathcal{N}^\infty_{\mathcal{H}}(\mathcal{G})\otimes R_n\longrightarrow \mathcal{G}\otimes R_n\end{align*}$$ factors through a morphism $$\begin{align*} & \mathcal{N}_{\mathcal{H}}^\infty(\mathcal{G})\otimes R_n\longrightarrow \mathcal{H}\otimes R_n, \end{align*}$$ and this is the central point, we conclude that |$\textbf{v}+\pi ^{n+1}V$| is |$\mathcal{N}_{\mathcal{H}}^\infty (\mathcal{G})$|-semi-invariant. (The arrow |$\mathcal{N}_{\mathcal{H}}^\infty (\mathcal{G})\otimes R_n\to \mathcal{H}\otimes R_n$| is even an isomorphism [10, Corollary 5.11].) As |$\mathcal{N}_{\mathcal{H}}^\infty (\mathcal{G})$| is assumed prudent, |$\textbf{v}$| is |$\mathcal{N}_{\mathcal{H}}^\infty (\mathcal{G})$|-semi-invariant as well. But then, Lemma 6.6-(3) guarantees that |$\textbf{v}\in \textrm{SI}_{\mathcal{G}}(V)$|⁠. This implies that |$R\textbf{v}$| is stable under |$\mathcal{G}$|⁠, so that |$\mathcal{G}=\mathcal{H}$|⁠. The next result is widespread in the case of a base field, see for example, [25, 16.1]. It is explicitly proved in [17, Proposition 1.2], but we include the details because the proof given in [25, 16.1]—which we follow—is quite transparent. Lemma 6.17. Let |$\mathcal{H}\to \mathcal{G}$| be a closed embedding in |$(\textbf{FGSch}/R)$|⁠. If |$\mathcal{G}$| is of finite type, then, there exists a certain |$V\in \textrm{Rep}^\circ _{R}(\mathcal{G})$| and an element |$\textbf{v}\in V$| such that $$\begin{align*} & \textbf{Stab}_{\mathcal{G}}(R\textbf{v})=\mathcal{H}. \end{align*}$$ Proof. Let |$V\subset R[\mathcal{G}]$| be a subcomodule of the right regular representation which, as an |$R$|-module, is of finite type and contains generators of the ideal |$I$| cutting out |$\mathcal{H}$|⁠. Let |$W:=V\cap I$| and note that |$W$| is saturated in |$V$| since |$I\subset R[\mathcal{G}]$| is. Let then |$\{\textbf{v}_1,\ldots ,\textbf{v}_s\}$| be a basis of |$V$| over |$R$| such that |$\{\textbf{v}_1,\ldots ,\textbf{v}_r\}$| is a basis of |$W$|⁠; in this case |$R[\mathcal{G}]\textbf{v}_1+\cdots +R[\mathcal{G}]\textbf{v}_r=I$|⁠. Write |$\Delta \textbf{v}_j=\sum \textbf{v}_i\otimes a_{ij}$|⁠, so that |$\textbf{Stab}_{\mathcal{G}}(W)$| is cut out by the ideal |$J=(a_{ij}\,:\,i>r,\,\, j\le r)$|⁠, see the proof of the last Lemma in [25, 12.4]. We now claim that |$J=I$|⁠, hence proving that |$\mathcal{H}=\textbf{Stab}_{\mathcal{G}}(W)$|⁠. First, since |$\textbf{v}_j=(\varepsilon \otimes \textrm{id})\circ \Delta \textbf{v}_j$|⁠, we see, using that |$\varepsilon (\textbf{v}_1)=\ldots =\varepsilon (\textbf{v}_r)=0$|⁠, that |$\textbf{v}_j=\sum _{i=r+1}^s\varepsilon (\textbf{v}_i)a_{ij}$|⁠, which shows that |$I\subset J$|⁠. Now, let |$\psi :R[\mathcal{G}]\to R[\mathcal{H}]$| be the projection and consider the |$R[\mathcal{H}]$|-comodule defined by |$(\textrm{id}\otimes \psi )\Delta :R[\mathcal{G}]\to R[\mathcal{G}]\otimes R[\mathcal{H}]$|⁠. (This corresponds to the action of |$\mathcal{H}$| on |$\mathcal{G}$| by multiplication on the right.) Then, we know that |$V$| and |$I$| are |$R[\mathcal{H}]-$|subcomodules, so that |$(\textrm{id}\otimes \psi )\Delta (I)\subset I\otimes R[\mathcal{H}]$| and |$(\textrm{id}\otimes \psi )\Delta (V)\subset V\otimes R[\mathcal{H}]$|⁠. This proves that |$(\textrm{id}\otimes \psi )\Delta (W)\subset W\otimes R[\mathcal{H}]$|⁠, because |$R[\mathcal{H}]$| is |$R$|-flat [17, Theorem 7.4(i)]. Therefore, since |$\textbf{v}_1,\ldots ,\textbf{v}_r\in W$|⁠, we see that |$(\textrm{id}\otimes \psi )\Delta (\textbf{v}_j)\in W\otimes R[\mathcal{H}]$| provided that |$1\le j\le r$|⁠. But |$(\textrm{id}\otimes \psi )\Delta (\textbf{v}_j)=\sum _{i\le r}\textbf{v}_i\otimes \psi (a_{ij})+\sum _{i>r}\textbf{v}_i\otimes \psi (a_{ij})$|⁠, and hence |$\psi (a_{ij})=0$|⁠, if |$i>r$| and |$j\le r$|⁠. This proves the inclusion |$J\subset I$|⁠, and our claim. The fact that |$W$| can be chosen of rank one follows from |$\textbf{Stab}_{\mathcal{G}}(W)=\textbf{Stab}_{\mathcal{G}}(\wedge ^s W)$| [25, Appendix 2]. Proof. of Theorem 6.14 Since |$G$| is prudent, Lemma 6.9 guarantees that |$\Psi ^{\prime}$| is prudent. Let |$H\subset \Psi $| be the closure of |$\textrm{Haus}(\Psi ^{\prime})\otimes K\subset \Psi \otimes K$| in |$\Psi $|⁠. Lemma 6.15 shows that |$\Psi ^{\prime}\to \Psi $| factors as $$\begin{align*} & \Psi^{\prime}\longrightarrow \mathcal{N}_{H}^\infty(\Psi)\longrightarrow\Psi. \end{align*}$$ Another application of Lemma 6.9 ensures that |$\mathcal{N}^\infty _H(\Psi )$| is prudent because |$\Psi ^{\prime}$| is. As |$\Psi $| is of finite type, Lemma 6.16 tells us that |$H=\Psi $|⁠. Hence, the closed embedding |$\textrm{Haus}(\Psi ^{\prime})\subset \Psi ^{\prime}$| becomes an isomorphism when base changed to |$K$|⁠. This is only possible if |$\textrm{Haus}(\Psi ^{\prime})=\Psi ^{\prime}$|⁠, and we are done. As a simple application of Theorem 6.11, we note that if |$\mathcal{N}$| is the group scheme studied in Section 4.3, then |$R[\mathcal{N}]$| is a free |$R$|-module. To see this, let |$\Pi $| stand for the Tannakian envelope of the infinite cyclic group |$\Gamma $| and |$u_\varphi :\Pi \to \mathcal{N}$| for the faithfully flat morphism constructed in Proposition 4.15. Since |$\Pi $| is a prudent group scheme (see Example 6.10) we conclude that |$(0)=\mathfrak{d}(\Pi )=\mathfrak{d}(\mathcal{N})$|⁠, which implies, due to Theorem 6.1, freeness of |$R[\mathcal{N}]$|⁠. 6.4 Prudence and some other characterizations Fundamental as it is, Kaplansky’s theorem has the drawback of needing the hypothesis on the rank. Using Raynaud–Gruson’s notion of Mittag–Leffler modules [20, Ch. 2, §2, Definition 3] (see also [19, Part 2]), we can put our knowledge in a more efficient structure and examine two other distinctive features that were singled out in [11] (repeated in Definition 6.18 below) and in [29, |$\textrm{VI}_B$|⁠, 11]. We recall that |$G$| is a flat group scheme over the complete discrete valuation ring |$R$|⁠. Definition 6.18 (cf. [11, Section 3.1]). A |$G$|-module |$M$| is called specially locally finite if for each |$G$|-submodule |$V\subset M$| whose underlying |$R$|-module is of finite type, the saturation |$V^{\textrm{sat}}\subset M$| is also of finite type (over |$R$|⁠). The group scheme |$G$| is specially locally finite if |$R[G]_{\textrm{right}}$| is specially locally finite. As remarked already in [11, Proposition 3.1.5], the above property is closely related to the Mittag–Leffler condition. The ensuing result straightens this relation. Proposition 6.19. The following conditions are equivalent. (1) The group scheme |$G$| is prudent. (2) The divisible ideal |$\mathfrak{d}(G)$| is null. (3) The group scheme |$G$| is specially locally finite. (4) The |$R$|-module |$R[G]$| satisfies the Mittag–Leffler condition of Gruson–Raynaud. (5) For each |$V\in \textrm{Rep}^\circ _{R}(G)$| and each finite subset |$F\subset V$|⁠, the set $$\begin{align*} & \{\text{{$W$} subrepresentation of {$V$} containing {$F$}}\}\end{align*}$$ has a least element. Proof. (1) |$\Rightarrow $| (2). This is the content of Theorem 6.11. (2) |$\Rightarrow $| (3). We know from [11, Theorem 4.1.1] that |$G$| is the limit in |$(\textbf{FGSch}/R)$| of a system $$\begin{align*} &\{G_\lambda,\varphi_{\lambda,\mu}:G_\mu\to G_\lambda\}\end{align*}$$ where (a) each |$G_\lambda \otimes K$| is of finite type, and (b) each |$\varphi _{\lambda \mu }$| is faithfully flat. Since |$\mathfrak{d}(G)=0$|⁠, we can certainly say that |$\mathfrak{d}(G_\lambda )=0$| for all |$\lambda $|⁠, so that, by Kaplansky’s Theorem (Theorem 6.1), |$R[G_\lambda ]$| is free as an |$R$|-module. Now, |$R[G]_{\textrm{right}}$| is the direct limit of a system of |$G$|-modules, |$R[G]_{\textrm{right}}=\varinjlim _\lambda R[G_\lambda ]$|⁠, and, by Proposition 3.1.5-(ii) of [11], |$R[G_\lambda ]$| is specially locally finite. As each |$R[G_\lambda ]$| is saturated in |$R[G]$|⁠, it becomes a simple matter to show that |$R[G]$| is also specially locally finite. (3) |$\Rightarrow $| (4). This is already explained in [11, Proposition 3.1.5-(i)]. (4) |$\Rightarrow $| (5). We write |$G=\varprojlim _\lambda G_\lambda $| as in the proof of “(2) |$\Rightarrow $| (3).” Since the canonical morphism |$G\to G_\lambda $| is faithfully flat, condition (4) and Corollary 2.1.6 of [19, Part 2, §1] guarantee that |$R[G_\lambda ]$| satisfies the Mittag–Leffler condition. The fact that |$K[G_\lambda ]$| is of finite type jointly with Proposition 1 of [20, Chapter 2, §2] assure in turn that, for each |$\lambda $|⁠, |$R[G_\lambda ]$| is a projective |$R$|-module. Let now |$V$| and |$F$| be as in condition (5). It is possible to find some |$\lambda $| such that |$V$| actually comes from a representation of |$G_\lambda $|⁠. Now [29, |$\textrm{VI}_B$|⁠, 11.8.1, page 418] assures the existence of a |$G_\lambda $|-subrepresentation |$W$| containing |$F$| and not properly containing any other such |$G_\lambda $|-subrepresentation. Since any |$G$|-subrepresentation of |$V$| must actually come from a representation of |$G_\lambda $| [11, Theorem 4.1.2(i)], we are done. (5) |$\Rightarrow $| (1). Let |$V\in \textrm{Rep}^\circ _{R}(G)$|⁠. If |$\textbf{v}\in V$| is such that its image in |$V/\pi ^{n+1}$| is a semi-invariant, it follows that |$W_n:=R\textbf{v}+\pi ^{n+1}V$| is a subrepresentation. Now, let |$W\subset V$| be a subrepresentation containing |$\textbf{v}$| and not properly containing any other such. It then follows that |$R\textbf{v}\subset W\subset \cap W_n=R\textbf{v}$|⁠, so that |$R\textbf{v}$| is a subrepresentation. By definition, |$\textbf{v}$| is semi-invariant, and |$G$| must be prudent. 6.5 Applications to the full differential Galois group Let |$f:X\to S$| be a smooth and proper morphism having geometrically connected fibres and admitting a section |$\xi \in X(R)$|⁠. Given |$\mathcal{E}\in \mathcal{D}(X/R)\text{-}\textbf{mod}^\circ $|⁠, let |$\textrm{Gal}^{\prime}(\mathcal{E})$| be the full differential Galois group. (See Section 7 of [12] or Section 2.3.) Theorem 6.20. The ring of functions of the group |$\textrm{Gal}^{\prime}(\mathcal{E})$| is a free |$R$|-module. Proof. For brevity, we write |$G^{\prime}=\textrm{Gal}^{\prime}(\mathcal{E})$| and |$E=\xi ^*\mathcal{E}$|⁠. We need to show that |$G^{\prime}$| is prudent to apply Theorem 6.11 and Theorem 6.1. So let |$\mathcal{V}\in \mathcal{D}(X/S)\text{-}\textbf{mod}^\circ $| belong to |$\langle \mathcal{E}\rangle _\otimes $| and write |$V=\xi ^*\mathcal{V}$|⁠; this free |$R$|-module affords a representation of |$G^{\prime}$|⁠. Let now |$\textbf{v}$| be a non-zero element of |$V$| whose image in |$V_n=V/\pi ^{n+1}$|⁠, call it |$\textbf{v}_n$|⁠, is semi-invariant. It is to prove that |$R\textbf{v}$| is also a subrepresentation of |$V$|⁠. As Lemma 6.8 guarantees, there is no loss of generality in supposing that |$\textbf{v}$| belongs to a basis of |$V$|⁠. We denote by |$\psi _n:L_n\to V_n$| the inclusion of |$R\textbf{v}_n$| (an |$R$|-module isomorphic to |$R_n$|⁠) into |$V_n$|⁠. The equivalence |$\xi ^*:\langle \mathcal{E} \rangle _\otimes \stackrel{\sim }{\to }\langle E\rangle _\otimes $| produces, from the commutative diagram with exact rows a commutative diagram with exact rows in |$\mathcal{D}(X/S)\text{-}\textbf{mod}$|⁠: By GFGA [14, Theorem 8.4.2], there exists an arrow of coherent sheaves $$\begin{align*} & \phi:\mathcal{L}\longrightarrow\mathcal{V} \end{align*}$$ inducing |$\phi _n$| for each |$n$|⁠. Using Lemma 5.4 and the fact that each |$\phi _n$| is a monomorphism, we conclude that |$\phi $| is also a monomorphism. Applying the claim made on the proof of essential surjectivity in Proposition 5.3, we conclude that |$\mathcal{L}$| carries a structure of |$\mathcal{D}(X/S)$|-module. In addition, |$\phi $| is then an arrow of |$\mathcal{D}(X/S)$|-modules, according to the claim made on the proof of fully faithfulness in Proposition 5.3. As we already remarked that |$\phi $| is a monomorphism, we conclude that |$\phi $| is really an arrow of |$\langle \mathcal{E}\rangle _\otimes $|⁠. Let |$\psi :L\to V$| be an arrow in |$\textrm{Rep}_{R}(G^{\prime})$| whose image under |$\xi ^*$| is |$\phi $| so that |$\psi $| induces |$\psi _n$| upon reduction modulo |$\pi ^{n+1}$|⁠. It only takes a moment’s thought to see that |$R\textbf{v}=\psi (L)$|⁠, so that |$R\textbf{v}$| is a subrepresentation of |$V$|⁠. The various differential Galois groups can be put together to form the relative fundamental group scheme of |$X$| based at |$\xi $|⁠, see [11, Section 5.1]. If we denote this group scheme by |$\Pi $|⁠, then |$\Pi =\varprojlim _\lambda G_\lambda $|⁠, where each |$R[G_\lambda ]$| is a free |$R$|-module (by Theorem 6.20) and the transition arrows |$G_\mu \to G_\lambda $| are faithfully flat (we use [11, Theorem 4.1.2(i)]). Then [22, Tag 0AS7] tells us that: Corollary 6.21. The ring of functions of the relative fundamental group scheme (cf. [11, Section 5.1]) of |$X/R$| based at |$\xi $| satisfies the Mittag–Leffler condition. Let us end this section by putting together Theorem 6.20 and the results on unipotent group schemes explained in Section 3. Corollary 6.22. Assume that |$\mathcal{E}_K$| is unipotent and that |$R$| is of characteristic |$(0,0)$|⁠. Then |$\textrm{Gal}^{\prime}(\mathcal{E})$| is of finite type. Proof. We consider the morphism |$\textrm{Gal}^{\prime}(\mathcal{E})\to \textrm{Gal}(\mathcal{E})$| from the full to the restricted differential Galois group (see Section 7 of [12]). Since |$\mathcal{E}_K$| is unipotent, |$\textrm{Gal}(\mathcal{E})\otimes K$| is unipotent and, a fortiori, connected [6, IV.2.2.5, page 487]. Hence, [24, Theorem 2.11, page 187] shows that |$\textrm{Gal}(\mathcal{E})$| is unipotent over |$R$|⁠. We now apply Corollary 3.6 and Theorem 6.20 to conclude that |$\textrm{Gal}^{\prime}(\mathcal{E})$| is of finite type. Remark 6.23. In Theorem 6.20, suppose that |$R$| is of characteristic |$(0,0)$|⁠. From Corollary 3.3, we know that |$\textrm{Gal}^{\prime}(\mathcal{E})$| is of the form |$\mathcal{N}^\infty _{\mathfrak{H}}(G)$| with |$G$| of finite type over |$R$|⁠. Theorem 6.20 then says that |$\mathfrak{H}$| is not algebraizable: (barring the case |$\mathfrak H = \widehat G$|⁠), that is, it is not the completion of some closed flat subgroup |$H\subset G$| (otherwise |$R[\textrm{Gal}^{\prime}(\mathcal{E})]$| would contain a copy of |$K$|⁠). Remark 6.24. Theorem 6.20 is certainly false in case |$X$| is simply affine. See Example 7.11 in [11]. Funding This work was supported by the Vietnam National Foundation for Science and Technology Development [101.04-2019.315 to P.H.H]. Parts of this work were developed while J. P. dos Santos was temporarily assigned to the Centre national de la recherche scientifique (CNRS). Acknowledgments We thank heartily the referee for a careful reading and for many comments that improved the structure of this work. References [1] Amoros , J. , M. Burger, K. Corlette, D. Kotschick, and D. Toledo Fundamental Groups of Compact Kähler Manifolds . Math. Surveys Monogr., 44 . Providence, RI : American Math. Soc. , 1996 . 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This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Patimo,, Leonardo

doi: 10.1093/imrn/rnz255pmid: N/A

Abstract We propose a combinatorial interpretation of the coefficient of |$q$| in Kazhdan–Lusztig polynomials and we prove it for finite simply-laced Weyl groups. For every pair of elements |$x,y$| in a Coxeter group |$W$|⁠, Kazhdan and Lusztig [13] introduced a polynomial |$P_{x,y}(q)\in{\mathbb{Z}}[q]$| known as the Kazhdan–Lusztig polynomial. The definition is elementary and can be given using a recursive formula. Over the course of the last decades, Kazhdan–Lusztig polynomials have played a central role in many different areas of representation theory, from semisimple Lie algebras in characteristic zero to algebraic groups in large characteristic or to quantum groups at roots of unity (see [18] for a history of the subject). A somewhat elementary but still open problem involving Kazhdan–Lusztig polynomials is the so-called combinatorial invariance conjecture, which was proposed by Dyer and Lusztig during the 1980s (cf. [4]). The conjecture states that the Kazhdan–Lusztig polynomial |$P_{x,y}(q)$| depends only on the poset structure of the Bruhat interval |$[x,y]$|⁠. Partial progress has been made towards this conjecture; most notably in [2], Brenti, Caselli, and Marietti proved the conjecture in the case |$x=e$|⁠. Even for the coefficient of |$q$|⁠, which is presumably the simplest coefficient of Kazhdan–Lusztig polynomials to study (e.g., the positivity of this coefficient was shown by Tagawa in [16], almost 20 years before the general proof of positivity in [9]), a general combinatorial interpretation is missing. In this paper we propose a new combinatorial interpretation for the coefficient of |$q$|⁠. In finite simply-laced type (i.e., if |$W$| is a Weyl group of type |$ADE$|⁠) we are able to show that our combinatorial formula holds. As a consequence we confirm the combinatorial invariance conjecture for the coefficient of |$q$| for finite simply-laced groups. We explain now how this paper is structured. After recalling some background on Coxeter groups and on Kazhdan–Lusztig polynomials in Sections 1 and 2, following [10] we introduce the moment graph |${\mathcal{G}}$| of a Coxeter group and the related sheaves in Section 3. Moment graphs provide a useful algebraic/combinatorial setup for the study of Kazhdan–Lusztig polynomials (cf. [14]). In Section 4 we explain how we can compute the coefficient of |$q$| of the polynomial |$P_{x,y}(q)$| (which we denote by |$q_{x,y}$|⁠) by looking at sections of degree |$2$| of the structure sheaf |${\mathcal{A}}$| on the moment graph. In formulas, we have $$\begin{equation*}q_{x,y}=c_{x,y}-\dim \Gamma_0({\mathcal{A}},[x,y])^2,\end{equation*}$$ where |$c_{x,y}$| is the number of coatoms of |$[x,y]$| and |$\Gamma _0({\mathcal{A}},[x,y])^2$| is the space of sections of degree |$2$| of the structure sheaf on |${\mathcal{G}}|_{[x,y]}$| considered up to global translations. This formula was actually already obtained by Dyer in [8] with different methods. Our next step is to give an upper bound on |$\dim \Gamma _0({\mathcal{A}},[x,y])^2$| as follows. For a subset |$F$| of the edges in |${\mathcal{G}}|_{[x,y]}$| we define an operation of “diamond closure” by taking |$F^\diamond$| to be the smallest subset of edges such that |$F\subseteq F^\diamond$| and such that whenever we have a “diamond” in |${\mathcal{G}}|_{[x,y]}$| of the form with |$A,B\in F^\diamond$|⁠, then |$C,D\in F^\diamond$| (the orientation of the edges does not matter here). We define |$g_{x,y}$| to be the minimal possible cardinality of a set |$F$| such that |$F^\diamond$| contains all the edges in |${\mathcal{G}}|_{[x,y]}$|⁠. Then we show the following: $$\begin{equation}\dim \Gamma_0({\mathcal{A}},[x,y])^2\leq g_{x,y}.\end{equation}$$ (1) To prove the equality in (1) we need an additional ingredient called the generalised lifting property, which is unfortunately available only for finite Weyl groups of type |$ADE$|⁠. The study of the generalised lifting property is the content of Section 5. The coefficient of |$q$| in Kazhdan–Lusztig polynomials can be also approached using the related family of |$R$|-polynomials [13]. Recall that if |$x<y$| are in |$W$| and |$s$| is a simple reflection satisfying |$xs>x$| and |$ys<y$|⁠, then we have |$y\geq xs$|⁠, |$x\leq ys$| and the |$R$|-polynomial |$R_{x,y}(q)$| can be obtained using the recursive formula |$R_{x,y}(q) = (q-1) R_{x,ys}(q) +q R_{xs,ys}(q)$|⁠. For arbitrary elements |$x<y$| in |$W$| the existence of such a simple reflection |$s$| is not guaranteed. The work of Tsukermann and Williams [17] in type |$A$|⁠, extended by Caselli and Sentinelli in arbitrary finite simply-laced type (i.e., in type |$D$| and |$E$|⁠) gives a workaround. They show that we can always find a (not necessarily simple) reflection |$t$| such that |$y\geq xt\gtrdot x$|⁠, |$y\gtrdot yt\geq x$| (here |$\gtrdot$| denotes the covering relation for the Bruhat order) and that $$\begin{equation}R_{x,y}(q) = (q-1) R_{x,yt}(q) +q R_{xt,yt}(q).\end{equation}$$ (2) Such a reflection |$t$| is called a minimal reflection for |$(x,y)$| and it is obtained by taking a minimal reflection among the reflections satisfying |$xt>x$| and |$yt<y$| (minimal with respect to the order induced by the dominance order on positive roots, cf. Definition 5.1). The importance of (2) lies in the fact that we can use it to relate the coefficient of |$q$| for the interval |$[x,y]$| to the coefficients of |$q$| relative to smaller intervals |$[x^{\prime},y^{\prime}]$|⁠, that is, such that |$\ell (y)-\ell (x)>\ell (y^{\prime})-\ell (x^{\prime})$|⁠. This finally enables us to prove by induction in Section 6 the combinatorial formula $$\begin{equation*}q_{x,y}=c_{x,y}-g_{x,y}.\end{equation*}$$ We remark that as an intermediate (and crucial) step in our proof we generalise to minimal reflections the following well-known property of simple reflections; let |$x<y$| and assume |$t$| is a minimal reflection for |$(x,y)$|⁠. Then there exists a maximal chain |$x\lessdot z_1\lessdot z_2\lessdot \ldots \lessdot z_{\ell (y)-\ell (x)-2}<yt$| such that |$z_it\gtrdot z_i$| for all |$i$|⁠. 1 Preliminaries on Coxeter Groups We refer to [12] for background material on Coxeter groups. Let |$(W,S)$| be a Coxeter system. For |$s,t\in S$|⁠, let |$m_{st}\in{\mathbb{N}}$| denote the order of |$st$|⁠. We say that |$W$| is simply-laced if |$m_{st}\leq 3$| for any |$s,t\in S$|⁠. We denote by |$\ell$| the length function and by |$\geq$| the Bruhat order. Let |$T=\bigcup _{w\in W} wSw^{-1}$| denote the set of reflections. For |$x,y\in W$| let $$\begin{align*}D(y) = \{ t \in T \mid yt <y\} \\ A(x) = \{ t \in T \mid xt>x\}\\ AD(x,y) = A(x) \cap D(y).\end{align*}$$ We have |$|D(y)|=\ell (y)$|⁠; hence, |$|AD(x,y)|\geq \ell (y)-\ell (x)$| and |$AD(x,y)$| is not empty for any |$x<y$|⁠. We denote by |$\lessdot$| the covering relation of the Bruhat order, that is, |$x\lessdot y$| if |$x<y$| and |$\ell (x)+1=\ell (y)$|⁠. We denote by |$[x,y]:=\{z\in W \mid x\leq z\leq y\}$| the Bruhat interval. The intervals |$[x,y)$| and |$(x,y]$| are similarly defined. We denote by $$\begin{align*}\textrm{at}\ (x,y):= \{z \in W | x\lessdot z \leq y\} \\ \textrm{coat }(x,y):= \{z \in W | x\leq z \lessdot y\}\end{align*}$$ the set of atoms and coatoms of |$[x,y]$|⁠. We also define $$\begin{align*}\textrm{at }{^T}(x,y):= \left\{x^{-1}z \middle| z \in \textrm{at}\ (x,y)\right\}\subseteq T\\ \textrm{coat }{^T}(x,y):= \left\{z^{-1}y \middle| z \in \textrm{coat }(x,y)\right\}\subseteq T.\end{align*}$$ We fix |${\mathfrak{h}}$| to be a finite-dimensional vector space over |${\mathbb{R}}$| with the property that there exist linearly independent subsets $$\begin{equation*}\{\alpha_s\}_{s\in S}\subseteq{\mathfrak{h}}^*\quad\textrm{and}\quad \{\alpha_s^\vee\}_{s\in S}\subseteq{\mathfrak{h}}\end{equation*}$$ such that |$\alpha _s(\alpha _t^\vee )=-2 \cos (\pi /m_{st})$|⁠. We further assume that |${\mathfrak{h}}$| is of minimal possible dimension among representations satisfying the above properties. Then the assignment |$s(v):=v-\alpha _s(v)\alpha _s^\vee$| for any |$s\in S$| defines a representation of |$W$| on |${\mathfrak{h}}$|⁠. Notice that if |$W$| is finite, then |$\dim{\mathfrak{h}} = |S|$| and |${\mathfrak{h}}$| is the geometric representation of |$W$| (cf. [12, §5.3]). As shown in [15, Proposition 2.1] the representation |${\mathfrak{h}}$| is reflection faithful, that is, the subset of fixed points of |$t\in W$| form a hyperplane in |${\mathfrak{h}}$| if and only if |$t\in T$|⁠. We call |$\{\alpha _s\}_{s\in S}$| and |$\{\alpha _s^\vee \}_{s\in S}$| the set of simple roots and simple coroots respectively. Let |$\Phi = \{ w(\alpha _s) \mid w \in W, s\in S\}\subseteq{\mathfrak{h}}^*$| denote the set of roots. Let |$\Phi ^+\subseteq \Phi$| be the subset of positive roots, that is, the subset of roots that are a positive linear combination of simple roots. For a reflection |$t\in T$| there exists |$w \in W$| and |$s \in S$| with |$ws> w$| such that |$t=wsw^{-1}$|⁠. Then we set |$\alpha _t:=w(\alpha _s) \in \Phi ^+$|⁠. Notice that |$\alpha _t$| vanishes on the hyperplane fixed by |$t$|⁠. The root |$\alpha _t$| is well defined and, if |$r\neq t$|⁠, the roots |$\alpha _r$| and |$\alpha _t$| are linearly independent. We introduce a partial order |$\succ$| on |$\Phi ^+$| by setting |$\alpha \succ \beta$| if |$\alpha -\beta$| is a positive linear combination of simple roots. This also induces a partial order |$\succ$| on |$T$| where, for |$r,t\in T$|⁠, we say |$r\succ t$| if |$\alpha _r \succ \alpha _t$|⁠. The following simple lemma describes Bruhat intervals of length |$2$| and it is applied several times in this paper. Lemma 1.1. Let |$x,y\in W$| and assume |$x< y$| and |$\ell (y)-\ell (x)=2$|⁠. Then there exists |$r,t\in A(x)$| with |$r\neq t$| such that |$[x,y]=\{x,xr,xt,y\}$|⁠. Moreover, if |$W$| is simply-laced, then |$y=xtr$| or |$y=xrt$|⁠. Proof. The 1st part is [1, Lemma 2.7.3]. Assume now that |$W$| is simply-laced. By [7, Lemma 3.1], the reflection subgroup |$W^{\prime}=\langle r,t,rx^{-1}y,tx^{-1}y\rangle$| is isomorphic to a simply-laced Coxeter subgroup of rank |$2$| (i.e., |$W^{\prime}$| is of type |$A_1\times A_1$| or of type |$A_2$|⁠) and, by [7, Proposition 2.1] the interval |$[x,y]$| is isomorphic to a Bruhat interval in |$W^{\prime}$|⁠. It is then enough to prove the claim for Coxeter groups of type |$A_1\times A_1$| or of type |$A_2$|⁠, and in these cases it is a trivial check. The following Lemma is needed in Section 5 (and it is crucial in Caselli and Sentinelli’s proof of the generalised lifting property). Lemma 1.2 ([6, Proposition 2.3]). Let |$t_1,t_2\in T$|⁠. i) If there exists |$x \in W$| such that |$t_1\in D(x)$| and |$t_2t_1t_2\in A(xt_2)$|⁠, then |$t_1\in D(t_2)$|⁠. ii) If there exists |$x \in W$| such that |$t_1\in D(x)$| and |$t_2t_1t_2\in D(xt_2)$|⁠, then |$t_1\in A(t_2)$|⁠. Assume |$W$| to be simply-laced. Let |$r,t\in T$| with |$r\neq t$|⁠. If |$r \in D(t)$|⁠, then |$rt\neq tr$| [6, Corollary 2.4]; hence, |$r,t$| generate a Coxeter group of type |$A_2$|⁠. Moreover, from [6, Corollary 3.4] we have $$\begin{equation}r\in D(t)\setminus \{t\} \implies \alpha_r+\alpha_{rtr}=\alpha_t,\; \text{and so} \;r\prec t.\end{equation}$$ (3) 2 Kazhdan–Lusztig Polynomial and R-Polynomials In [13] Kazhdan–Lusztig polynomials are originally introduced as coefficients of a certain canonical basis in the Iwahori–Hecke algebra. Since the Iwahori–Hecke algebra does not play any direct role in this work, we follow [1, §5.1] and we give instead an equivalent definition which emphasises the relations of Kazhdan–Lusztig polynomials with the |$R$|-polynomials. Definition/Theorem 2.1. Let |$W$| be a Coxeter group. There exists a unique family of polynomials with integral coefficients |$\{R_{x,y}(q)\}_{x,y\in W}$|⁠, called |$R-polynomials$|⁠, satisfying the following conditions: i) |$R_{x,x}(q)=1$| ii) |$R_{x,y}(q)=0$| if |$x\not \leq y$|⁠, iii) for any |$s\in S$| such that |$ys<y$| we have $$\begin{equation*}R_{x,y}(q)=\begin{cases} R_{xs,ys}(q)& \textrm{if}\ xs<x,\\ qR_{xs,ys}(q)+(q-1)R_{x,ys}(q) & \textrm{if}\ xs>x.\\\end{cases}\end{equation*}$$ Definition/Theorem 2.2. Let |$W$| be a Coxeter group. There exists a unique family of polynomials with integral coefficients |$\{P_{x,y}(q)\}_{x,y\in W}$|⁠, called Kazhdan–Lusztig polynomials, satisfying the following conditions: i) |$P_{x,x}(q)=1$|⁠, ii) |$P_{x,y}(q)=0$| if |$x\not \leq y$|⁠, iii) |$\deg (P_{x,y}(q))\leq \frac 12 (\ell (y)-\ell (x)-1)$| if |$x<y$|⁠, iv) for any |$x\leq y$| we have $$\begin{equation}q^{\ell(y)-\ell(x)}P_{x,y}(q^{-1}) = \sum_{z\in[x,y]} R_{x,z}(q)P_{z,y}(q).\end{equation}$$ (4) It is an immediate consequence of the definitions that for any |$x\leq y$| the polynomial |$R_{x,y}(q)$| is monic of degree |$\ell (y)-\ell (x)$| and the polynomial |$P_{x,y}(q)$| has constant term equal to |$1$|⁠. We introduce some further notation. For |$x\leq y$| let |$q_{x,y},d_{x,y}\in{\mathbb{Z}}$| be such that $$\begin{align*}P_{x,y}(q) = 1 + q_{x,y}q + \textrm{``higher terms in q}^{\prime\prime},\\ R_{x,y}(q) = q^{\ell(y)-\ell(x)}- d_{x,y}q^{\ell(y)-\ell(x)-1} + \textrm{``lower terms in q}^{\prime\prime}.\end{align*}$$ Lemma 2.3. Let |$W$| be a Coxeter group. Then i) |$d_{x,x}=0$| for any |$x\in W$|⁠, ii) for any |$x\leq y$| and for any |$s\in S$| such that |$ys<y$| we have $$\begin{equation*}d_{x,y} = \begin{cases} d_{xs,ys} & \textrm{if}\ xs<x,\\ d_{x,ys} +1 & \textrm{if}\ xs>x \ \textrm{and}\ xs\not\leq ys, \\ d_{x,ys} & \textrm{if}\ xs>x\ \textrm{ and } xs\leq ys. \\\end{cases}\end{equation*}$$ Proof. This is an immediate consequence of Definition/Theorem 2.1. The only terms contributing to the coefficient of |$q^{\ell (y)-\ell (x)-1}$| in the RHS of (4) are |$z=y$| and |$z\in{\operatorname{coat}}(x,y)$|⁠. If |$c_{x,y}:= |{\operatorname{coat}}(x,y)|$| we obtain $$\begin{equation}q_{x,y}=-d_{x,y}+\sum_{z\in{\operatorname{coat}}(x,y)} 1=c_{x,y}-d_{x,y}.\end{equation}$$ (5) The main purpose of the next sections is to provide a combinatorial interpretation of the term |$d_{x,y}$|⁠. 3 The Moment Graph of Coxeter Groups We recall the definition of moment graphs and of the related sheaves from [10]. Recall that |${\mathfrak{h}}$| is the reflection faithful representation of |$W$| introduced in Section 1. Definition 3.1. The moment graph|${\mathcal{G}}:={\mathcal{G}}(W,{\mathfrak{h}})$| of a Coxeter group |$W$| is a labelled directed graph defined as follows. The set of vertices is the set of elements of |$W$|⁠. Two vertices |$v,w$| are connected by an arrow |$v\rightarrow w$| if |$w>v$| and there exists a reflection |$t\in T$| such that |$w=vt$|⁠. We label this edge by |$\alpha _t\in{\mathfrak{h}}^*$|⁠, where |$\alpha _t$| is the positive root corresponding to |$t$|⁠. Note that this differs from [10] since we use right multiplication by reflections (instead of left) to label the edges of |${\mathcal{G}}$|⁠. The moment graph of Coxeter group is also often denoted Bruhat graph (cf. [1, Definition 2.1.1]). For a subset |$A\subseteq W$|⁠, we denote by |${\mathcal{G}}|_{A}$| the full subgraph of |${\mathcal{G}}$| whose vertices are the elements of |$A$|⁠. We denote by |$R={\operatorname{Sym}}_{\mathbb{R}}({\mathfrak{h}}^*)$| the symmetric algebra of |${\mathfrak{h}}^*$|⁠. We regard |$R$| as a graded algebra with |${\mathfrak{h}}^*$| sitting in degree |$2$|⁠. Definition 3.2. A sheaf |${\mathcal{M}}$| on the moment graph of |$W$| is given by a graded |$R$|-module |${\mathcal{M}}_x$| for any |$x\in W$|⁠, for any arrow |$x\rightarrow xt$| in |${\mathcal{G}}$| a graded |$R$|-module |${\mathcal{M}}_{x\rightarrow xt}$| such that |$\alpha _t\cdot{\mathcal{M}}_{x\rightarrow xt}=0$|⁠, for any arrow |$x\rightarrow xt$| in |${\mathcal{G}}$| two morphisms of graded |$R$|-modules |$\pi _{x,xt}:{\mathcal{M}}_x\rightarrow{\mathcal{M}}_{x\rightarrow xt}$| and |$\pi _{xt,x}:{\mathcal{M}}_{xt}\rightarrow{\mathcal{M}}_{x\rightarrow xt}$|⁠. If |$A\subseteq W$|⁠, the space of sections of a sheaf |${\mathcal{M}}$| over |$A$| is $$\begin{equation*}\Gamma({\mathcal{M}},A):=\left\{(m_x)\in \prod_{x\in A}{\mathcal{M}_x}\mid\begin{array}{c} \pi_{x,xt}(m_x)=\pi_{xt,x}(m_{xt})\\ \textrm{ for all}\ x\in A \ \textrm{and} \ t \in T\textrm{ such that}\ xt\in A\end{array}\right\}. \end{equation*}$$ The space of global sections of a sheaf |${\mathcal{M}}$| is |$\Gamma ({\mathcal{M}}):=\Gamma ({\mathcal{M}},W)$|⁠. We simply write |$\Gamma ({\mathcal{M}},>x)$| for |$\Gamma ({\mathcal{M}},\{w \in W \mid w>x\})$| and similarly for |$\geq x,\leq x,$| and |$<x$|⁠. Example 3.3. A trivial example of a sheaf on the moment graph is the structure sheaf |${\mathcal{A}}$|⁠. It is defined by setting |${\mathcal{A}}_x:= R$| for any |$x\in W$| and |${\mathcal{A}}_{x\rightarrow xt}:=R/(\alpha _t)$| for any arrow |$x\rightarrow xt$| in |${\mathcal{G}}$|⁠. The maps |$\pi _{x,tx}$| and |$\pi _{xt,x}$| are the natural projections |$R\twoheadrightarrow R/(\alpha _t)$|⁠. Notice that the orientation of the arrows in |${\mathcal{G}}$| does not matter in the definition of |${\mathcal{A}}$|⁠. Let |$y\in W$|⁠. We quickly recall the construction of the canonical sheaf |${\mathcal{B}}(y)$| from [3]. We start by setting |${\mathcal{B}}(y)_y=R$| and |${\mathcal{B}}(y)_z= 0$| for any |$z\not \leq y$|⁠. Fix |$x \leq y$| and assume that |${\mathcal{B}}(y)_z$| is already defined for all |$z>x$|⁠. Then for any |$t\in A(x)$| we define |${\mathcal{B}}(y)_{x\rightarrow xt}:= {\mathcal{B}}(y)_{xt}/(\alpha _t)$| and |$\pi _{xt,x}:{\mathcal{B}}(y)_{xt}\rightarrow{\mathcal{B}}(y)_{x\rightarrow xt}$| to be the natural projection. Let $$\begin{equation*}{\mathcal{B}}(y)_{\delta x}:= {\operatorname{Im}}\left(p:\Gamma({\mathcal{B}}(y),>x)\rightarrow \bigoplus_{t\in A(x)} {\mathcal{B}}(y)_{x\rightarrow xt}\right),\end{equation*}$$ where |$p$| is the restriction map. Define |${\mathcal{B}}(y)_x$| to be the projective cover of |${\mathcal{B}}(y)_{\delta x}$| as a graded |$R$|-module. Notice that |${\mathcal{B}}(y)_x$| is a free |$R$|-module. Finally, for any |$t\in A(x)$|⁠, let the morphism |$\pi _{x,xt}$| be defined as the following composition: $$\begin{equation*}{\mathcal{B}}(y)_x\rightarrow{\mathcal{B}}(y)_{\delta x} \hookrightarrow \bigoplus_{t\in A(x)} {\mathcal{B}}(y)_{x\rightarrow xt}\twoheadrightarrow{\mathcal{B}}(y)_{x\rightarrow xt}.\end{equation*}$$ Remark 3.4. Assume |$W$| is the Weyl group of a complex algebraic group |$G$| with maximal torus |$T$|⁠. Let |$X$| be the flag variety of |$G$| and for |$w\in W$| we denote by |$X_w$| the corresponding Schubert variety. Then we can use the sheaves |${\mathcal{A}}$| and |${\mathcal{B}}(w)$| to compute the |$T$|-equivariant cohomology and |$T$|-equivariant intersection cohomology of |$X_w$|⁠. More precisely, we have isomorphisms of |$R$|-modules (cf. [11, Theorem 1.2.2] and [3, Theorem 1.5]) $$\begin{equation*}\Gamma({\mathcal{A}},\leq w)\cong H_T(X_w,{\mathbb{R}})\quad\textrm{and}\quad\Gamma({\mathcal{B}}(w))\cong IH_T(X_w,{\mathbb{R}}).\end{equation*}$$ If |$V$| is a graded vector space, we denote by |$V^i$| its graded component of degree |$i$|⁠. We regard |${\mathbb{R}}$| as a |$R$|-module via the isomorphism |${\mathbb{R}}\cong R/R^+$|⁠, where |$R^+$| denotes the ideal of elements of positive degree. Theorem 3.5. The Poincaré polynomial of |$\overline{{\mathcal{B}}(y)_x}={\mathcal{B}}(y)_x\otimes _R {\mathbb{R}}$| is the Kazhdan–Lusztig polynomial |$P_{x,y}(q^{2})$|⁠, that is, we have $$\begin{equation*}P_{x,y}(q) = \sum_{i\geq 0}\dim \overline{{\mathcal{B}}(y)_x}^{2i} q^i.\end{equation*}$$ The proof of Theorem 3.5 is due to Braden and MacPherson [3] for finite and affine Weyl groups. Their proof employs geometric techniques and it ultimately relies on Saito’s theory of Hodge modules. For arbitrary Coxeter groups the theorem follows by combining the work of Fiebig [10] (which links sheaves on the moment graphs to Soergel bimodules) and the work of Elias and Williamson; in [9] they prove Theorem 3.5 in the setting of Soergel bimodules by developing an algebraic version of Hodge theory. Remark 3.6. A remarkable consequence of Theorem 3.5 is that the polynomial |$P_{x,y}(q)$| only depends on the subgraph |${\mathcal{G}}|_{[x,y]}$|⁠. On the other hand the poset structure of |$[x,y]$| determines |${\mathcal{G}}|_{[x,y]}$| as an unlabelled directed graph [7]. Thus, the missing step in the combinatorial invariance conjecture is to show that the labels in |${\mathcal{G}}$| are actually a superfluous piece of data for the computations of the Kazhdan–Lusztig polynomials. 4 Moment Graphs and the Coefficient of q For a graded |$R$|-module |$M$| we denote by |$M^i$| its graded component of degree |$i$| and we denote by |$M[i]$| its grading shift, that is, |$M[i]^k=M^{i+k}$|⁠. If |$x\leq y$| the constant term of |$P_{x,y}(q)$| is |$1$|⁠; hence, the |$R$|-module |${\mathcal{B}}(y)_x$| contains a unique summand isomorphic to |$R$|⁠. This can be obtained as the |$R$|-submodule generated by any nonzero element in degree |$0$|⁠. Moreover, from Theorem 3.5 it follows that |${\mathcal{B}}(y)_x$| contains the summand |$R[-2]$| with multiplicity |$q_{x,y}$|⁠. Fix |$y\in W$|⁠. For every |$x\leq y$| we choose |$1_x\in R^0\subseteq{\mathcal{B}}(y)_x^0$| compatibly so that if |$xt\leq y$| for some |$t\in T$|⁠, then |$1_x$| and |$1_{xt}$| are identified via the maps |$\pi _{x,xt}$| and |$\pi _{xt,x}$|⁠. For every |$x$| the vector space |$\Gamma ({\mathcal{B}}(y),>x)^0$| is one-dimensional and it is generated by the section |$(1_z)_{z\in (x,y]}$|⁠. We define |$\Gamma _0({\mathcal{B}}(y),>x)$| to be the subspace of sections |$(f_z)_{z\in (x,y]}\in \Gamma ({\mathcal{B}}(y),>x)$| such that |$f_y=0$|⁠. Let |$i_0:\Gamma _0({\mathcal{B}}(y),>x)\hookrightarrow \Gamma ({\mathcal{B}}(y),>x)$| be the inclusion. Consider the following diagram: where |$j$| is the morphism of |$R$|-modules defined by |$j(1)=p((1_z)_{z\in (x,y]})$|⁠. Let |$p_0:=p\circ i_0$|⁠. Recall that if |$M$| is a graded |$R$|-module, then the projective cover of |$M$| is isomorphic to |$R\otimes _{{\mathbb{R}}} \overline{M}$|⁠, with |$\overline{M}=M\otimes _R {\mathbb{R}}$|⁠. Therefore, it is easy to see that $$\begin{equation*}q_{x,y} =\dim \left(\overline{{\mathcal{B}}(y)_{\delta x}}^2\right)={\operatorname{codim}} \left({\operatorname{Im}}(j)^2 \subseteq{\mathcal{B}}(y)_{\delta x}^2\right)=\dim\left({\mathcal{B}}(y)_{\delta x}/{\operatorname{Im}}(j)\right)^2.\end{equation*}$$ We have The map |$\overline{i_0}$| is also surjective: in fact, if |$f=(f_z)_{z\in (x,y]}\in \Gamma ({\mathcal{B}}(y),>x)$|⁠, we can write $$\begin{equation*}f = (f-\overline{f}) + \overline{f,}\end{equation*}$$ where |$\overline{f}$| is the constant section |$\overline{f}:=(f_y)_{z\in (x,y]}\in p^{-1}(j(f_y))$|⁠. It follows that $$\begin{equation}q_{x,y} = \dim \Gamma_0({\mathcal{B}}(y),>x)^2 -\dim p_0^{-1}({\operatorname{Im}}(j)^2).\end{equation}$$ (6) The 1st term in (6) has an immediate combinatorial interpretation. Lemma 4.1. Let |$x,y\in W$| with |$x<y$| i) If |$\ell (y)-\ell (x)\geq 2$|⁠, then |$\Gamma _0({\mathcal{B}}(y),>x)\cong \Gamma _0({\mathcal{B}}(y),\geq x)$|⁠. ii) The dimension of |$\Gamma _0({\mathcal{B}}(y),\geq x)^2$| is equal to the number of coatoms |$c_{x,y}= |{\operatorname{coat}}(x,y)|$| of |$[x,y]$|⁠. Proof. First assume |$\ell (y)-\ell (x)\geq 2$|⁠. From Lemma 1.1 it follows that there are (at least) two reflections |$t_1,t_2\in A(x)$| such that |$xt_1,xt_2\leq y$|⁠. We claim that the morphism |$j: R \rightarrow{\mathcal{B}}(y)_{\delta x}$| is injective in degrees |$0$| and |$2$|⁠. Let |$f\in R$| such that |$j(f)=0$|⁠, so we have |$\alpha _{t_1}\mid f$| and |$\alpha _{t_2}\mid f$|⁠. Since |$\alpha _{t_1}$| and |$\alpha _{t_2}$| are linearly independent we conclude that |$f=0$| or |$\deg (f)\geq 4$|⁠. Since |${\mathcal{B}}(y)_x$| is the projective cover of |${\mathcal{B}}(y)_{\delta x}$|⁠, it follows that the map |${\mathcal{B}}(y)_{x}\rightarrow{\mathcal{B}}(y)_{\delta x}$| is bijective in degrees |$0$| and |$2$|⁠. Hence, if |$\ell (y)-\ell (x)\geq 2$|⁠, every section in |$\Gamma ({\mathcal{B}}(y),>x)^2$| extends uniquely to a section in |$\Gamma ({\mathcal{B}}(y),\geq x)^2$| and i) follows. By repeating this argument we see that every section in |$\Gamma _0({\mathcal{B}}(y),{\operatorname{coat}}(x,y) \cup \{y\})^2$| extends uniquely to a section in |$\Gamma _0({\mathcal{B}}(y),\geq x)^2$|⁠. Moreover, we have $$\begin{equation*}\Gamma_0({\mathcal{B}}(y),\geq x)^2 \cong\Gamma_0({\mathcal{B}}(y),{\operatorname{coat}}(x,y) \cup \{y\})^2\cong \bigoplus_{z\in{\operatorname{coat}}(x,y)} \Gamma_0({\mathcal{B}}(y),\geq z)^2.\end{equation*}$$ Hence, it is enough to consider the case |$\ell (y)-\ell (x)=1$|⁠. Then |${\mathcal{B}}(y)_x\cong R$| and a section in |$\Gamma _{0}({\mathcal{B}}(y),\geq x)$| can be thought of as a polynomial |$f\in{\mathcal{B}}(y)_x$| such that |$\alpha _{x^{-1}y}\mid f$|⁠. It follows that |$\dim \Gamma _0({\mathcal{B}}(y),\geq x)^2=1$|⁠. We give now a more insightful description of the vector space |$p_0^{-1}({\operatorname{Im}}(j)^2)$|⁠. Proposition 4.2. Let |$x,y\in W$| with |$x<y$| and |$\ell (y)-\ell (x)\geq 2$|⁠. We have $$\begin{equation*}\Gamma_0({\mathcal{A}},[x,y])^2\cong p_0^{-1}({\operatorname{Im}}(j))^2.\end{equation*}$$ Proof. Since |$\ell (y)-\ell (x)\geq 2$|⁠, the same argument as in the proof of Lemma 4.1 shows that the restriction map |$\Gamma _0({\mathcal{A}},[x,y])^2\rightarrow \Gamma _0({\mathcal{A}},(x,y])^2$| is injective. Moreover, the map |$\Gamma _0({\mathcal{A}},(x,y])\rightarrow p_0^{-1}({\operatorname{Im}}(j))$| induced by the inclusions |$R\hookrightarrow{\mathcal{B}}(y)_z$| for |$z\in (x,y]$| is clearly injective. It remains to show that the composition map |$\varphi :\Gamma _0({\mathcal{A}},[x,y])^2\rightarrow p_0^{-1}({\operatorname{Im}}(j))^2$| is surjective. Let |$f=(f_z)_{z\in (x,y]}\in \Gamma _0({\mathcal{B}}(y),>x)^2$| and assume that |$f\in p_0^{-1}({\operatorname{Im}}(j)^2)$|⁠. For every |$z\in (x,y]$|⁠, we have |${\mathcal{B}}(y)_z=R\oplus R[-2]^{q_{z,y}}\oplus \ldots$|⁠, so we can write |$f_z=f_{z,0}+f_{z,2}$| with |$f_{z,0}\in R$| and |$f_{z,2}\in R[-2]^{q_{z,y}}$|⁠. Let |$a$| be an atom in |$[x,y]$| with |$t=x^{-1}a\in T$|⁠. The image of the map $$\begin{equation*}R\rightarrow{\mathcal{B}}(y)_{x\rightarrow a}={\mathcal{B}}(y)_a/(\alpha_t)=R/(\alpha_t)\oplus R/(\alpha_t)[-2]^{q_{a,y}}\oplus \ldots\end{equation*}$$ is contained in the summand |$R/(\alpha _t)$| for degree reasons. Then the assumption |$f\in p_0^{-1}({\operatorname{Im}}(j)^2)$| forces to have |$f_{a,2}=0$|⁠. Fix |$z\in (x,y]$| and assume that |$f_{w,2}=0$| for all |$w<z$|⁠. If |$z$| is not an atom in |$[x,y]$|⁠, we can choose |$t\in D(z)$| such that |$zt\in (x,z]$|⁠. Since |$f_{zt}=f_{zt,0}$| the same argument as above shows |$f_{z,2}=0$|⁠. Hence, by induction we obtain that |$f_{z,2}=0$| for any |$z\in (x,y]$| or, equivalently, that |$f_z$| is contained in the summand |$R\subseteq{\mathcal{B}}(y)_z$|⁠. This implies that every section |$f\in p_0^{-1}({\operatorname{Im}}(j)^2)$| can be thought as a section of the structure sheaf |${\mathcal{A}}$|⁠; hence, it is in the image of |$\varphi$|⁠. Notice that, after a trivial check in the case |$\ell (y)-\ell (x)\leq 1$|⁠, we can rewrite (6) as $$\begin{equation*}q_{x,y} = \dim \Gamma_0({\mathcal{B}}(y),\geq x)^2 - \dim \Gamma_0({\mathcal{A}},[x,y])^2= c_{x,y} -\dim \Gamma_0({\mathcal{A}},[x,y])^2.\end{equation*}$$ As a consequence of (5) and Lemma 4.1 we obtain |$\dim \Gamma _0({\mathcal{A}},[x,y])^2=d_{x,y}$|⁠. We identify |$R^2$| with the vector space |${\mathfrak{h}}^*$|⁠. Under this identification |$\Gamma _0({\mathcal{A}},[x,y])$| corresponds to the vector space $$\begin{equation*}V_{x,y} := \left\{(v_z)_{z\in [x,y]}\in \bigoplus_{z\in [x,y]}{\mathfrak{h}}^\ast\bigg|\; \begin{array}{ccc} v_y=0 \ \ and\\ v_{zt}\in v_z+{\mathbb{R}}\alpha_t\ \textrm{for any}\ z\in [x,y]\ \textrm{and}\ t\in T\\ \textrm{such that}\ zt\in [x,y]\end{array}\right\}.\end{equation*}$$ We use this identification to give an upper bound to |$\dim \Gamma _0({\mathcal{A}},[x,y])^2=\dim V_{x,y} =d_{x,y}$|⁠. Remark 4.3. The formula |$q_{x,y}=c_{x,y}-\dim V_{x,y}$| already appeared in [8]. In his work Dyer proved this formula using different and more elementary techniques as moment graphs and Theorem 3.5 were still not available at that time. From this formula he derived a new proof of the positivity of the coefficient |$q_{x,y}$| for arbitrary Coxeter groups (the positivity was originally proved by Tagawa in [16]). We remark that one could in principle rely on Dyer’s work and avoid the recourse to Theorem 3.5 also in the remainder of this work. We choose to reprove this formula in the moment graph setting since we hope (and believe) similar methods can be used to investigate also other coefficients of Kazhdan–Lusztig polynomials. In what follows we regard the moment graph solely as an undirected labelled graph, as the orientation of the arrows will not play any role since we are working with |${\mathcal{A}}$| (cf. Example 3.3). We denote by |$E_{x,y}$| the set of edges in |${\mathcal{G}}|_{[x,y]}$|⁠. We denote an edge between |$z$| and |$w$| by |$(z - w)$|⁠. For |$e=(z - zt)\in E_{x,y}$| we define the function |$\lambda _e(v): V_{x,y}\rightarrow{\mathbb{R}}$| by |$\lambda _e(v)=(v_z-v_{zt})/(\alpha _t)$|⁠. We will sometimes denote the function |$\lambda _e$| simply by |$\lambda _{z,zt}$|⁠. Definition 4.4. We call diamond any |$4$|-cycle in |${\mathcal{G}}|_{[x,y]}$| consisting of four edges between four different vertices. Notice that in every diamond the labels of a pair of adjacent edges are linearly independent while the span of all its labels has dimension |$2$| (cf. [7, Lemma 3.1]). The following key observation is inspired by [3, Example 2.3]. Assume we have a diamond in |${\mathcal{G}}|_{[x,y]}$| as in Figure 1, that is, |$z,zt,zr,ztu\in [x,y]$| with |$t,r,u,rtu\in T$|⁠. Since |$\alpha _t$| and |$\alpha _r$| are linearly independent, for any |$(v_z)\in V_{x,y}$| the intersection of the lines |$v_{zt}+{\mathbb{R}} \alpha _t$| and |$v_{zr}+{\mathbb{R}} \alpha _r$| is the point |$v_z$|⁠. This means that |$v_{zt}$| and |$v_{zr}$| uniquely determine |$v_z$| (and |$v_{ztu}$|⁠). Equivalently, the real numbers |$\lambda _{z,zt}(v)$| and |$\lambda _{z,zr}(v)$| uniquely determine |$\lambda _{zt,ztu}(v)$| and |$\lambda _{zr,ztu}(v)$|⁠. Fig. 1. Open in new tabDownload slide A diamond. Fig. 1. Open in new tabDownload slide A diamond. Definition 4.5. Let |$F$| be a subset of |$E_{x,y}$|⁠. We say that |$F$| is diamond closed when for any |$z,zt,zr,ztu$| forming a diamond as in Figure 1, if |$(z - zt), (z - zr)\in F$|⁠, then |$(zt - ztu), (ztu - zr)\in F$|⁠. We denote by |$F^\diamond$| the smallest diamond closed subset of |$E_{x,y}$| such that |$F\subseteq F^\diamond$|⁠. We call |$F^\diamond$| the diamond closure of |$F$|⁠. We call a subset of edges |$F$|diamond generating if |$F^\diamond =E_{x,y}$|⁠. The diamond closure |$F^\diamond$| is well defined for any |$F$| and it can be simply obtained as follows. If |$F\neq F^\diamond$|⁠, there exists a diamond |${\mathcal{D}}$| in |${\mathcal{G}}|_{[x,y]}$| such that two adjacent edges of |${\mathcal{D}}$| are in |$F$| but not all the edges of |${\mathcal{D}}$| are contained in |$F$|⁠. Then we build a new set |$F^{\prime}=F\cup \{$|edges of |${\mathcal{D}}\}$|⁠. Clearly we have |$F\subsetneqq F^{\prime}\subseteq F^\diamond$|⁠, so we can replace |$F$| with |$F^{\prime}$| and repeat this operation until we obtain |$F=F^\diamond$|⁠. Definition 4.6. We define $$\begin{equation*}g_{x,y}:=\min\{ |F| \mid F \textrm{ diamond generating subset of}\ E_{x,y}\}.\end{equation*}$$ In Section 6 we show that for simply-laced Weyl groups |$g_{x,y}=d_{x,y}$|⁠, and doing so we assign a combinatorial meaning to the coefficient |$d_{x,y}$|⁠. Here we show first in full generality one inequality. Proposition 4.7. Let |$x,y\in W$| with |$x<y$|⁠. We have |$d_{x,y}\leq g_{x,y}$|⁠. Proof. We need to show that for every diamond generating set |$F$| we have |$d_{x,y}\leq |F|$|⁠. Assume |$F$| is a diamond generating subset of |$E_{x,y}$| and let |$v\in V_{x,y}$|⁠. We immediately see from the construction of |$F^\diamond$| given above that the numbers |$\lambda _f(v)$| for every |$f\in F^\diamond =E_{x,y}$| are uniquely determined by |$(\lambda _e(f))_{e\in F}$|⁠. Moreover, recall that we have |$v_y=0$|⁠. Thus, |$v_z$| for every |$z\in [x,y]$| is uniquely determined by |$(\lambda _e(v))_{e\in F}$|⁠. Let |${\mathbb{R}}^F$| be a vector space of dimension |$|F|$|⁠. Then from the discussion above it follows that the linear map $$\begin{equation*}V_{x,y}\rightarrow{\mathbb{R}}^F\end{equation*}$$ $$\begin{equation*}v\mapsto (\lambda_e(v))_{e\in F}\end{equation*}$$ is injective. Hence, |$d_{x,y}=\dim V_{x,y}\leq |F|$|⁠. Furthermore, we can restrict ourselves to look at the Hasse diagram of |$[x,y]$|⁠, that is, it is enough to consider only edges |$(z-zt)$| in |$E_{x,y}$| with |$| \ell (zt)-\ell (z)|=1$|⁠. In fact, as it is shown in the the proof of [7, Proposition 3.3], if |$z\in W$| and |$t\in A(z)$| are such that |$\ell (zt)-\ell (z)>1$|⁠, then there exists a subgraph of |${\mathcal{G}}_{[z,zt]}$| of the following form. Fig. 2. Open in new tabDownload slide A long arrow |$(z-zt)$| in |${\mathcal{G}}|_{[x,y]}$|⁠. Fig. 2. Open in new tabDownload slide A long arrow |$(z-zt)$| in |${\mathcal{G}}|_{[x,y]}$|⁠. Lemma 4.8. If |$F$| is a subset of edges such that |$F^\diamond$| contains all the edges |$(z-zt)$| in |$E_{x,y}$| with |$|\ell (zt)-\ell (z)|=1$| then |$F$| is diamond generating. Proof. If |$F$| is not diamond generating, we can take an edge |$(z-zt)$| such that |$(z-zt)\not \in F^\diamond$| with |$|\ell (z)-\ell (zt)|$| minimal amongst all the edges not in |$F^\diamond$|⁠. Then using (e.g.) the diamond with edges |$z,zt, w_1$| and |$w_3$| in Figure 2 we see that |$F^\diamond$| must also contain the edge |$(z-zt)$|⁠. Remark 4.9 (Upper bounds on |$g_{x,y}$|⁠). The set of edges $$\begin{equation*}F=\{(w-y)\mid w\in{\operatorname{coat}}(x,y)\}\end{equation*}$$ is a diamond generating set. In fact, assume |$z\in [x,y]$| with |$\ell (y)-\ell (z)\geq 2$| and consider an edge |$(z-zt)$| with |$zt\gtrdot z$|⁠. Then if we take any |$w\in [x,y]$| such that |$w\gtrdot zt$|⁠, we see by Lemma 1.1 that the the elements in the interval |$[z,w]$| form a diamond. Hence, by induction on |$\ell (y)-\ell (z)$|⁠, we get |$(z-zt)\in F^\diamond$| and, by Lemma 4.8, it follows that |$F$| is diamond generating. The same also holds for the set $$\begin{equation*}F=\{(x-z)\mid z\in{\operatorname{at}}(x,y)\}.\end{equation*}$$ We obtain |$g_{x,y}\leq \min \{ |{\operatorname{at}}(x,y)|,|{\operatorname{coat}}(x,y)|\}$|⁠. If |$x=z_0\lessdot z_1\lessdot \ldots \lessdot z_{\ell (y)-\ell (x)}=y$| is a maximal chain from |$x$| to |$y$| then the set $$\begin{equation*}F = \{ (z_i -z_{i+1}) \mid 0\leq i \leq \ell(y)-\ell(x)-1\}\end{equation*}$$ is diamond generating. This is an immediate consequence of the shellability of Bruhat intervals [5]. It follows |$g_{x,y}\leq \ell (y)-\ell (x)$|⁠. Example 4.10. Let |$W$| be a Weyl group of type |$A_3$| with simple reflections |$S=\{s,t,u\}$|⁠. We draw the graph |${\mathcal{G}}|_{[t,tsut]}$|⁠. If we take as |$F_1$| the set |$\{(t-st), (t-tu)\}$| then |$F_1=F_1^\diamond$| as there is no diamond containing the two edges in |$F_1$|⁠. If we take as |$F_2$| the set |$\{(t-st), (t-tu), (t-ts)\}$| then |$F_2$| is diamond generating. For example, we can look at the diamonds with vertices |$t,st,ts,sts$| and |$t,ts,tu,tsu$| to see that |$(ts-sts)$| and |$(ts-tsu)$| are in |$F_2^\diamond$|⁠. Hence, also |$(sts-tsut)\in F_2^\diamond$|⁠, so |$F_2^\diamond$| contains a maximal chain in |$[x,y]$| and we can conclude by Remark 4.9. Since |$d_{t,tsut}=3$|⁠, this also shows that |$g_{x,y}=|F_2|=3$|⁠. 5 The Generalised Lifting Property In this section we assume that |$W$| is finite and simply-laced, that is, that |$W$| is a Weyl group of type |$A$|⁠, |$D,$| or |$E$|⁠. Recall the partial order |$\succ$| on |$T$| from Section 1. Definition 5.1. Assume |$\ell (y)>\ell (x)$|⁠. Then a minimal element |$t\in AD(x,y)$| with respect to |$\prec$| is called a minimal reflection for |$(x,y)$|⁠. Theorem 5.2 ([17, Prop. 5.3] and [6, Prop. 5.4]). Let |$x<y$| in |$W$| and let |$t$| be a minimal reflection for |$(x,y)$|⁠. Then we have |$x\leq yt\lessdot y$| |$x\lessdot xt\leq y$|⁠. |$R_{x,y}(q) = (q-1) R_{x,yt}(q) +q R_{xt,yt}(q)$|⁠. As a corollary, Lemma 2.3 immediately generalises to minimal reflections. If |$x<y$| and |$t$| is a minimal reflection for |$(x,y)$|⁠, we have $$\begin{equation}d_{x,y} = \begin{cases} d_{x,yt} +1 & \textrm{if}\ xt\not\leq yt \\ d_{x,yt} & \textrm{if}\ xt\leq yt. \\\end{cases}\end{equation}$$ (7) Example 5.3. Assume |$W$| is the symmetric group |$S_n$|⁠. We write |$x=(x(1)x(2)\ldots x(n))$| and |$y=(y(1)y(2)\ldots y(n))$| for the corresponding permutations. Then the minimal reflections for |$(x,y)$| are the transpositions |$t=(i,j)$| with |$i<j$| such that |$[i,j]$| is a minimal interval (with respect to the inclusion order) satisfying |$y(i)>y(j)$| and |$x(i)<x(j)$|⁠. Let |$x<y$| and let |$t$| be a minimal reflection for |$(x,y)$|⁠. As a corollary of Theorem 5.2, the reflection |$t$| is also a minimal element of the set |${\operatorname{at}}^T(x,y)\cap{\operatorname{coat}}^T(x,y)$|⁠. Lemma 5.4. Assume |$x<y$| with |$\ell (y)-\ell (x)\geq 2$| and let |$t$| be a minimal reflection for |$(x,y)$|⁠. Then there exists |$r\in{\operatorname{at}}^T(x,y)\cup{\operatorname{coat}}^T(x,y)$| such that |$r\not \preceq t$|⁠. Proof. Take |$u$| a minimal reflection for |$(x,yt)$|⁠. Clearly |$u\neq t$|⁠. If |$u\not \prec t$|⁠, then the claim follows because |$u\in{\operatorname{at}}^T(x,y)$|⁠. So we can assume |$u\prec t$|⁠. We have |$y\gtrdot yt \gtrdot ytu$| and, by applying Lemma 1.1 to |$[ytu,y]$|⁠, we have |$yu\in [ytu,y]$| or |$ytut\in [ytu,y]$|⁠; hence, |$u\in D(y)$| or |$tut\in D(y)$|⁠. But if |$u\in D(y)$|⁠, then also |$u\in AD(x,y)$|⁠, contradicting the minimality of |$t$|⁠. It remains to consider the case |$tut\in D(y)$| with |$t$| and |$u$| not commuting, hence |$tut=utu$|⁠. Since |$t\in D(ytut)$| and |$u=(tut)t(tut)\in A(y)$|⁠, we can apply Lemma 1.2.i) for |$x=ytut$|⁠, |$t_1=t$| and |$t_2=tut$| to deduce |$t\in D(tut)$|⁠. Hence, by (3), |$t\prec tut$| and we conclude since |$tut\in{\operatorname{coat}}^T(x,y)$|⁠. Remark 5.5. We can slightly strengthen Lemma 5.4 for groups of type |$A$|⁠, that is, for the symmetric group |$S_n$|⁠. As in the proof of Lemma 5.4 let |$x<y$|⁠, let |$t$| be a minimal reflection for |$(x,y)$| and |$u$| be a minimal reflection for |$(x,yt)$|⁠. If |$u\prec t$|⁠, then the same proof as above shows that there exists |$r\in{\operatorname{coat}}^T(x,y)$| with |$r\not \preceq t$|⁠. If |$u\not \preceq t$|⁠, then, by Lemma 1.1, either |$u\in{\operatorname{coat}}^T(x,y)$|⁠, or |$u$| and |$t$| do not commute and |$tut\in{\operatorname{coat}}^T(x,y)$|⁠. We claim that in the latter case we have |$tut\not \prec t$|⁠. In fact, we have |$t(\alpha _u)=\alpha _u\pm \alpha _t$| and, since |$\alpha _u\not \preceq \alpha _t$|⁠, |$t(\alpha _u)$| must be a positive root, namely |$t(\alpha _u)=\alpha _{tut}$|⁠. If |$\alpha _u+ \alpha _t=\alpha _{tut}$| then |$tut\succ t$|⁠. Assume now |$\alpha _u- \alpha _t=\alpha _{tut}$|⁠. Notice that if |$u$| is the transposition |$(i,j)\in S_n$| and |$t=(i^{\prime},j^{\prime})\in S_n$| then |$\alpha _u-\alpha _t$| is a positive root if and only if |$i=i^{\prime}$| and |$j^{\prime}<j$| or |$i^{\prime}>i$| and |$j=j^{\prime}$|⁠, so it is easy to see that |$\alpha _{tut}=\alpha _u-\alpha _t\not \preceq \alpha _t$|⁠. Hence, in type |$A$| we can always find |$r\in{\operatorname{coat}}^T(x,y)$| such that |$r\not \preceq t$|⁠. By a symmetric argument, we can also always find |$r^{\prime}\in{\operatorname{at}}^T(x,y)$| such that |$r^{\prime}\not \preceq t$|⁠. Lemma 5.6. Assume |$x<y$| with |$\ell (y)-\ell (x)\geq 2$| and let |$t$| be a minimal reflection for |$(x,y)$|⁠. Then there exists |$z\in{\operatorname{at}}(x,y)$| such that |$t$| is a minimal reflection for |$(z,y)$| or |$w\in{\operatorname{coat}}(x,y)$| such that |$t$| is a minimal reflection for |$(x,w)$|⁠. Proof. From Lemma 5.4, the set $$\begin{equation*}U:= \{ r \in{\operatorname{at}}^T(x,y)\cup{\operatorname{coat}}^T(x,y)\mid r\not\preceq t\}\end{equation*}$$ is not empty. Let |$r$| be a maximal element in |$U$| with respect to |$\succ$|⁠. We can assume |$r\in{\operatorname{coat}}^T(x,y)$| as the case |$r\in{\operatorname{at}}^T(x,y)$| is completely symmetric. Since |$r\not \prec t$|⁠, by (3), we have |$r\in A(t)$|⁠. Then, by Lemma 1.2.i) for |$x=y$|⁠, |$t_1=r$| and |$t_2=t$|⁠, since |$r\in D(y)$| we have |$trt\in D(yt)$|⁠. In other words, we have |$\ell (yrt)<\ell (yt)=\ell (yr)$|⁠, thus also |$t\in D(yr)$|⁠. Since |$t\in A(x)$|⁠, we obtain |$t\in AD(x,yr)$|⁠. Assume that |$t$| is not minimal in |$AD(x,yr)$|⁠, thus there exists a minimal reflection |$u$| for |$(x,yr)$| with |$u\prec t$|⁠. Applying Lemma 1.1 to the interval |$[yru,y]$| we get |$yu\in [yru,y]$| or |$yrur\in [yru,y]$|⁠. But we cannot have |$u\in D(y)$| since |$t$| is minimal in |$AD(x,y)$|⁠, thus we deduce that |$yrur\in [yru,y]$| and that |$r$| and |$u$| do not commute. Now we have |$r\in D(yrur)$| and |$u=(rur)r(rur)\in A(y)$|⁠. Lemma 1.2.i) for |$x=yrur$|⁠, |$t_1=r$| and |$t_2=rur$| implies that |$r\in D(rur)$| and in particular, by (3), we have |$r\prec rur$|⁠, contradicting the maximality of |$r$| in |$U$|⁠. Proposition 5.7. Assume |$x<y$| and let |$t$| be a minimal reflection for |$(x,y)$|⁠. Then there exists a maximal chain |$x=z_0\lessdot z_1\lessdot \ldots \lessdot z_{\ell (y)-\ell (x)-1} = yt$| such that |$z_it\in [x,y]$| and |$z_it \gtrdot z_i$| for all |$i$|⁠. Proof. We prove the claim by induction on |$\ell (y)-\ell (x)$|⁠. The case |$\ell (y)-\ell (x)=1$| is trivial, so we can assume |$\ell (y)-\ell (x)\geq 2$|⁠. From Lemma 5.6 we can assume that there exists |$r\in{\operatorname{coat}}^T(x,y)$| such that |$t$| is a minimal reflection for |$(x,yr)$| (the case |$r\in{\operatorname{at}}^T(x,y)$| is completely symmetric). By induction there exists a chain |$x=z_0\lessdot z_1\lessdot \ldots \lessdot z_{\ell (y)-\ell (x)-2} = yrt$| with |$z_it\in [x,yr]$| and |$z_it\gtrdot z_i$| for all |$i\leq \ell (y)-\ell (x)-2$|⁠. Since |$yrt=yt(trt)$| we have |$yrt\lessdot yt$|⁠, so we conclude by setting |$z_{\ell (y)-\ell (x)-1} = yt$|⁠. 6 The Coefficient of q in Finite Simply-Laced Type We are ready to finally put together the results of Sections 4 and 5 to obtain the main theorem of this paper. Theorem 6.1. Let |$W$| be a Weyl group of type ADE. Then for any |$x,y\in W$| with |$x<y$| we have |$d_{x,y}=g_{x,y}$|⁠. Proof. If |$\ell (y)-\ell (x)=1$| the claim is clear since |$d_{x,y}=g_{x,y}=1$|⁠. Let |$x,y\in W$| with |$\ell (y)-\ell (x)>1$|⁠. By induction, assume the claim for any |$x^{\prime},y^{\prime}\in W$| with |$\ell (y^{\prime})-\ell (x^{\prime})<\ell (y)-\ell (x)$|⁠. Let |$t$| be a minimal reflection for |$(x,y)$|⁠; hence, |$x\lessdot xt< y$| and |$x<yt\lessdot y$|⁠. We divide the proof into two cases as in Equation (7). Case 1: Assume |$xt\leq yt$|⁠. We have |$g_{x,y}\geq d_{x,y}=d_{x,yt}=g_{x,yt}$|⁠. Then it is enough to show |$g_{x,yt}\geq g_{x,y}$|⁠, or that any diamond-generating subset |$F\subseteq E_{x,yt}$| is also diamond generating as a subset of |$E_{x,y}$|⁠. Since |$xt\leq yt$| the edge |$(x - xt)$| belongs to |$F^\diamond$|⁠. Let |$x=z_0\lessdot z_1\lessdot \ldots \lessdot z_{\ell (y)-\ell (x)-1}=yt$| be a maximal chain with |$z_it\gtrdot z_i$| and |$z_it\in [x,y]$| as in Lemma 5.7. Then since |$F^\diamond$| contains |$(x - xt)$| and |$(z_i - z_{i+1})$| for any |$0\leq i\leq \ell (y)-\ell (x)-2$|⁠, it must also contain the edges |$(z_i-z_it)$| and |$(z_it - z_{i+1}t)$| for any |$i$| (see Figure 3). In particular, |$(yt - y)\in F^\diamond$|⁠, so |$F^\diamond$| contains a maximal chain from |$x$| to |$y$| and, as in Remark 4.9, this implies that |$F$| is a diamond generating subset of |$E_{x,y}$|⁠. Case 2: Assume |$xt\not \leq yt$|⁠. We have |$g_{x,y}\geq d_{x,y}=d_{x,yt}+1=g_{x,yt}+1$|⁠. It is enough to show that |$g_{x,yt}+1\geq g_{x,y}$|⁠. Let |$F$| be any diamond generating subset of |$E_{x,yt}$|⁠. Let |$F^{\prime}=F \cup \{x-xt\}$|⁠. Then the same argument of the previous case shows that |$F^{\prime}$| is a diamond generating subset of |$E_{x,y}$|⁠. The claim follows. Fig. 3. Open in new tabDownload slide A “ladder” between |$x$| and |$y.$| Fig. 3. Open in new tabDownload slide A “ladder” between |$x$| and |$y.$| As a byproduct of the proof we also obtain that there always exists a diamond generating subset |$F\subseteq E_{x,y}$| such that for any |$(z - w)\in F$| we have |$|\ell (z)-\ell (w)|=1$|⁠. Hence to compute |$g_{x,y}$| in type |$ADE$| it is enough to look at the Hasse diagram of |$[x,y]$|⁠. Corollary 6.2. The coefficient of |$q$| of Kazhdan-Lusztig polynomials in type ADE is a combinatorial invariant, that is, |$q_{x,y}$| depends only on the poset structure of the Bruhat interval |$[x,y]$|⁠. Moreover, the coefficient |$q_{x,y}$| can be explicitly computed using the formula $$\begin{equation*}q_{x,y} = c_{x,y} - g_{x,y}.\end{equation*}$$ Acknowledgments This paper was born as a consequence of many insightful discussions with Geordie Williamson, whom I would like to warmly thank. I would also like to thank Ben McDonnell and Lars Thorge Jensen for their comments on a preliminary version of this paper. Communicated by Prof. Simon Donaldson References [1] Björner , A. and F. Brenti Combinatorics of Coxeter groups Graduate Texts in Mathematics 231. New York : Springer , 2005 . Google Preview WorldCat COPAC [2] Brenti , F. , F. Caselli , and M. 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Google Scholar Crossref Search ADS WorldCat © The Author(s) 2018. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)

Moon, Han-Bom; Yoo, Sang-Bum

doi: 10.1093/imrn/rnaa264pmid: N/A

In the original version of this manuscript Sang-Bum Yoo’s affiliation was Division of General Studies, Ulsan National Institute of Science and Technology, Ulsan, Republic of Korea. This has now been corrected to Department of Mathematics Education, Gongju National University of Education, Gongju-si, Chungcheongnam-do, 32553, Republic of Korea. Communicated by Prof. Dragos Oprea © The Author(s) 2020. Published by Oxford University Press. All rights reserved. For permissions, please e-mail: [email protected]. This article is published and distributed under the terms of the Oxford University Press, Standard Journals Publication Model (https://academic.oup.com/journals/pages/open_access/funder_policies/chorus/standard_publication_model)