Derived categories of resolutions of cyclic quotient singularities

Derived categories of resolutions of cyclic quotient singularities Abstract For a cyclic group G acting on a smooth variety X with only one character occurring in the G-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold [X/G] and the blow-up resolution Y˜→X/G. Some results generalize known facts about X=An with diagonal G-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals |G|, we study the induced tensor products under the equivalence Db(Y˜)≅Db([X/G]) and give a ‘flop–flop = twist’ type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities. 1. Introduction For geometric, homological and other reasons, it has become commonplace to study the bounded derived category of a variety. One of the many intriguing aspects are connections, some of them conjectured, some of them proven, to birational geometry. One expected phenomenon concerns a birational correspondence   of smooth varieties. Then we should have A fully faithful embedding Db(X)↪Db(X′), if q*KX≤p*KX′. A fully faithful embedding Db(X′)↪Db(X), if q*KX≥p*KX′. An equivalence Db(X′)≅Db(X), in the flop case q*KX=p*KX′.This is proven in many instances, see [12, 14, 28, 38]. Another very interesting aspect of derived categories is their occurrence in the context of the McKay correspondence. Here, one of the key expectations is that the derived category of a crepant resolution Y˜→X/G of a Gorenstein quotient variety is derived equivalent to the corresponding quotient orbifold: Db(Y˜)≅Db([X/G])=DGb(X). In [9], this expectation is proven in many cases under the additional assumption that Y˜≅HilbG(X) is the fine moduli space of G-clusters on X. It is enlightening to view the derived McKay correspondence as an orbifold version of the conjecture on derived categories under birational correspondences described above; for more information on this point of view, see [29, Section 2], where the conjecture is called the DK-Hypothesis. Indeed, if we denote the universal family of G-clusters by Z⊂Y˜×X, we have the following diagram of birational morphisms of orbifolds:   (1.1)Since the pullback of the canonical sheaf of X/G under π is the canonical sheaf of [X/G], the condition that ϱ is a crepant resolution amounts to saying that (1.1) is a flop of orbifolds. In many situations, a crepant resolution of X/G does not exist. However, given a resolution ϱ:Y˜→X, the DK-Hypothesis still predicts the behaviour of the categories Db(Y˜) and DGb(X) if ϱ*KX/G≥KY˜ or ϱ*KX/G≤KY˜. Another related idea is that even though a crepant resolution does not exist in general, there should always be a categorical crepant resolution of Db(X/G), see [35]. The hope is to find such a categorical resolution as an admissible subcategory of the derived category Db(Y˜) of a geometric resolution. Besides dimensions 2 and 3, one of the most studied testing grounds for the above, and related, ideas is the isolated quotient singularity An/μm. Here, the cyclic group μm of order m acts on the affine space by multiplication with a primitive mth root of unity ζ. In this paper, we consider the following straight-forward generalization of this set-up. Namely, let X be a quasi-projective smooth complex variety acted upon by the finite cyclic group μm. We assume that only 1 and μm occur as the isotropy groups of the action and write S≔𝖥𝗂𝗑(μm)⊂X for the fixed point locus. Fix a generator g of μm and assume that g acts on the normal bundle N≔NS/X by multiplication with some fixed primitive mth root of unity ζ. Then the blow-up Y˜→X/μm with centre S is a resolution of singularities; see Section 3 for further details. There are four particular cases we have in mind: X=An with the diagonal action of any μm. X=Z2, where Z is a smooth projective variety of arbitrary dimension, and μ2=S2 acts by permuting the factors. Then Y˜≅Z[2], the Hilbert scheme of two points. X is an abelian variety, μ2 acts by ±1. In this case, Y˜ is known as the Kummer resolution. X→Y=X/μm is a cyclic covering of a smooth variety Y, branched over a divisor. Here, n=1 and X˜=X, Y˜=Y. This case has been studied in [32]. First, we prove the following result in Section 3.1. This is probably well known to experts, but we could not find it in the literature. Write G≔μm. Proposition A (=Proposition 3.2) The resolution obtained by blowing up the fixed point locus in X/Gis isomorphic to the G-Hilbert scheme: Y˜≅HilbG(X). We set n≔codim(S↪X) and find the following dichotomy, in accordance with the DK-Hypothesis. We keep the notation from diagram (1.1). In particular, for n=m, we obtain new instances of BKR-style derived equivalences between orbifold and resolution. Theorem B (=Theorem4.1) The functor Φ≔p*q*:Db(Y˜)→DGb(X)is fully faithful for m≥nand an equivalence for m=n. For m>n, there is a semi-orthogonal decomposition of DGb(X)consisting of Φ(Db(Y˜))and m−npieces equivalent to Db(S). The functor Ψ≔q*p*:DGb(X)→Db(Y˜)is fully faithful for n≥mand an equivalence for n=m. For n>m, there is a semi-orthogonal decomposition of Db(Y˜)consisting of Ψ(DGb(X))and n−mpieces equivalent to Db(S). For a more exact statement with an explicit description of the embeddings of the Db(S) components into Db(Y˜) and DGb(X), see Section 4. In particular, for m>n, the push-forward a*:Db(S)→DGb(X) along the embedding a:S↪X of the fixed point divisor is fully faithful. In the basic affine case (a), the result of the theorem is also stated in [29, Example 4] and there are related results in the more general toroidal case in [29, Section 3]. Proofs, in the basic case, are given in [1, Section 4] for n≥m and in [27] for n=2. If n=1, the quotient is already smooth and we have Y˜=X/G — here the semi-orthogonal decomposition categorifies the natural decomposition of the orbifold cohomology; compare [43]. The n=1 case is also proven in [36, Theorem 3.3.2]. We study the case m=n, where Φ and Ψ are equivalences, in more detail. On both sides of the equivalence, we have distinguished line bundles. The line bundle OY˜(Z) on Y˜, corresponding to the exceptional divisor, admits an mth root L. On [X/G], there are twists of the trivial line bundle by the group characters OX⊗χi. For i=−m+1,…,−1,0, we have Ψ(OX⊗χi)≅Li. Furthermore, we see that the functors Db(S)→Db(Y˜) and Db(S)→DGb(X), which define fully faithful embeddings in the n>m and m<n cases, respectively, become spherical for m=n, and hence induce twist autoequivalences; see Section 2.8 for details on spherical functors and twists. We show that the tensor products by the distinguished line bundles correspond to the spherical twists under the equivalences Ψ and Φ. In particular, one part of Theorem 4.26 is the following formula. Theorem C There is an isomorphism Ψ−1(Ψ(̲)⊗L−1)≅𝖳a*−1(̲⊗χ−1)of autoequivalences of DGb(X)where the inverse spherical twist 𝖳a*−1is defined by the exact triangle of functors  𝖳a*−1→id→a*(a*(̲)G)→. The tensor powers of the line bundle L form a strong generator of Db(Y˜), thus Theorem C, at least theoretically, completely describes the tensor product   ̲⊗^̲≔Ψ−1(Ψ(̲)⊗Ψ(̲)):DGb(X)×DGb(X)→DGb(X)induced by Ψ on DGb(X). There is related unpublished work on induced tensor products under the McKay correspondence in dimensions 2 and 3 by T. Abdelgadir, A. Craw, J. Karmazyn, and A. King. In Corollary 4.27, we also get a formula which can be seen as a stacky instance of the ‘flop–flop = twist’ principle as discussed in [4]. In Section 5, we introduce a general candidate for a weakly crepant categorical resolution (see [35] or Section 5.1 for this notion), namely the weakly crepant neighbourhood WCN(ϱ)⊂Db(Y˜), inside the derived category of a given resolution ϱ:Y˜→Y of a rational Gorenstein variety Y. The idea is pretty simple: by Grothendieck duality, there is a canonical section s:OY˜→Oϱ of the relative dualizing sheaf, and this induces a morphism of Fourier–Mukai transforms t≔ϱ*(̲⊗s):ϱ*→ϱ!. Set ϱ+≔cone(t) and WCN(ϱ)≔ker(ϱ+). Then, by the very construction, we have ϱ*∣WCN(ϱ)≅ϱ!∣WCN(ϱ) which amounts to the notion of categorical weak crepancy. There is one remaining condition needed to ensure that WCN(ϱ) is a categorical weakly crepant resolution: whether it is actually a smooth category; this holds as soon as it is an admissible subcategory of Db(Y˜) which means that its inclusion has adjoints. We prove that, in the Gorenstein case m∣n of our set-up of cyclic quotients, WCN(ϱ)⊂Db(Y˜) is an admissible subcategory; see Theorem 5.4. In Section 5.4, we observe that there are various weakly crepant resolutions inside Db(Y˜). However, a strongly crepant categorical resolution inside Db(Y˜) is unique, as we show in Proposition 5.8. Our concept of weakly crepant neighbourhoods was motivated by the idea that some non-CY objects possess ‘CY neighbourhoods’ (a construction akin to the spherical subcategories of spherelike objects in [25]), that is full subcategories in which they become Calabi–Yau. This relationship is explained in Section 5.5. In the final Section 6, we construct Bridgeland stability conditions on Kummer three-folds as an application of our results, see Corollary 6.2. Conventions. We work over the complex numbers. All functors are assumed to be derived. We write Hi(E) for the ith cohomology object of a complex E∈Db(Z) and H*(E) for the complex ⊕iHi(Z,E)[−i]. If a functor Φ has a left/right adjoint, they are denoted ΦL, ΦR. There are a number of spaces, maps and functors repeatedly used in this text. For the convenience of the reader, we collect our notation as the following:   G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα    G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα    G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα    G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα  2. Preliminaries 2.1. Fourier–Mukai transforms and kernels Recall that given an object E in Db(Z×Z′), where Z and Z′ are smooth and projective, we get an exact functor Db(Z)→Db(Z′), F↦pZ′*(E⊗pZ*F). Such a functor, denoted by 𝖥𝖬E, is called a Fourier–Mukai transform (or FM transform) and E is its kernel. See [26] for a thorough introduction to FM transforms. For example, if Δ:Z→Z×Z is the diagonal map and L is in Pic(Z), then 𝖥𝖬Δ*L(F)=F⊗L. In particular, 𝖥𝖬OΔ is the identity functor. Convention. We will write 𝖬L for the functor 𝖥𝖬Δ*L. The calculus of FM transforms is, of course, not restricted to smooth and projective varieties. Note that f* maps Db(Z) to Db(Z′) as soon as f:Z→Z′ is proper. In order to be able to control the tensor product and pullbacks, one can restrict to perfect complexes. Recall that a complex of sheaves on a quasi-projective variety Z is called perfect if it is locally quasi-isomorphic to a bounded complex of locally free sheaves. The triangulated category of perfect complexes on Z is denoted by Dperf(Z). It is a full subcategory of Db(Z). These two categories coincide if and only if Z is smooth. We will sometimes take cones of morphisms between FM transforms. Of course, one needs to make sure that these cones actually exist. Luckily, if one works with FM transforms, this is not a problem, because the maps between the functors come from the underlying kernels and everything works out, even for (reasonable) schemes which are not necessarily smooth and projective, see [5]. 2.2. Group actions and derived categories Let G be a finite group acting on a smooth variety X. Recall that a G-equivariant coherent sheaf is a pair (F,λg), where F∈Coh(X) and λg:F→∼g*F are isomorphisms satisfying a cocycle condition. The category of G-equivariant coherent sheaves on X is denoted by CohG(X). It is an abelian category. The equivariant-derived category, denoted by DGb(X), is defined as Db(CohG(X)), see, for example [42] for details. Recall that for every subgroup G′⊂G, the restriction functor 𝖱𝖾𝗌:DGb(X)→DG′b(X) has the induction functor 𝖨𝗇𝖽:DG′b(X)→DGb(X) as a left and right adjoint (see for example [42, Section 1.4]). It is given for F∈Db(Z) by   𝖨𝗇𝖽(F)=⨁[g]∈G′⧹Gg*F (2.1)with the G-linearization given by the G′-linearization of F together with appropriate permutations of the summands. If G acts trivially on X, there is also the functor 𝗍𝗋𝗂𝗏:Db(X)→DGb(X) which equips an object with the trivial G-linearization. Its left and right adjoint is the functor (̲)G:DGb(X)→Db(X) of invariants. Given an equivariant morphism f:X→X′ between varieties endowed with G-actions, there are equivariant pushforward and pullback functors, see, for example [42, Section 1.3] for details. We will sometimes write f*G for (̲)G◦f*. It is also well known that the category DGb(X) has a tensor product and the usual formulas, for example the adjunction formula, hold in the equivariant setting. Finally, we need to recall that a character κ of G acts on the equivariant category by twisting the linearization isomorphisms with κ. If F∈DGb(X), we will write F⊗κ for this operation. We will tacitly use that twisting by characters commutes with the equivariant pushforward and pullback functors along G-equivariant maps. 2.3. Semi-orthogonal decompositions References for the following facts are, for example, [12, 13]. Let T be a Hom-finite triangulated category. A semi-orthogonal decomposition of T is a sequence of full triangulated subcategories A1,…,Am such that (a) if Ai∈Ai and Aj∈Aj, then Hom(Ai,Aj[l])=0 for i>j and all l, and (b) the Ai generate T, that is, the smallest triangulated subcategory of T containing all the Ai is already T. We write T=⟨A1,…,Am⟩. If m=2, these conditions boil down to the existence of a functorial exact triangle A2→T→A1 for any object T∈T. A subcategory A of T is right admissible if the embedding functor ι has a right adjoint ιR, left admissible if ι has a left adjoint ιL, and admissible if it is left and right admissible. Given any triangulated subcategory A of T, the full subcategory A⊥⊆T consists of objects T such that Hom(A,T[k])=0 for all A∈A and all k∈Z. If A is right admissible, then T=⟨A⊥,A⟩ is a semi-orthogonal decomposition. Similarly, T=⟨A,⊥A⟩ is a semi-orthogonal decomposition if A is left admissible, where ⊥A is defined in the obvious way. Examples typically arise from so-called exceptional objects. Recall that an object E∈Db(Z) (or any C-linear triangulated category) is called exceptional if Hom(E,E)=C and Hom(E,E[k])=0 for all k≠0. The smallest triangulated subcategory containing E is then equivalent to Db(Spec(C)) and this category, by abuse of notation again denoted by E, is admissible, leading to a semi-orthogonal decomposition Db(Z)=⟨E⊥,E⟩. A sequence of objects E1,…,En is called an exceptional collection if Db(Z)=⟨(E1,…,En)⊥,E1,…,En⟩ and all Ei are exceptional. The collection is called full if (E1…,En)⊥=0. Note that any fully faithful FM transform Φ:Db(X)→Db(X′) gives a semi-orthogonal decomposition Db(X′)=⟨Φ(Db(X))⊥,Φ(Db(X))⟩, because any FM transform has a right and a left adjoint, see [26, Proposition 5.9]. 2.4. Dual semi-orthogonal decompositions Let T be a triangulated category together with a semi-orthogonal decomposition T=⟨A1,…,An⟩ such that all Ai are admissible. Then there is the left-dual semi-orthogonal decomposition T=⟨Bn,…,B1⟩ given by Bi≔⟨A1,…,Ai−1,Ai+1,…,An⟩⊥. There is also a right-dual decomposition, but we will always use the left-dual and refer to it simply as the dual semi-orthogonal decomposition. We summarize the properties of the dual semi-orthogonal decomposition needed later on in the following: Lemma 2.1 Let T=⟨A1,…,An⟩be a semi-orthogonal decomposition with dual semi-orthogonal decomposition T=⟨Bn,…,B1⟩. ⟨A1,…,Ar⟩=⟨Br,…,B1⟩and ⟨A1,…,Ar⟩⊥=⟨Bn,…,Br+1⟩for 1≤r≤n. If ⟨A1,…,An⟩is given by an exceptional collection that is, Ai=⟨Ei⟩, then its dual is also given by an exceptional collection Bi=⟨Fi⟩such that Hom*(Ei,Fj)=δijC[0]. Proof Part (i) is [21, Proposition 2.7(i)]. Part (ii) is then clear.□ An important classical example is the following: Lemma 2.2 There are dual semi-orthogonal decompositions  Db(Pn−1)=⟨O,O(1),…,O(n−1)⟩,Db(Pn−1)=⟨Ωn−1(n−1)[n−1],…,Ω1(1)[1],O⟩. Proof The fact that both sequences are indeed full goes back to Beilinson, see [26, Section 8.3] for an account. The fact that they are dual is classical and follows by a direct computation, for instance using [16, Lemma 2.5].□ The following relative version is the example of dual semi-orthogonal decompositions which we will need throughout the text. Lemma 2.3 Let ν:Z→Sbe a Pn−1-bundle. There is the semi-orthogonal decomposition  Db(Z)=⟨ν*Db(S),ν*Db(S)⊗Oν(1),…,ν*Db(S)⊗Oν(n−1)⟩whose dual decomposition is given by  Db(Z)=⟨ν*Db(S)⊗Ωνn−1(n−1),…,ν*Db(S)⊗Ων1(1),ν*Db(S)⟩. Proof Part (i) is [40, Theorem 2.6]. Part (ii) follows from Lemma 2.2.□ The following consequence will be used in Section 4.5. Corollary 2.4 If m<n, there is the equality of subcategories of Db(Z)  ⟨ν*Db(S)⊗Oν(m−n),…,ν*Db(S)⊗Oν(−1)⟩=⟨ν*Db(S)⊗Ωνn−1(n−1),…,ν*Db(S)⊗Ωνm(m)⟩. Proof Applying Lemma 2.1(i) to the dual decompositions of Lemma 2.3 gives the equalities   ⟨ν*Db(S)⊗Ωνn−1(n−1),…,ν*Db(S)⊗Ωνm(m)⟩=⟨ν*Db(S),…,ν*Db(S)⊗Oν(m−1)⟩⊥=⟨ν*Db(S)⊗Oν(m−n),…,ν*Db(S)⊗Oν(−1)⟩.□ 2.5. Linear functors and linear semi-orthogonal decompositions Let T be a tensor triangulated category, that is a triangulated category with a compatible symmetric monoidal structure. Moreover, let X be a triangulated module category over T, that is there is an exact functor π*:T→X and a tensor product ⊗:T×X→X, that is an assignment π*(A)⊗E functorial in A∈T and E∈X. We will take T=DGperf(Y) for some variety Y with an action by a finite group G. Note that DGperf(Y) has a (derived) tensor product, and it is compatible with G-linearizations. For X, we have several cases in mind. If X is a smooth G-variety X with a G-equivariant morphism π:X→Y, then we take X=DGb(X)=DGperf(X); this is a tensor-triangulated category itself and π* preserves these structures. If Λ is a finitely generated OY-algebra, then let X=Db(Λ) be the bounded derived category of finitely generated right Λ-modules with π*(A)=A⊗OYΛ and π*(A)⊗E=A⊗OYΛ⊗ΛE=A⊗OYE∈X. Note that if Λ is not commutative, then X is not a tensor category. We say that a full triangulated subcategory A⊂X is T-linear (since in our cases we have T=Dperf(Y), we will also speak of Y-linearity) if   π*(A)⊗E∈AforallA∈TandE∈A.We say that a semi-orthogonal decomposition X=⟨A1,…,An⟩ is T-linear, if all the Ai are T-linear subcategories. We call a class of objects S⊂X (left/right) spanning over T if π*T⊗S is a (left/right) spanning class of X in the non-relative sense. Recall that a subset C⊂X is generating if X=⟨C⟩ is the smallest triangulated category closed under direct summands containing C. The subset C⊂X is called generating over T if C⊗π*T generates Db(X). Let X′ be a further tensor triangulated category together with a tensor-triangulated functor π′*:T→X′. We say that an exact functor Φ:X→X′ is T-linear if there are functorial isomorphisms   Φ(π*(A)⊗E)≅π′*(A)⊗Φ(E)forallA∈TandE∈X.The verification of the following lemma is straight-forward. Lemma 2.5 If Φ:X→X′is T-linear, then Φ(X)is a T-linear subcategory of X′. Let A⊂Db(Y)be a T-linear (left/right) admissible subcategory. Then the essential image of Ais Db(Y)if and only if Acontains a (left/right) spanning class over T. For the following, we consider the case that X=Db(X) for some smooth variety X together with a proper morphism π:X→Y. Lemma 2.6 Let A,B⊂Db(X)be Y-linear full subcategories. Then  A⊂B⊥⟺π*Hom(B,A)=0∀A∈A,B∈B. Proof The direction ⟸ follows immediately from Hom*(B,A)≅Γ(Y,π*Hom(B,A)); recall that all our functors are the derived versions. Conversely, assume that there are A∈A and B∈B such that π*Hom(B,A)≠0. Since Dperf(Y) spans D(QCoh(Y)), this implies that there is an E∈Dperf(Y) such that   0≠Hom*(E,π*Hom(B,A))≅Γ(Y,π*Hom(B,A)⊗E∨)≅Γ(Y,π*(Hom(B,A)⊗π*E∨))≅Γ(Y,π*Hom(B⊗π*E,A))≅Hom*(B⊗π*E,A).By the Y-linearity, we have B⊗π*E∈B and hence A⊂B⊥.□ 2.6. Relative Fourier–Mukai transforms Let π:X→Y and π′:X′→Y be proper morphisms of varieties with X and X′ being smooth. We denote the closed embedding of the fibre product into the product by i:X×YX′↪X×X′. We call Φ:Db(X)→Db(X′) a relative FM transform if Φ=𝖥𝖬ι* for some object ∈Db(X×YX). It is a standard computation that a relative FM transform is linear over Y, with respect to the pullbacks π* and π′*. Furthermore, we have Φ≅p*(q*(̲)⊗), where p and q are the projections of the fibre diagram   (2.2) The right adjoint of Φ is given by ΦR≔q*(p!(̲)⊗∨):Db(X′)→Db(X). We also have the following slightly stronger statement which one could call relative adjointness. Lemma 2.7 For E∈Db(X)and F∈Db(X′), there are functorial isomorphisms  π*′Hom(Φ(E),F)≅π*Hom(E,ΦR(F)). Proof This follows by Grothendieck duality, commutativity of (2.2) and projection formula:   π*′Hom(Φ(E),F)≅π*′Hom(p*(q*E⊗),F)≅π*′p*Hom(q*E⊗,p!F)≅π*q*Hom(q*E,p!F⊗∨)≅π*Hom(E,q*(p!F⊗∨))≅π*Hom(E,ΦR(F)).□ For E,F∈Db(X), using the isomorphism of the previous lemma, we can construct a natural morphism Φ˜:π*Hom(E,F)→π*′Hom(Φ(E),Φ(F)) as the composition   Φ˜=Φ˜(E,F):π*Hom(E,F)→π*′Hom(E,ΦRΦ(F))≅π*′Hom(Φ(E),Φ(F)), (2.3)where the first morphism is induced by the unit of adjunction F→ΦRΦ(F). Note that taking global sections gives back the functor Φ on morphisms, that is Φ=Γ(Y,Φ˜) as maps   Hom*(E,F)≅Γ(Y,π*Hom(E,F))→Γ(Y,π*′Hom(Φ(E),Φ(F)))≅Hom*(Φ(E),Φ(F)).More generally, Φ induces functors for open subsets U⊆Y,   ΦU:Db(W)→Db(W′),whereW=π−1(U)⊆XandW′=π′−1(U)⊆X′,given by restricting the FM kernel ι* to W×W′ and we have ΦU=Γ(U,Φ˜). From this, we see that Φ˜ is compatible with composition which means that the following diagram, for E,F,G∈Db(X), commutes   (2.4) 2.7. Relative tilting bundles Let π:X→Y be a proper morphism of varieties and let X be smooth. Later on, X and Y will have G-actions, and Db(X) will be replaced by DGb(X). We say that V∈Coh(X) is a relative tilting sheaf if ΛV≔Λ≔π*Hom(V,V) is cohomologically concentrated in degree 0 and V is a spanning class over Y. For a more general theory of relative tilting bundles, see [8]. Note that Λ is a finitely generated OY-algebra. We denote the bounded derived category of coherent right modules over Λ by Db(Λ). It is a triangulated module category over Dperf(Y) via π*A=A⊗OYΛ, and Λ is a relative generator. In particular, for A∈Db(X) and M∈Db(Λ), the tensor product A⊗M is over the base OY. The functor π*Hom(V,̲):Coh(X)→Coh(Y) factorizes over Coh(Λ). Since it is left exact, we can consider its right-derived functor π*Hom(V,̲):Db(X)→Db(Λ). This yields a relative tilting equivalence: Proposition 2.8 Let V∈Db(X)be a relative tilting sheaf over Y. Then Vgenerates Db(X)over Y, and the following functor defines a Y-linear exact equivalence:   tV≔π*Hom(V,̲):Db(X)→∼Db(Λ). Proof The Y-linearity of tV is due to the projection formula   tV(π*A⊗E)=π*(π*A⊗Hom(V,E))≅A⊗π*Hom(V,E)=A⊗tV(E).Consider the restricted functor tV′:V≔⟨V⊗π*Dperf(Y)⟩→Db(Λ). We show that tV′ is fully faithful, using the adjunctions π*⊣π* and ̲⊗OYΛ⊣𝖥𝗈𝗋, where 𝖥𝗈𝗋:Db(Λ)→Db(Y) is scalar restriction, the projection formula, and the Y-linearity of tV′:   HomOX(π*A⊗V,π*B⊗V)≅HomOX(π*A,π*B⊗Hom(V,V))≅HomOY(A,π*(π*B⊗Hom(V,V)))≅HomOY(A,B⊗Λ)≅HomΛ(A⊗Λ,B⊗Λ)≅HomΛ(tV′(π*A⊗V),tV′(π*B⊗V)).Since objects of type π*A⊗V generate V, this shows that tV′ is fully faithful. We have tV′(V)=Λ. Since Λ is a relative generator, hence a relative spanning class, of Db(Λ), we get an equivalence V≅Db(Λ), see Lemma 2.5. We now claim that the inclusion V↪Db(X) has a right adjoint, namely   tV′−1tV:Db(X)→Db(Λ)→V.For this, take A∈Dperf(Y), F∈Db(X) and compute   HomOX(π*A⊗V,F)≅HomOY(A,π*Hom(V,F))≅HomΛ(A⊗Λ,tV(F))≅HomV(tV′−1(A⊗Λ),tV′−1tV(F))≅HomV(π*A⊗V,tV′−1tV(F)),where we use the projection formula, the adjunction Λ⊗OY̲⊣𝖥𝗈𝗋, the fact that tV′−1 is an equivalence, hence fully faithful, and the Y-linearity of tV′−1. Since the right-admissible Y-linear subcategory V⊂Db(X) contains the relative spanning class V, we get V=Db(X) by Lemma 2.5. This shows that V is a relative generator and that tV=tV′ is an equivalence.□ Let π′:X′→Y be a second proper morphism and let Φ:Db(X)→∼Db(X′) be a relative FM transform. Lemma 2.9 If  Φ˜Λ≔Φ˜(V,V):ΛV=π*Hom(V,V)→π*′Hom(Φ(V),Φ(V))=ΛΦ(V)is an isomorphism, then the following diagram of functors commutes:   (2.5) Proof We first show that Φ˜(V,E):tV(E)→tΦ(V)(Φ(E)) is an isomorphism in Db(Y) for every E∈D(X). Assume first that there is an exact triangle π*A⊗V→E→π*B⊗V for some A,B∈Dperf(Y) and consider the induced morphism of triangles   The outer vertical arrows are isomorphisms because they decompose as   π*Hom(V,V⊗π*A)→∼π*Hom(V,V)⊗A→Φ˜Λ∼π*′Hom(Φ(V),Φ(V))⊗A→∼π*′Hom(Φ(V),Φ(V)⊗π′*A)→∼π*′Hom(Φ(V),Φ(V⊗π*A)).Therefore, the middle vertical arrow is an isomorphism as well. Since V is a relative generator, we can show that Φ˜(V,E) is an isomorphism for arbitrary E∈Db(X) by repeating the above argument. Using the commutativity of (2.4) with E plugged in for G and V plugged in for all other arguments, we see that Φ˜(V,E) induces an ΛΦ(V)-linear isomorphism π*Hom(V,E)⊗ΛVΛΦ(V)→∼π*′Hom(Φ(V),Φ(E)).□ Lemma 2.10 The functor Φis fully faithful if and only if Φ˜Λ:ΛV→ΛΦ(V)is an isomorphism. Proof If Φ is fully faithful, the unit id→ΦRΦ is an isomorphism. Hence, Φ˜Λ is an isomorphism, see (2.3). Conversely, let Φ˜Λ be an equivalence. By Lemma 2.9, we get a commutative diagram   In this diagram, the horizontal functors are tilting equivalences. The right-hand vertical functor is an equivalence, too, by assumption on Φ˜Λ. Hence, Φ:Db(X)→⟨Φ(V)⟩ is an equivalence, which implies that Φ:Db(X)→Db(X′) is fully faithful.□ Lemma 2.11 Let V∈Db(X)be a relative tilting sheaf, Φ:Db(X)→∼Db(X)a relative FM autoequivalence, and ν:V→∼Φ(V)an isomorphism such that  Φ˜Λ=ν◦̲◦ν−1:π*Hom(V,V)→π*Hom(Φ(V),Φ(V)),that is ΦU(φ)◦ν=ν◦φfor all open subsets U⊂Yand φ∈ΛV(U). Then there exists an isomorphism of functors id→∼Φrestricting to ν. Proof We claim that, under our assumptions, the following diagram of functors commutes:   (2.6) We construct a natural isomorphism η:tΦV→∼tV⊗ΛVΛΦV as follows. For E∈Db(X), there is a natural OY-linear isomorphism π*Hom(Φ(V),E)→∼π*Hom(V,E)⊗ΛVΛΦ(V) given by f↦fν⊗1; the inverse map is g⊗1↦gν−1. This map is linear over ΛΦV because, for a local section λ∈π*Hom(Φ(V),Φ(V)), we have by our assumption, setting φ=Φ˜−1(λ):   η(fλ)=fλν⊗1=fνΦ˜−1(λ)⊗1=fν⊗λ=(fν⊗1)λ.Comparing the diagrams (2.6) and (2.5) shows that Φ≅id.□ Corollary 2.12 Let V∈Db(X)be a relative tilting sheaf, Φ1,Φ2:Db(X)→∼Db(X′)relative FM equivalences, and ν:Φ1(V)→∼Φ2(V)an isomorphism such that Φ2,U(φ)◦ν=ν◦Φ1,U(φ)for all φ∈ΛV(U)and U⊂Yopen. Then there exists a isomorphism of functors Φ1→∼Φ2restricting to ν. Moreover, if V=L1⊕⋯⊕Lkdecomposes as a direct sum, then the above condition is satisfied by specifying isomorphisms νi:Φ1(Li)→∼Φ2(Li)inducing functor isomorphisms Φ1,U→∼Φ2,Uon the full finite subcategory {L1∣U,…,Lk∣U}of Db(π−1(U)). Remark 2.13 All the results of this subsection remain valid in an equivariant setting, where a finite group G acts on X and π:X→Y is G-invariant. Then the correct sheaf of OY-algebras is ΛV=π*GHom(V,V). 2.8. Spherical functors An exact functor φ:C→D between triangulated categories is called spherical if it admits both adjoints, if the cone endofunctor F[1]≔cone(idC→φRφ) is an autoequivalence of C, and if the canonical functor morphism φR→FφL[1] is an isomorphism. A spherical functor is called split if the triangle defining F is split. The proper framework for dealing with functorial cones are dg-categories; the triangulated categories in this article are of geometric nature, and we can use Fourier–Mukai transforms. See [6] for proofs in great generality. Given a spherical functor φ:C→D, the cone of the natural transformation 𝖳=𝖳φ≔cone(φφR→idD) is called the twist around φ; it is an autoequivalence of D. The following lemma follows immediately from the definition, since an equivalence has its inverse functor as both left and right adjoint. Lemma 2.14 Let φ:C→Dbe a spherical functor and let δ:D→D′be an equivalence. Then δ◦φ:C→D′is a spherical functor with associated twist functor 𝖳δφ=δ𝖳φδ−1. 3. The geometric setup Let X be a smooth quasi-projective variety together with an action of a finite group G. Let S≔fix(G) be the locus of fixed points. Then S⊂X is a closed subset, which is automatically smooth since, locally in the analytic topology, the action can be linearized by Cartan’s lemma, see [17, Lemma 2]. Also note that X/G has rational singularities, like any quotient singularity over C [31]. Condition 3.1 We make strong assumptions on the group action: G≅μm is a cyclic group. Fix a generator g∈G. Only the trivial isotropic groups 1 and μm occur. The generator g acts on the normal bundle N≔NS/X by multiplication with some fixed primitive mth root of unity ζ. Condition (ii) obviously holds if m is prime. Condition (iii) can be rephrased: there is a splitting TX∣S=TS⊕NS/X because TS is the subsheaf of G-invariants of TX∣S and we work over characteristic 0. By (iii), this is even the splitting into the eigenbundles corresponding to the eigenvalues 1 and ζ. We denote by χ:G→C* the character with χ(g)=ζ−1. Hence, we can reformulate (iii) by saying that G acts on N via χ−1. From these assumptions, we deduce the following commutative diagram:   (3.1)where a, b, i, and j are closed embeddings and π is the quotient morphism. The G-action on X lifts to a G-action on X˜. Since, by assumption, G acts diagonally on N, it acts trivially on the exceptional divisor Z=P(N). In particular, the fixed point locus of the G-action on X˜ is a divisor. Hence, the quotient variety Y˜ is again smooth and the quotient morphism q is flat due to the Chevalley–Shephard–Todd theorem. Since the composition π◦p is G-invariant, it induces the morphism ϱ:Y˜→Y which is easily seen to be birational, hence a resolution of singularities. The preimage ϱ−1(S) of the singular locus is a divisor in Y˜. Hence, by the universal property of the blow-up, we get a morphism Y˜→BlSY which is easily seen to be an isomorphism. 3.1. The resolution as a moduli space of G-clusters The result of this section might be of independent interest. Let X be a smooth quasi-projective variety and G a finite group acting on X. A G-cluster on X is a closed zero-dimensional G-invariant subscheme W⊂X such that the G-representation H0(W,OW) is isomorphic to the regular representation of G. There is a fine moduli space HilbG(X) of G-clusters, called the G-Hilbert scheme. It is equipped with the equivariant Hilbert–Chow morphism τ:HilbG(X)→X/G,W↦𝗌𝗎𝗉𝗉(W), mapping G-clusters to their underlying G-orbits. Proposition 3.2 Let Gbe a finite cyclic group acting on Xsuch that all isotropy groups are either 1 or G, and such that Gacts on the normal bundle N𝖥𝗂𝗑(G)/Xby scalars which means that Condition3.1is satisfied. Then there is an isomorphism  φ:Y˜→≅HilbG(X)withτ◦φ=ϱ. Proof We use the notation from (3.1). One can identify X˜ with the reduced fibre product (Y˜×YX)red which gives a canonical embedding X˜⊂Y˜×X. Under this embedding, the generic fibre of q is a reduced free G-orbit of the action on X. In particular, it is a G-cluster. By the flatness of q, every fibre is a G-cluster and we get the classifying morphism φ:Y˜→HilbG(X) which is easily seen to satisfy τ◦φ=ϱ. Let s∈S and z∈Z with ν(z)∈s. Let ℓ⊂N(s) be the line corresponding to z. Then, one can check that the tangent space of the G-cluster q−1(i(z))⊂X is exactly ℓ. Hence, the G-clusters in the family X˜ are all different so that the classifying morphism φ is injective. For the bijectivity of φ, it is only left to show that the G-orbits supported on a given fixed point s∈S are parametrized by P(N(s)). Let ξ⊂X be such a G-cluster. In particular, ξ is a length m=∣G∣ subscheme concentrated in s, and hence can be identified with an ideal I⊂OX,s/mX,sm of codimension m. By Cartan’s lemma, the G-action on X can be linearized in an analytic neighbourhood of s. Hence, there is an G-equivariant isomorphism   OX,s/mX,sm≅C[x1,…,xk,y1,…,yn]/(x1,…,xk,y1,…,yn)m≕R,where G acts trivially on the xi and by multiplication by ζ−1 on the yi. Furthermore, n=rankNS/X and k=rankTS=dimX−n. By assumption, O(ξ) is the regular μm-representation. In other words,   O(ξ)≅R/I≅χ0⊕χ⊕⋯⊕χm−1, (3.2)where χ is the character given by multiplication by ζ−1. In particular, R/I has a one-dimensional subspace of invariants. It follows that every xi is congruent to a constant polynomial modulo I. Hence, we can make an identification O(ξ)≅R′/J where J is a G-invariant ideal in R′=C[y1,…,yk]/nm where n=(y1,…,yn). The decomposition of the G-representation R′ into eigenspaces is exactly the decomposition into the spaces of homogeneous polynomials. Hence, an ideal J⊂R′ is G-invariant if and only if it is homogeneous. Furthermore, (3.2) implies that   dimC(ni/(J∩ni+ni+1))=1foralli=0,…,m−1,which means that ξ is curvilinear. In summary, ξ can be identified with a homogeneous curvilinear ideal J in R′. The choice of such a J corresponds to a point in P((n/n2)∨)≅P(N(s)), see [22, Remark 2.1.7]. Hence, φ is a bijection and we only need to show that HilbG(X) is smooth. The smoothness in points representing free orbits is clear since the G-Hilbert–Chow morphism is an isomorphism on the locus of these points. So it is sufficient to show that   HomDGb(X)1(Oξ,Oξ)=dimX=n+kfor a G-cluster ξ supported on a fixed point. Following the above arguments, we have   HomDGb(X)*(Oξ,Oξ)≅HomDGb(Ak×An)*(Oξ′,Oξ′),where G acts trivially on Ak and by multiplication by ζ on An. Furthermore, by a transformation of coordinates, we may assume that   ξ′=V(x1,…,xk,y1m,y2,…,yn)⊂Ak×An.We have Oξ′≅O0⊠Oη, where   η=V(y1m,y2,…,yn)⊂An.By Künneth formula, we get   HomDGb(Ak×An)*(Oξ′,Oξ′)≅HomDb(Ak)*(O0,O0)⊗HomDGb(An)*(Oη,Oη)≅∧*(Ck)⊗HomDGb(An)*(Oη,Oη).Furthermore, HomDGb(An)0(Oη,Oη)≅H0(Oη)G≅C. Hence, it is sufficient to show that HomDGb(An)1(Oη,Oη)≅Cn. Note that η is contained in the line ℓ=V(y2,…,yn). On ℓ, we have the Koszul resolution   0→Oℓ→·y1mOℓ→Oη→0.Using this, we compute   HomDb(ℓ)*(Oη,Oη)≅Oη[0]⊕Oη[−1].Note that the normal bundle of ℓ, as an equivariant bundle, is given by Nℓ/An≅(Oℓ⊗χ−1)⊕n−1. By [2, Theorem 1.4], we have   HomDb(An)*(Oη,Oη)≅HomDb(ℓ)*(Oη,Oη⊗∧*Nℓ/An)≅HomDb(ℓ)*(Oη,Oη)⊗∧*((Oℓ⊗χ−1)⊕n−1).Evaluating in degree 1 gives   HomDb(An)1(Oη,Oη)≅Oη⊕(Oη⊗χ−1)⊕n−1.Since, as a G-representation, Oη≅χ0⊕χ1⊕…⊕χm−1, we get an n-dimensional space of invariants   HomDGb(An)1(Oη,Oη)≅HomDb(An)1(Oη,Oη)G≅Cn.□ The following lemma is needed later in Section 4.4, but its proof fits better into this section. Lemma 3.3 Assume that m=∣G∣≥n=codim(S↪X). Let ξ1,ξ2⊂Xbe two different G-clusters supported on the same point s∈S. Then HomDGb(X)*(Oξ1,Oξ2)=0. Proof By the same arguments as in the proof of the previous proposition we can reduce to the claim that   HomDGb(An)*(Oη1,Oη2)=0,where η1=V(y1m,y2,…,yn) and η2=V(y1,y2m,y3,…,yn). Set ℓ1=V(y2,…,yn), ℓ2=V(y1,y3,…,yn), E=⟨ℓ1,ℓ2⟩=V(y3,…,yn) and consider the diagram of closed embeddings   where Nt≅(OE⊗χ−1)⊕n−2. By [33, Lemma 3.3] (alternatively, one may consult [23] or [3] for more general results on derived intersection theory), we get   HomDb(An)*(Oη1,Oη2)=HomDb(An)*(ι1*Oη1,ι2*Oη2)≅HomDb(ℓ2)*(ι2*ι1*Oη1,Oη2)≅HomDb(ℓ2)*(u*v*Oη1,Oη2)⊗∧*Nt∣ℓ2≅HomDb(ℓ2)*(u*v*Oη1,Oη2)⊗∧*(Oℓ2⊗χ−1)⊕n−2. (3.3)We consider the Koszul resolution 0→Oℓ1→y1mOℓ1→Oη1→0 of Oη1. Note that this is an equivariant resolution when we consider Oℓ1 equipped with the canonical linearization since y1m is a G-invariant function. Applying u*v*, we get an equivariant isomorphism   u*v*Oη1≅O0⊕O0[1]. (3.4)Similarly, we have the equivariant Koszul resolution 0→Oℓ⊗χ→·yOℓ→O0→0 of O0, where we set ℓ≔ℓ2 and y≔y2. Applying Hom(̲,Oη2) to the resolution, we get   0→C[y]/ym⊗→·yC[y]/ym⊗χ−1→0and taking cohomology yields   HomDb(ℓ2)*(O0,Oη2)≅C⟨ym−1⟩[0]⊕C⟨1⟩⊗χ−1[−1]≅O0⊗χ−1[0]⊕O0⊗χ−1[−1]. (3.5) Plugging (3.4) and (3.5) into (3.3) gives   HomDb(An)*(Oη1,Oη2)≅(O0⊗χ−1[0]⊕O0⊕2⊗χ−1[−1]⊕O0⊗χ−1[−2])⊗∧*(χ−1)⊕n−2.The irreducible representations occurring are χ−1,χ−2,…,χ−(n−1), hence the invariants vanish (recall that m≥n).□ 4. Proof of the main result In this section, we will study the derived categories Db(Y˜) and DGb(X) in the setup described in the previous section. In particular, we will prove Theorems B and C. We set n=codim(S↪X) and m=∣G∣, in other words G=μm. We consider, for α∈Z/mZ and β∈Z, the exact functors   Φ≔p*◦q*◦𝗍𝗋𝗂𝗏:Db(Y˜)→DGb(X)Ψ≔(−)G◦q*◦p*:DGb(X)→Db(Y˜)Θβ≔i*(ν*(̲)⊗Oν(β)):Db(S)→Db(Y˜)Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα:Db(S)→DGb(X).With this notation, the precise version of Theorem B is Theorem 4.1 The functor Φis fully faithful for m≥nand an equivalence for m=n. For m>n, all the functors Ξαare fully faithful and there is a semi-orthogonal decomposition  DGb(X)=⟨Ξn−m(Db(S)),…,Ξ−1(Db(S)),Φ(Db(Y˜))⟩. The functor Ψis fully faithful for n≥mand an equivalence for n=m. For n>m, all the functors Θβare fully faithful and there is a semi-orthogonal decomposition  Db(Y˜)=⟨Θm−n(Db(S)),…,Θ−1(Db(S)),Ψ(DGb(X))⟩. Remark 4.2 We will see later in Lemma 4.14 that KY˜≤ϱ*KY for m≥n and KY˜≥ϱ*KY for n≥m. Hence, Theorem 4.1 is in accordance with the DK-Hypothesis as described in the introduction. For the proof, we first need some more preparations. 4.1. Generators and linearity Lemma 4.3 The bundle V≔OX⊗C[G]=OX⊗(χ0⊕⋯⊕χm−1)is a relative tilting sheaf for DGb(X)over Dperf(Y). Proof If L∈Pic(Y)⊂Dperf(Y) is an ample line bundle, then so is π*(L). Hence, Db(X) has a generator of the form E≔π*(OY⊕L⊕⋯⊕L⊗k) for some k≫0; see [41]. In particular, E is a spanning class of Db(X). Using the adjunction 𝖱𝖾𝗌⊣𝖨𝗇𝖽⊣𝖱𝖾𝗌, it follows that 𝖨𝗇𝖽(E)≅E⊕E⊗χ⊕⋯⊕E⊗χm−1 is a spanning class of DGb(X). Hence, V=𝖨𝗇𝖽OX is a relative spanning class of DGb(X) over Dperf(Y). Since V is a vector bundle, so is Hom(V,V)=V∨⊗V. The map π is finite, hence π* is exact (does not need to be derived). Finally, taking G-invariants is exact because we work in characteristic 0. Altogether, π*GHom(V,V) is a sheaf concentrated in degree 0.□ Lemma 4.4 The functors Φand Ψ, and for all α,β∈Zthe subcategories  Ξα(Db(S))=a*(Db(S))⊗χα⊂DGb(X)andΘβ(Db(S))=i*ν*Db(S)⊗OY˜(β)⊂Db(Y˜)are Y-linear for π*𝗍𝗋𝗂𝗏:Dperf(Y)→DGb(X)and ϱ*:Dperf(Y)→Db(Y˜), respectively. Proof We first show that Φ is Y-linear. Recall that in our setup this means   Φ(ϱ*(E)⊗F)≅π*𝗍𝗋𝗂𝗏(E)⊗Φ(F)for any E∈Dperf(Y) and F∈Db(Y˜). But this holds, since   π*𝗍𝗋𝗂𝗏(E)⊗Φ(F)≅π*𝗍𝗋𝗂𝗏(E)⊗p*q*𝗍𝗋𝗂𝗏(F)≅p*(p*π*𝗍𝗋𝗂𝗏(E)⊗q*𝗍𝗋𝗂𝗏(F))≅p*(q*ϱ*𝗍𝗋𝗂𝗏(E)⊗q*𝗍𝗋𝗂𝗏(F))≅p*q*𝗍𝗋𝗂𝗏(ϱ*(E)⊗F).The proof that Ψ is Y-linear is similar and is left to the reader. The Y-linearity of the image categories follows from Lemma 2.5(i).□ Lemma 4.5 The set of sheaves S≔{OY˜}∪{is*Ωr(r)∣s∈S,r=0,…,n−1}forms a spanning class of Db(Y˜)over Y, where is:Pn−1≅ϱ−1(s)↪Y˜denotes the fibre embedding. Proof We need to show that Sˆ≔ϱ*Dperf(Y)⊗S is a spanning class of Db(Y˜). Let y˜∈Y˜⧹Z. Then y=ϱ(y˜) is a smooth point of Y. Hence, Oy∈Dperf(Y) and Oy˜∈ϱ*Dperf(Y)=ϱ*Dperf(Y)⊗OY˜⊂Sˆ. Thus, an object E∈Db(Y˜) with 𝗌𝗎𝗉𝗉E∩(Y˜⧹Z)≠∅ satisfies Hom*(E,Sˆ)≠0≠Hom*(Sˆ,E); see [26, Lemma 3.29]. Let now 0≠E∈Db(Y˜) with 𝗌𝗎𝗉𝗉E⊂Z. Then there exists s∈S such that is*E≠0≠is!E; see again [26, Lemma 3.29]. Since the Ωr(r) form a spanning class of Pn−1, we get by adjunction Hom*(E,S)≠0≠Hom*(S,E).□ 4.2. On the equivariant blow-up Recall that the blow-up morphism q:X˜→X is G-equivariant. Let LX˜∈PicG(X˜) (we will sometimes simply write L instead of LX˜) be the equivariant line bundle OX˜(Z) equipped with the unique linearization whose restriction to Z gives the trivial action on OZ(Z)≅Oν(−1). We consider a point z∈Z with ν(z)=s corresponding to a line ℓ⊂NS/X(s). Then the normal space NZ/X˜(z) can be equivariantly identified with ℓ. It follows by Condition 3.1 that NZ/X˜≅(LX˜⊗χ−1)∣Z as an equivariant bundle. Hence, in CohG(X˜), there is the exact sequence   0→LX˜−1⊗χ→OX˜→OZ→0, (4.1)where both OX˜ and OZ are equipped with the canonical linearization, which is the one given by the trivial action over Z. Lemma 4.6 For ℓ=0,…,n−1we have p*LX˜ℓ=OX⊗χℓ. Proof We have p*OX˜≅OX, both, OX˜ and OX, equipped with the canonical linearizations. Hence, the assertion is true for ℓ=0. By induction, we may assume that p*LX˜ℓ−1≅OX⊗χℓ−1. We tensor (4.1) by LX˜ℓ to get   0→LX˜ℓ−1⊗χ→LX˜ℓ→Oν(−ℓ)→0.Since 0≤ℓ≤n−1, we have p*Oν(−ℓ)=0. Hence, we get an isomorphism   p*(LX˜ℓ)≅p*(LX˜ℓ−1⊗χ)≅p*(LX˜ℓ−1)⊗χ≅OX⊗χℓ−1⊗χ≅OX⊗χℓ.□ Lemma 4.7 The smooth blow-up p:X˜→Xhas G-linearized relative dualizing sheaf  ωp≅LX˜n−1⊗χ1−n∈PicG(X˜). Proof The non-equivariant relative dualizing sheaf of the blow-up is ωp≅OX˜((n−1)Z). Since p is G-equivariant, ωp has a unique linearization such that p!=p*(̲)⊗ωp:DGb(X)→DGb(X˜) is the right-adjoint of p*:DGb(X˜)→DGb(X). We now compute this linearization of ωp. As the equivariant pull-back p* is fully faithful, p!:DGb(X)→DGb(X˜) is fully faithful, too. Hence, adjunction gives an isomorphism of equivariant sheaves, p*ωp≅p*p!OX≅OX. The claim now follows from Lemma 4.6.□ We denote by is:Pn−1≅ϱ−1(s)↪Y˜ the embedding of the fibre of ϱ and by js:Pn−1≅p−1(s)↪X˜ the embedding of the fibre of p over s∈S. Lemma 4.8 For s∈Sand r=0,…,n−1, the cohomoloy sheaves of p*Os∈DGb(X˜)are  H−r(p*Os)≅js*(Ωr(r)⊗χr). Proof It is well known that, for the underlying non-equivariant sheaves, we have H−r(p*Os)≅js*Ωr(r); see [26, Proposition 11.12]. Since the sheaves Ωr(r) are simple, that is End(Ωr(r))=C, we have H−r(p*Os)≅js*(Ωr(r)⊗χαr) for some αr∈Z/mZ. So we only need to show αr=r. Let r∈{0,…,n−1}. We have p!L−r≅p*(L−r+n−1⊗χ1−n) by Lemma 4.7. Since −r+n−1∈{0,…,n−1}, Lemma 4.6 gives p!L−r≅OX⊗χ−r. By adjunction,   C[0]≅HomDGb(X)*(OX⊗χ−r,Os⊗χ−r)≅HomDGb(X˜)*(L−r,p*Os⊗χ−r).By Lemma 2.3, for r≠v, we have   HomDb(X˜)*(OX˜(−rZ),js*Ωv(v))≅HomDb(Pn−1)*(O(r),Ωv(v))=0.Using the spectral sequence in DGb(X˜)  E2u,v=Homu(L−r,Hv(p*Os⊗χ−r))⇒Eu+v=Homu+v(L−r,p*Os⊗χ−r),it follows that   C[0]≅HomDGb(X˜)*(L−r,p*Os⊗χ−r)≅HomDGb(X˜)*(L−r,H−r(p*Os)⊗χ−r)[r]≅(HomDb(Pn−1)*(O(r),Ωr(r))⊗χαr⊗χ−r)G[r]≅(C[−r]⊗χαr−r)G[r],where the last isomorphism is again due to Lemma 2.3. Comparing the first and the last term of the above chain of isomorphisms, we get C≅(χαr−r)G which implies αr=r.□ Corollary 4.9 Let n≥mand ℓ∈{0,…,m−1}. Let λ≥0be the largest integer such that ℓ+λm≤n−1. Then  H*(Ψ(Os⊗χ−ℓ))≅is*(⨁t=0λΩℓ+tm(ℓ+tm)[ℓ+tm]). Proof Since the (non-derived) functor q*G:CohG(X˜)→Coh(Y˜) is exact, we have   H−r(Ψ(Os⊗χ−ℓ))0≅q*G(H−r(p*Os)⊗χ−ℓ)and the claim follows from Lemma 4.8.□ 4.3. On the cyclic cover The morphism q:X˜→Y˜=X˜/G is a cyclic cover branched over the divisor Z. This geometric situation and the derived categories involved are studied in great detail in [32]. However, we will only need the following basic facts, all of which can be found in [32, Section 4.1]. Lemma 4.10 The sheaf of invariants q*G(OX˜⊗χ−1)is a line bundle which we denote LY˜−1∈Pic(Y˜). LY˜m≅OY˜(Z). q*G(OX˜⊗χα)≅LY˜αfor α∈{−m+1,…,0}. q*◦𝗍𝗋𝗂𝗏:Db(Y˜)↪DGb(X˜)is fully faithful, due to q*G(OX˜)≅OY˜. q*(𝗍𝗋𝗂𝗏(LY˜))≅LX˜are isomorphic G-equivariant line bundles. In particular, LY˜∣Z≅LX˜∣Z≅Oν(−1). Corollary 4.11 Ψ(OX⊗χα)≅LY˜αfor α∈{−m+1,…,0}. Lemma 4.12 The relative dualizing sheaf of q:X˜→Y˜=X˜/Gis ωq≅OX˜((m−1)Z). Proof Since the G-action on W≔X˜⧹Z is free, we have ωq∣W≅OW. Hence, ωq≅OX˜(αZ) for some α∈Z. We have Hom(OZ,OX˜)≅OZ(Z)[−1]≅j*Oν(−1)[−1], and hence   i*Oν(−1)[−1]≅q*j*Oν(−1)[−1]≅q*Hom(OZ,OX˜)≅q*Hom(OZ,q*OY˜)≅q*Hom(OZ,q!LY˜−α)byLemma4.10(v)≅Hom(q*OZ,LY˜−α)byGrothendieckduality≅OZ(Z)⊗LY˜−α[−1]≅i*Oν(−m+α)[−1]byLemma4.10(ii)+(vi),and thus we conclude α=m−1.□ Remark 4.13 As an equivariant bundle, we have ωq≅LX˜m−1⊗χ, but we will not use this. Corollary 4.14 We have ωY˜∣Z≅Oν(m−n). Proof We have ωX˜∣Z≅Oν(−n+1); compare Lemma 4.7. Furthermore, ωY˜∣Z≅(q*ωY˜)∣Z. Hence,   Oν(1−m)≅4.12ωq∣Z≅ωX˜∣Z⊗ωY˜∣Z∨≅Oν(1−n)⊗ωY˜∣Z∨.□ 4.4. The case m≥n. Throughout this subsection, let m≥n Proposition 4.15 (i) If m>n, then the functor Ξαis fully faithful for any α∈Z/mZ. (ii) Let m−n≥2and α≠β∈Z/mZ. Then  ΞβRΞα=0⟺α−β∈{n−m+1¯,n−m+2¯,…,−1¯}. Proof Recall that Ξβ=(a*◦𝗍𝗋𝗂𝗏(̲))⊗χβ:Db(S)→DGb(X). Hence, the right-adjoint of Ξβ is given by ΞβR≅(a!(̲)⊗χ−β)G. By [2, Theorem 1.4 and Section 1.20],   ΞβRΞα≅(a!a*(̲)⊗χα−β)G≅((̲)⊗∧*N⊗χα−β)G≅(̲)⊗(∧*N⊗χα−β)G,where by Condition 3.1, the G-action on ∧ℓN is given by χ−ℓ. We see that (∧*N)G≅∧0N[0]≅OS[0];here we use that m>n. This shows that, in the case α=β, we have ΞαRΞα≅id which proves (i). Furthermore, since the characters occurring in ∧*N are χ0, χ−1,…, χ−n, we obtain (ii) from   ΞβRΞα≠0⟺(∧*N⊗χα−β)G≠0⟺0¯∈{α−β¯,α−β−1¯,…,α−β−n¯},thatisΞβRΞα=0⟺α−β∈{n+1¯,…,m−1¯}={n−m+1¯,n−m+2¯,…,−1¯}.□ Corollary 4.16 For m>n, there is a semi-orthogonal decomposition  DGb(X)=⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S)),A⟩,where A=⊥⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S))⟩. Proposition 4.17 The functor Φ=p*q*𝗍𝗋𝗂𝗏:Db(Y˜)→DGb(X)is fully faithful. Proof By [26, Proposition 7.1], we only need to show for x,y∈Y˜ that   HomDGb(X)i(Φ(Ox),Φ(Oy))={Cifx=yandi=00ifx≠yori∉[0,dimX].By Proposition 3.2, Φ(Ox)=Oξ for some G-cluster ξ. Hence,   HomDGb(X)0(Φ(Ox),Φ(Ox))≅H0(Oξ)G≅C.Furthermore, since Φ(Ox) is a sheaf, the complex Hom*(Φ(Ox),Φ(Ox)) is concentrated in degrees 0,…,dim(X). It remains to prove the orthogonality for x≠y. If ϱ(x)≠ϱ(y), the corresponding G-clusters are supported on different orbits. Hence, their structure sheaves are orthogonal. If ϱ(x)=ϱ(y), but x≠y, the orthogonality was shown in Lemma 3.3.□ Lemma 4.18 The functor Φfactors through A. Proof By Corollary 4.16, this statement is equivalent to ΦRΞα=0 for α∈{n−m,…,−1}, where ΦR:DGb(X)→Db(Y˜) is the right adjoint of Φ. Since the composition ΦRΞα is a Fourier–Mukai transform, it is sufficient to test the vanishing on skyscraper sheaves of points, see [34, Section 2.2]. So we have to prove that   ΦRΞα(Os)≅ΦR(Os⊗χα)=0for every s∈S and every α∈{n−m,…,−1}. We have ΦR≅q*Gp!; recall that q*G stands for (̲)G◦q*. By Lemma 4.7 together with Lemma 4.8, we have   H−r(p!Os)≅is*(Ωr(r+1−n)⊗χr+1−n),where the non-vanishing cohomologies occur for r∈{0,…,n−1}. Thus, the linearizations of the cohomologies of p!(Os⊗χα) are given by the characters χγ for γ∈{α+1−n,…,α}. We see that, for α∈{n−m,…,−1}, the trivial character does not occur in H*(p!Os⊗χα). This implies that q*p!(Os⊗χα) has vanishing G-invariants.□ We denote by B⊂DGb(X) the full subcategory generated by the admissible subcategories Ξα(Db(S)) for α∈{n−m,…,−1} and Φ(Db(Y˜)). By the above, these admissible subcategories actually form a semi-orthogonal decomposition   B=⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S)),Φ(Db(Y˜))⟩. Proposition 4.19 We have the (essential) equalities B=DGb(X)and Φ(Db(Y˜))=A. For the proof, we need the following: Lemma 4.20 We have p*LX˜r⊗χ−λ∈Bfor r∈Zand λ∈{0,…,m−n}. Proof By Lemma 4.10, LX˜r≅q*(𝗍𝗋𝗂𝗏(LY˜r)). Hence,   p*(LX˜r)≅p*q*(𝗍𝗋𝗂𝗏(LY˜r))=Φ(LY˜r)∈Φ(Db(Y˜))⊂Bwhich proves the assertion for λ=0. We now proceed by induction over λ. Tensoring (4.1) by LX˜r⊗χ−λ and applying p*, we get the exact triangle   p*LX˜r−1⊗χ−(λ−1)→p*LX˜r⊗χ−λ→p*j*OZ(−r)⊗χ−λ→, (4.2)where OZ(−r) carries the trivial G-action. The first term of the triangle is an object of B by induction. Furthermore, by diagram (3.1), we have p*j*OZ(−r)≅a*ν*OZ(−r). Hence, the third term of (4.2) is an object of a*DGb(S)⊗χ−λ=Ξ−λ(Db(S))⊂B. Thus, also the middle term is an object of B which gives the assertion.□ Proof of Proposition 4.19 The second assertion follows from the first one since, if B=DGb(X) holds, both, Φ(Db(Y˜)) and A, are given by the left-orthogonal complement of   ⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S))⟩in DGb(X). The subcategories Ξα(Db(S)) and Φ(Db(Y˜)) of DGb(X) are Y-linear by Lemma 4.4. Hence, for the equality B=DGb(X) it suffices to show that   OX⊗χℓ∈B=⟨Db(S)⊗χn−m,…,Db(S)⊗χ−1,Φ(Db(Y˜))⟩for every ℓ∈Z/mZ; see Lemma 4.3. Combining Lemmas 4.10 and 4.6, we see that   Φ(LY˜ℓ)≅p*(q*𝗍𝗋𝗂𝗏(LY˜)ℓ)≅OX⊗χℓforℓ=0,…,n−1.In particular, OX⊗χℓ∈Φ(Db(Y˜))⊂B for ℓ=0,…,n−1. Setting r=0 in the previous lemma, we find that also OX⊗χℓ for ℓ=n−m,…,−1 is an object of B.□ Combining the results of this subsection gives Theorem 4.1(i). 4.5. The case n≥m. Throughout this subsection, let n≥m Proposition 4.21 Let n>m. Then the functors Θβ:Db(S)→Db(Y˜)are fully faithful for every β∈Zand there is a semi-orthogonal decomposition  Db(Y˜)=⟨C(m−n),C(m−n+1),…,C(−1),D⟩,where C(ℓ)≔Θℓ(Db(S))=i*ν*Db(S)⊗OY˜(ℓ)and  D={E∈Db(Y˜)∣i*E∈⊥⟨ν*Db(S)⊗OY˜(m−n),…,ν*Db(S)⊗OY˜(−1)⟩}={E∈Db(Y˜)∣i*E∈⟨ν*Db(S),…,ν*Db(S)⊗OY˜(m−1)⟩}. Proof This follows from [35, Theorem 1]. However, for convenience, we provide a proof for our special case. By construction, ΘβR≅ν*𝖬Oν(−β)i!. We start with the standard exact triangle of functors id→i!i*→𝖬OZ(Z)[−1]→ (see for example [26, Corollary 11.4]). By Lemma 4.10, OZ(Z)≅Oν(−m), and thus the above triangle induces for any α,β∈Z  ν*𝖬Oν(α−β)ν*→ΘβRΘα→ν*𝖬Oν(α−β−m)ν*→.By projection formula, we can rewrite this as   (̲)⊗ν*Oν(α−β)→ΘβRΘα→(̲)⊗ν*Oν(α−β−m)→.Now, ν*Oν≅OS and ν*Oν(γ)=0 for γ∈{−n+1,…,−1}. Hence, ΘβRΘβ≅id and ΘβRΘα=0 if α−β∈{m−n+1,…,−1}. Therefore, we get a semi-orthogonal decomposition   Db(Y˜)=⟨C(m−n),C(m−n+1),…,C(−1),D⟩.The description of the left-orthogonal D follows by the adjunction i*⊣i*.□ Lemma 4.22 The functor Ψ:DGb(X)→Db(Y˜)factors through D. Proof By Lemma 4.3, the equivariant bundles OX,OX⊗χ,…OX⊗χm−1 generate DGb(X) over Dperf(Y), and therefore so do the bundles OX⊗χ−m+1,…,OX⊗χ−1,OX obtained by twisting with χ1−m. Hence, it is sufficient to prove that Ψ(OX⊗χα)∈D for α∈{−m+1,…,0} as Ψ and D are Y-linear, see Lemma 4.4. Indeed, by Lemma 4.10 we have i*LY˜α=LY˜∣Zα≅Oν(−α), hence   Ψ(OX⊗χα)≅4.11q*G(OX⊗χα)≅LY˜α∈Dforα∈{−m+1,…,0}.□ Proposition 4.23 The functor Ψ:DGb(X)→Db(Y˜)is fully faithful. Proof We first observe that V≔OX⊗C[G]=OX⊗(χ0⊕⋯⊕χm−1) is a relative tilting bundle for DGb(X) over Dperf(Y); see Lemma 4.3. For the fully faithfulness, we follow Lemma 2.10. So we need to show that Ψ induces an isomorphism ΛV=π*GHom(V,V)→∼ϱ*Hom(Ψ(V),Ψ(V)). In turn, it suffices to consider the direct summands of V. Thus, let α,β∈{−m+1,…,0} and compute   π*GHom*(OX⊗χα,OX⊗χβ)≅π*GHom*(OX⊗χα+n−1⊗χ1−n,OX⊗χβ)≅4.6π*GHom*(p*LX˜α+n−1⊗χ1−n,OX⊗χβ)≅4.7π*GHom*(p*(LX˜α⊗ωp),OX⊗χβ)≅π*Gp*HomX*(LX˜α⊗ωp,p!OX⊗χβ)≅π*Gp*HomX˜*(LX˜α,p*OX⊗χβ)≅4.10ϱ*q*GHomX˜*(q*q*G(OX˜⊗χα),OX˜⊗χβ)≅ϱ*HomY˜*(q*G(OX˜⊗χα),q*G(OX˜⊗χβ))=ϱ*HomY˜*(Ψ(OX˜⊗χα),Ψ(OX˜⊗χβ)).□ We denote by E⊂Db(Y˜) the full subcategory generated by the admissible subcategories Ψ(DGb(X)) and Θℓ(Db(S))=i*ν*Db(S)⊗OY˜(ℓ) for ℓ∈{m−n,…,−1}. By the above, these admissible subcategories actually form a semi-orthogonal decomposition   E=⟨Θm−n(Db(S)),…,Θ−1(Db(S)),Ψ(DGb(X))⟩⊆Db(Y˜). Proposition 4.24 We have the (essential) equalities E=Db(Y˜)and Ψ(DGb(X))=D. Proof Analogously to Proposition 4.19, it is sufficient to prove the equality E=Db(Y˜). As E is constructed from images of fully faithful FM transforms (which have both adjoints), it is admissible in Db(Y˜). Therefore, it suffices to show that E contains a spanning class for Db(Y˜). Moreover, because all functors and categories involved are Y-linear, it suffices to prove that the relative spanning class S of Lemma 4.5 is contained in E. We already know that OY˜≅Ψ(OX)∈Ψ(DGb(X))⊂E. By Corollary 2.4, we get for s∈S and r∈{m,…,n−1}  is*Ωr(r)∈⟨Θm−n(Db(S)),…,Θ−1(Db(S))⟩⊂E.By Corollary 4.9, we have, for ℓ∈{0,…,m−1}, an exact triangle   E→Ψ(Os⊗χ−ℓ)→is*Ωℓ(ℓ)[ℓ]→,where E is an object in the triangulated category spanned by is*Ωr(r) for r∈{m,…,n−1}. In particular, the first two terms of the exact triangle are objects in E. Hence, also is*Ωℓ(ℓ)∈E for ℓ∈{0,…,m−1}.□ Combining the results of this subsection gives Theorem 4.1(ii). 4.6. The case m=n: spherical twists and induced tensor products Throughout this subsection, m=n, so that both functors Φ and Ψ are equivalences. We will show that the functors Θβ and Ξα, which were fully faithful in the cases n>m and m>n, respectively, are now spherical. Furthermore, the spherical twists along these functors allow to describe the transfer of the tensor structure from one side of the derived McKay correspondence to the other. We set Θ≔Θ0 and Ξ≔Ξ0. Proposition 4.25 For every α∈Z/mZ, the functor Ξα:Db(S)→DGb(X)is a split spherical functor with cotwist 𝖬ωS/X[−n]. Proof Since Ξα≅𝖬χαΞ, it is sufficient to prove the assertion for α=0, see Lemma 2.14. Following the proof of Proposition 4.15, we have ΞRΞ≅(̲)⊗(∧*N)G, where G acts on ∧ℓN by χ−ℓ. From rankN=n=m=ordχ, we get   (∧*N)G≅OS[0]⊕detN[−n]≅OS[0]⊕ωS/X[−n].Hence, ΞRΞ≅id⊕C with C≔𝖬ωS/X[−n]. Moreover, ΞR≅CΞL follows from a!≅Ca*.□ We introduce autoequivalences 𝖬L:Db(Y˜)→Db(Y˜) and 𝖬χ:DGb(X)→DGb(X) given by the tensor products with the line bundle LY˜ and the character χ, respectively. Theorem 4.26 There are the following relations between functors: Ψ−1≅𝖬χΦ𝖬Ln−1; ΨΞ≅Θ, in particular, the functors Θβare spherical too; 𝖳Θ≅Ψ𝖳ΞΨ−1; Ψ−1𝖬LΨ≅𝖬χ𝖳Ξand Ψ−1𝖬L−1Ψ≅𝖳Ξ−1𝖬χ−1. Proof In the verification of (i), we use ωp≅LX˜n−1⊗χ, from Lemma 4.7 and m=n:   Ψ−1≅ΨL≅p!q*≅4.7p*𝖬LX˜n−1⊗χq*≅4.10𝖬χp*q*𝖬Ln−1≅𝖬χΦ𝖬Ln−1.For (ii), first note that, since the G-action on Z⊂X˜ is trivial, we have   Θ≅i*ν*≅q*j*ν*≅q*Gj*ν*𝗍𝗋𝗂𝗏.Hence, the base change morphism ϑ:p*a*→j*ν* induces a morphism of functors   ϑˆ:ΨΞ≅q*Gp*a*𝗍𝗋𝗂𝗏→q*Gj*ν*𝗍𝗋𝗂𝗏≅Θwhich in turn is induced by a morphism between the Fourier–Mukai kernels, see [34, Section 2.4]. Hence, it is sufficient to show that ϑ induces an isomorphism ΨΞ(Os)≅Θ(Os) for every s∈S, see [34, Section 2.2]. The morphism ϑ induces an isomorphism on degree zero cohomology L0p*a*(Os)≅Op−1(a(s))≅j*L0ν*(Os). But there are no cohomologies in non-zero degrees for j*ν* since ν is flat and j a closed embedding. Furthermore, the non-zero cohomologies of p*a* vanish after taking invariants, see Corollary 4.9. Hence, ϑˆ(Os) is indeed an isomorphism. The second assertion of (ii) and (iii) is direct consequences of Proposition 4.25 and the formula ΨΞ≅Θ, see Lemma 2.14. For (iv), it is sufficient to prove the second relation, and we employ Corollary 2.12 with Li=OX⊗χi, see also Lemma 4.3. Recall that 𝖳Ξ−1=cone(id→ΞΞL)[−1], and ΞL≅(̲)Ga*. For 1≠α∈Z/nZ, we get   ΞL𝖬χ−1(OX⊗χα)≅(OS⊗χα−1)G≅0.Hence, 𝖳Ξ−1𝖬χ−1(OX⊗χα)≅OX⊗χα−1. We have ΞL(OX)=OS. Therefore, ΞΞL(OX)≅a*OS and 𝖳Ξ−1(OX)≅IS. In summary,   𝖳Ξ−1𝖬χ−1(OX⊗χα)≅{OX⊗χα−1forα≠1,ISforα=1.On the other hand, for α∈{−n+1,…,0}, we have Ψ(OX⊗χα)≅Lα, see Corollary 4.11. Hence, we have   Ψ−1𝖬L−1Ψ(OX⊗χα)≅OX⊗χα−1forα∈{−n+2,…,0}.For α=−n+1, we use (i) to get   Ψ−1𝖬L−1Ψ(OX⊗χ1−n)≅Ψ−1(LY˜−n)≅𝖬χΦ(LY˜−1)≅4.10p*(LX˜−1⊗χ)≅IS,where we get the last isomorphism by applying p* to the exact sequence (4.1). Therefore, for every α∈Z/nZ, we obtain isomorphisms   κα:F1(Lα)≔𝖳Ξ−1𝖬χ−1(OX⊗χα)→∼F2(Lα)≔Ψ−1𝖬L−1Ψ(OX⊗χα).Finally, we have to check that the isomorphisms κα can be chosen in such a way that they form an isomorphism of functors κ:F1,V∣{L0,…,Ln−1}→∼F2,V∣{L0,…,Ln−1} over every open set V⊂Y. Let U≔Y⧹S⊂Y the open complement of the singular locus. We claim that F1,U≅𝖬χ−1≅F2,U. This is clear for F2=𝖳Ξ−1𝖬χ−1. Furthermore, the map p:X˜→X is an isomorphism and q:X˜→Y˜ is a free quotient when restricted to W≔π−1(U). Since also LX˜=q*G(OX˜⊗χ), we get ΨU≅𝖬χ−1∣U. Hence, over W, the κi∣W can be chosen functorially. By the above computations, each κi∣W is given by a section of the trivial line bundle. As S has codimension at least 2 in X, the sections κi∣W over W uniquely extend to sections κi over X. The commutativity of the diagrams relevant for the functoriality now follows from the commutativity of the diagrams restricted to the dense subset W.□ The relations of Theorem 4.26 allow to transfer structures between Db(Y˜) and DGb(X). For example, we can deduce the formula Ψ𝖬χ−1Ψ−1≅𝖳Θ𝖬L−1. Since OX⊗χα for α∈{−(n−1),…,0} form a relative generator of DGb(X), their images Lα under Ψ do as well. Hence, at least theoretically, our formulas give a complete description of the tensor products induced by Ψ (and also Φ) on both sides. Note that Φ and Ψ are both equivalences, but not inverse to each other. Hence, they induce non-trivial autoequivalences ΨΦ∈Aut(DGb(X)) and ΦΨ∈Aut(Db(Y˜)). Considering the setup of the McKay correspondence as a flop of orbifolds as in diagram (1.1), it makes sense to call them flop–flop autoequivalences. These kinds of autoequivalences were widely studied for flops of varieties, see [4, 7, 19, 20, 45]. The general picture seems to be that the flop–flop autoequivalences can be expressed via spherical and P-twists induced by functors naturally associated to the centres of the flops. This picture is called the ‘flop–flop = twist’ principle, see [4]. The following can be seen as the first instance of an orbifold ‘flop–flop = twist’ principle which we expect to hold in greater generality. Corollary 4.27 ΨΦ≅𝖳Θ𝖬L−n≅𝖳Θ𝖬OY˜(−Z). Remark 4.28 Let us assume m=n=2 so that χ−1=χ. Then, for every k∈N, we get   Φ(L−k)≅ISk⊗χk, (4.3)where ISk denotes the k th power of the ideal sheaf of the fixed point locus. Indeed,   Φ(L−k)≅4.26(i)𝖬χΨ−1(L−k−1)≅𝖬χ(Ψ−1𝖬L−1Ψ)k(L−1)≅4.11𝖬χ(Ψ−1𝖬L−1Ψ)k(O⊗χ)≅4.26(iv)(𝖬χ𝖳Ξ−1)k(OX)≅ISk⊗χk.The last isomorphism follows inductively using the short exact sequences   0→ISk+1→ISk→ISk/ISk+1→0and the fact that the natural action of μ2 on ISk/ISk+1 is given by χk. Let now S be a surface and X=S2 with μ2 acting by permutation of the factors. Then Y˜=S[2] is the Hilbert scheme of two points and LY˜ is the square root of the boundary divisor Z parametrizing double points. For a vector bundle F on S of rank r, we have   detF[2]≅LY˜−r⊗DdetF,where F[2] denotes the tautological rank 2r bundle induced by F and, for L∈PicS, we put DL≔ϱ*π*(L⊠L)G∈PicS[2]. Hence, by the OY-linearity of Φ, formula (4.3) recovers the n=2 case of [44, Theorem 1.8]. 5. Categorical resolutions 5.1. General definitions Recall from [35] that a categorical resolution of a triangulated category T is a smooth triangulated category T˜ together with a pair of functors P*:T˜→T and P*:Tperf→T˜ such that P* is left adjoint to P* on Tperf and the natural morphism of functors idTperf→P*P* is an isomorphism. Here, Tperf is the triangulated category of perfect objects in T. Moreover, a categorical resolution (T˜,P*,P*) is weakly crepant if the functor P* is also right adjoint to P* on Tperf. For the notion of smoothness of a triangulated category, see for example [30]. For us, it is sufficient to notice that every admissible subcategory of Db(Z) for some smooth variety Z is smooth. In fact, we will always consider categorical resolutions of Db(Y), for some variety Y with rational Gorenstein singularities, inside Db(Y˜) for some fixed (geometric) resolution of singularities ϱ:Y˜→Y. By this, we mean an admissible subcategory T˜⊂Db(Y˜) such that ϱ*:Dperf(Y)→Db(Y˜) factorizes through T˜. By Grothendieck duality, we get a canonical isomorphism OY≅ϱ*OY˜≅ϱ*ωϱ. This induces a global section s of ωϱ, unique up to a global unit (that is scalar multiplication by an element of OY(Y)×), and hence a morphism of functors   t≔ϱ*(̲⊗s):ϱ*→ϱ!.Since this morphism can be found between the corresponding Fourier–Mukai kernels, we may define the cone of functors ϱ+≔cone(t):Db(Y˜)→Db(Y). Definition 5.1 The weakly crepant neighbourhood of Yinside Db(Y˜) is the full triangulated subcategory   WCN(ϱ)≔ker(ϱ+)⊂Db(Y˜). Proposition 5.2 If WCN(ϱ) is a smooth category (which is the case if it is an admissible subcategory of Db(Y˜)), it is a categorical weakly crepant resolution of singularities. Proof By adjunction formula, tϱ*:ϱ*ϱ*→ϱ!ϱ* is an isomorphism. Hence, ϱ+ϱ*=0 and ϱ*:Dperf(Y)→Db(Y˜) factors through WCN(ϱ). By definition, ϱ! is the left adjoint to ϱ*. Since ϱ* and ϱ! agree on WCN(ϱ), we also have the adjunction ϱ*⊣ϱ* on WCN(ϱ).□ Remark 5.3 We think of WCN(ϱ) as the biggest weakly crepant categorical resolution inside the derived category Db(Y˜) of a given geometric resolution ϱ:Y˜→Y. The only thing that prevents us from turning this intuition into a statement is the possibility that, for a given weakly crepant resolution T⊂Db(Y˜), there might be an isomorphism ϱ*∣T≅ϱ!∣T which is not the restriction of t (up to scalars). 5.2. The weakly crepant neighbourhood in the cyclic setup In the case of the resolution of the cyclic quotient singularities discussed in the earlier sections, WCN(ϱ) is indeed a categorical resolution by the following result. We use the notation of Section 3; recall G=μm. Theorem 5.4 Let Y=X/G, ϱ:Y˜→Yand i:Z=ϱ−1(S)↪Y˜be as in Section3. Assume m∣n=codim(S↪X)and n>m. Then there is a semi-orthogonal decomposition  WCN(ϱ)=⟨i*(E),Ψ(Dμmb(X))⟩,where  E=⟨A(−m+1),A(−m+2)…,A(−1),A⊗Ωn−m−1(n−m−1),A⊗Ωn−m−2(n−m−2),…,A⊗Ωm(m)⟩with A≔ν*Db(S)and A(i)≔A⊗O(i); the A⊗Ωi(i)parts of the decomposition do not occur for n=2m. In particular, WCN(ϱ)is an admissible subcategory of Db(Y˜). Proof We first want to show that Ψ(Dμmb(X))⊂WCN(ϱ). For this, by Lemma 4.3, it is sufficient to show that LY˜a=Ψ(OX⊗χa)∈WCN(ϱ) for every a∈{−m+1,…,0}. The equivariant derived category Dμmb(X) is a strongly (hence also weakly) crepant categorical resolution of the singularities of Y via the functors π*,π*μm, see [1, Theorem 1.0.2]. Since Ψ◦π*≅ϱ* (see Lemma 4.4), C≔Ψ(Dμmb(X)) is a crepant resolution via the functors ϱ*,ϱ*. Hence, ϱ*LY˜a≅ϱ!LY˜a for a∈{−m+1,…,0} and it is only left to show that this isomorphism is induced by t. Again by the Y-linearity of Ψ, we have ϱ*LY˜a≅π*(OX⊗χa)μm which is a reflexive sheaf on the normal variety Y (this follows for example by [24, Corollary 1.7]). By construction, t induces an isomorphism over Y⧹S. Since the codimension of S is at least 2, t:ϱ*LY˜a→ϱ!LY˜a is an isomorphism of reflexive sheaves over all of Y, see [24, Proposition 1.6]. By Theorem 4.1(ii), we have Db(Y˜)≅⟨B,C⟩ with   B≅i*⟨A(m−n),…,A(−1)⟩≅i*(⟨A,A(1),…,A(m−1)⟩⊥).We have ϱ*B=0. It follows that WCN(ϱ)=⟨B∩ker(ϱ!),C⟩. Indeed, consider an object A∈Db(Y˜). It fits into an exact triangle C→A→B→ with C∈C and B∈B. From the morphism of triangles   we see that t(A) is an isomorphism if and only if ϱ!B=0. It is left to compute B∩ker(ϱ!). Let F∈Db(Z) and B=i*F. By Lemma 4.14,   ϱ!B≅ϱ!i*F≅b*ν*(F⊗Oν(m−n)).We see that B∈kerϱ! if and only if ν*(F⊗Oν(m−n))=0 if and only if F∈ν*Db(S)(n−m)⊥. Hence, B∩kerϱ!=i*(F⊥) with   F=⟨A,A(1),…,A(m−1),A(n−m)⟩⊂Db(Z).Carrying out the appropriate mutations within the semi-orthogonal decomposition   Db(Z)=⟨A(−m+1),A(−m+2),…,A(n−m−1),A(n−m)⟩,we see that F⊥=E; compare Lemma 2.3. Since E⊂⟨A(m−n),…,A(−1)⟩ is an admissible subcategory, we find that i*:E→Db(Y˜) is fully faithful and has adjoints. Hence, WCN(ϱ)⊂Db(Y˜) is admissible.□ Remark 5.5 We have Db(Y˜)=⟨i*(A⊗Ωn−1(n−m)),WCN(ϱ)⟩. In other words, we can achieve categorical weak crepancy by dropping only one Db(S) part of the semi-orthogonal decomposition of Db(Y˜). 5.3. The discrepant category and some speculation Let Y be a variety with rational Gorenstein singularities and ϱ:Y˜→Y a resolution of singularities. Then, ϱ is a crepant resolution if and only if Db(Y˜)=WCN(ϱ); compare [1, Proposition 2.0.10]. We define the discrepant category of the resolution as the Verdier quotient   disc(ϱ)≔Db(Y˜)/WCN(ϱ).By [39, Remark 2.1.10], since WCN(ϱ) is a kernel, and hence a thick subcategory, we have disc(ϱ)=0 if and only if Db(Y˜)=WCN(ϱ). Therefore, we can regard disc(ϱ) as a categorical measure of the discrepancy of the resolution ϱ:Y˜→Y. In our cyclic quotient setup, where Y˜≅HilbG(X) is the simple blow-up resolution, we have disc(ϱ)≅Db(S) by Remark 5.5 and [37, Lemma A.8]. Hence, in this case, disc(ϱ) is the smallest non-zero category that one could expect (this is most obvious in the case that S is a point). This agrees with the intuition that the blow-up resolution is minimal in some way. Question 1 Given a variety Y with rational Gorenstein singularities, is there a resolution ϱ:Y˜→Y of minimal categorical discrepancy in the sense that, for every other resolution ϱ′:Y˜′→Y, there is a fully faithful embedding disc(ϱ)↪disc(ϱ′)? Often, in the case of a quotient singularity, a good candidate for a resolution of minimal categorical discrepancy should be the G-Hilbert scheme. At least, we can see that disc(ϱ) grows if we further blow up the resolution away from the exceptional locus. Proposition 5.6 Let ϱ:Y˜→Ybe a resolution of singularities and let f:Y˜′→Y˜be the blow-up in a smooth centre C⊂Y˜which is disjoint from the exceptional locus of ϱ. Set ϱ′≔ϱf:Y˜′→Y. Then there is a semi-orthogonal decomposition  disc(ϱ′)=⟨Db(C),disc(ϱ)⟩. We first need the following general. Lemma 5.7 Let Dbe a triangulated category, C⊂Da triangulated subcategory, and D=⟨A,B⟩a semi-orthogonal decomposition so that the right-adjoint iB!of the inclusion iB:B↪Dsatisfies iB!(C)⊂B∩C. Then there is a semi-orthogonal decomposition  D/C≅⟨A/(A∩C),B/(B∩C)⟩. Proof For every object D∈D, we have an exact triangle   iB!D→D→iA*D→, (5.1)where iA* is the left-adjoint to the embedding iA:A→D. Considering an object C∈C shows that our assumption iB!(C)⊂B∩C implies iA*(C)⊂A∩C. Let C∈C and A∈A. Then, using the long exact Hom-sequence associated to the triangle (5.1), we see that every morphism C→A factors as C→ιA*C→A. Hence, the embedding iA descends to a fully faithful embedding i¯A:A/(A∩C)→D/C, by [37, Proposition B.2] (set W=A, V=A∩C and use (ff2) of loc. cit.). Similarly, we get an induced fully faithful embedding i¯B:B/(B∩C)→D/C (use (ff2)op instead of (ff2)). Now let us show that HomD/C(B/(B∩C),A/(A∩C))=0. For B∈B and A∈A, a morphism B→A in D/C is represented by a roof   B←βD→αA,where β:D→B is a morphism in D with cone(β)∈C and α:D→A is any morphism in D, see [39, Definition 2.1.11]. Put C≔cone(β)[−1]∈C. We apply the triangle of functors iA*→id→iB!→ (formally, iA* has to be replaced by iAiA* and iB! by iBiB!) to the triangle of objects C→D→B→ and obtain the diagram   where we have used iB!B=B and iA*B=0. Now iB!C∈C∩B by assumption. The left column thus forces iA*C≅iA*D∈C. We get that coneβγ∈C since coneβ,coneγ∈C, see [39, Lemma 1.5.6]. Therefore, we get another roof representing the same morphism in D/C, replacing D by iB!D:   B←βγiB!D→αγA.However, iB!D∈B and HomD(B,A)=0, so the morphism is 0 in D/C. Finally, we need to show that A/(A∩C) and B/(B∩C) generate D/C, but this is clear, because A and B generate D.□ Proof of Proposition 5.6 We have a semi-orthogonal decomposition Db(Y˜′)=⟨A,B⟩ with B=f*Db(Y˜) and   A=⟨ι*(g*Db(C)⊗Og(−c+1)),…,ι*(g*Db(C)⊗Og(−1))⟩.Here, c=codim(C↪Y˜) and g and ι are the Pn−1-bundle projection and the inclusion of the exceptional divisor of the blow-up f:Y˜′→Y˜. Let U≔Y˜⧹C. For F∈Db(Y˜′), we have (f*F)∣U≅(f!F)∣U. Hence, if F∈WCN(ϱ′), we must have (f*F)∣U∈WCN(ϱ∣U). Since ϱ is an isomorphism in a neighbourhood of C, an object E∈Db(Y˜) is contained in WCN(ϱ) if and only if its restriction E∣U is contained in WCN(ϱ∣U). In summary,   f*F∈WCN(ϱ)foreveryF∈WCN(ϱ′).We have f*OY˜′≅OY˜≅f*ωf. By the projection formula, it follows that f*∣B≅f!∣B. Hence, we have B∩WCN(ϱ′)=f*WCN(ϱ) and B/(WCN(ϱ′)∩B)≅disc(ϱ). Now, we can apply Lemma 5.7 with C=WCN(ϱ′) to get a semi-orthogonal decomposition   disc(ϱ′)=⟨A/(WCN(ϱ′)∩A),disc(ϱ)⟩.We have f*(A)=0, hence ϱ*′(A)=0. Accordingly,   WCN(ϱ′)∩A=ker(ϱ!′)∩A=ker(f!)∩A.The second equality is due to the fact that all objects of f!A are supported on C, where ϱ is an isomorphism. Now, in analogy to the computations of the proof of Theorem 5.4 and Remark 5.5, we get a semi-orthogonal decomposition   A=⟨ι*(g*Db(C)⊗Ωc−1(c−1)),ker(f!)∩A⟩.Hence, A/(WCN(ϱ′)∩A)≅ι*(g*Db(C)⊗Ωc−1(c−1))≅Db(C).□ 5.4. (Non-)unicity of categorical crepant resolutions Let Y˜→Y be a resolution of rational Gorenstein singularities and let D⊂Dperf(Y˜) be an admissible subcategory which is a weakly crepant resolution, that is ϱ*:Dperf(Y)→Db(Y˜) factors through D and ϱ*∣D≅ϱ!∣D. Then every admissible subcategory D′⊂D with the property that ϱ*:Dperf(Y)→Db(Y˜) factors through D′ is a weakly crepant resolution, too. In particular, in our setup of cyclic quotients, there is a tower of weakly crepant resolutions of length n−m given by successively dropping the Db(S) parts of the semi-orthogonal decomposition of WCN(ϱ). We see that weakly crepant categorical resolutions are not unique, even if we fix the ambient derived category Db(Y˜) of a geometric resolution Y˜→Y. In contrast, strongly crepant categorical resolutions are expected to be unique up to equivalence, see [35, Conjecture 4.10]. A strongly crepant categorical resolution of Db(Y) is a module category over Db(Y) with trivial relative Serre functor, see [35, Section 3]. For an admissible subcategory D⊂Db(Y˜) of the derived category of a geometric resolution of singularities ϱ:Y˜→Y this condition means that D is Y-linear and there are functorial isomorphisms   ϱ*Hom(A,B)∨≅ϱ*Hom(B,A) (5.2)for A,B∈D. In our cyclic setup, Ψ(DG(X))⊂Db(Y˜) is a strongly crepant categorical resolution, see [35, Theorem 1] or [1, Theorem 10.2]. We require strongly crepant categorical resolutions to be indecomposable, which means that they do not decompose into direct sums of triangulated categories or, in other words, they do not admit both-sided orthogonal decompositions. Under this additional assumption, we can prove that strongly crepant categorical resolutions are unique if we fix the ambient derived category of a geometric resolution. Proposition 5.8 Let Y˜→Ybe a resolution of Gorenstein singularities and D,D′⊂Db(Y˜)admissible indecomposable strongly crepant subcategories. Then D=D′. Proof The intersection D∩D′ is again an admissible Y-linear subcategory of Db(Y˜) containing ϱ*(Dperf(Y)). Furthermore, condition (5.2) is satisfied for every pair of objects of D∩D′; so the intersection is again a strongly crepant resolution. Hence, we can assume D′⊂D. Let A be the right-orthogonal complement of D′ in D, so that we have a semi-orthogonal decomposition D=⟨A,D′⟩. By Lemma 2.6, this means that ϱ*Hom(D,A)=0 for A∈A and D∈D′. But then, by (5.2), we also get ϱ*Hom(A,D)=0 so that D=A⊕D′.□ 5.5. Connection to Calabi–Yau neighbourhoods In [25], spherelike objects and their spherical subcategories were introduced and studied. The paper hinted at a role of these notions for birationality questions of Calabi–Yau varieties. One of the starting points for our project was to consider Calabi–Yau neighbourhoods (a generalization of spherical subcategories) as candidates for categorical crepant resolutions of Calabi–Yau quotient varieties. In this subsection, we describe the connection to the weakly crepant resolutions considered above. We recall some abstract homological notions. Let T be a Hom-finite C-linear triangulated category and E∈T an object. We say that 𝖲E∈T is a Serre dual object for E if the functors Hom*(E,−) and Hom*(−,𝖲E)∨ are isomorphic. By the Yoneda lemma, 𝖲E is then uniquely determined. Fix an integer d. We call the object E a d-Calabi–Yau object, if E[d] is a Serre dual of E, d-spherelike if Hom*(E,E)=C⊕C[−d], and d-spherical if E is d-spherelike and a d-Calabi–Yau object.Note a smooth compact variety X of dimension d is a strict Calabi–Yau variety precisely if the structure sheaf OX is a d-spherical object of Db(X). In [25], the authors show that if E is a d-spherelike object, there exists a unique maximal triangulated subcategory of T in which E becomes d-spherical. In the following, we will imitate this construction for a larger class of objects. Definition 5.9 Let E∈T be an object in a triangulated category having a Serre dual 𝖲E. We call E a d-selfdual object if Hom(E,E[d])≅C, that is by Serre duality there is a morphism w:E→ω(E)≔𝖲E[−d] unique up to scalars, and the induced map w*:Hom*(E,E)→∼Hom*(E,ω(E)) is an isomorphism.In particular, a d-selfdual object satisfies Hom*(E,E)≅Hom(E,E)∨[−d], hence the name. Remark 5.10 If an object is d-spherelike, then it is d-selfdual; compare [25, Lemma 4.2]. For a d-selfdual object E, there is a triangle E→wω(E)→QE→E[1] induced by w. By our assumption, we get Hom*(E,QE)=0. Thus, following an idea suggested by Martin Kalck after discussing [25, Section 7] with Michael Wemyss, we propose the following: Definition 5.11 The Calabi–Yau neighbourhood of a d-selfdual object E∈T is the full triangulated subcategory   CY(E)≔⊥QE⊆T. Proposition 5.12 If E∈Tis a d-selfdual object then E∈CY(E)is a d-Calabi-Yau object. Proof If T∈CY(E), apply Hom*(T,−) to the triangle E→ω(E)→QE.□ Using the same proof as for [25, Theorem 4.6], we see that the Calabi–Yau neighbourhood is the maximal subcategory of T in which a d-selfdual object E becomes d-Calabi–Yau. Proposition 5.13 If U⊂Tis a full triangulated subcategory and E∈Uis d-Calabi–Yau, then U⊂CY(E). Proposition 5.14 Let Ybe a projective variety with rational Gorenstein singularities and trivial canonical bundle of dimension d=dimYand consider a resolution of singularities ϱ:Y˜→Y. Then, for every line bundle L∈PicY, the pull-back ϱ*L∈Db(Y˜)is d-selfdual. Furthermore, we have  WCN(ϱ)=⋂L∈PicYCY(ϱ*L). (5.3) Proof Note that, by our assumption that ωY is trivial, we have ωY˜≅ωϱ. Hence, by Grothendieck duality, there is a morphism wL:ϱ*L→ϱ*L⊗ωY˜ unique up to scalar multiplication, namely wL=idϱ*L⊗s, where s is the non-zero section of ωY˜≅ωϱ; compare the previous Section 5.1. Furthermore, wL*:Hom*(ϱ*L,ϱ*L)→Hom*(ϱ*L,ϱ*L⊗ωY˜) is an isomorphism, still by Grothendieck duality, which means that ϱ*L is d-selfdual. Recall that WCN(ϱ)=ker(ϱ+), where ϱ+ is defined as the cone   ϱ*→tϱ!→ϱ+→.By adjunction, we get WCN(ϱ)=⊥(ϱ+(Dperf(Y))), where ϱ+=ϱ+R is given by the triangle   ϱ+→ϱ*→tRϱ!→.Note that tR=(̲)⊗s. Hence, tR(L)=wL:ϱ*L→ϱ*L⊗ωY˜ and ϱ+(L)=Qϱ*L[−1]; compare Definition 5.11. Since the line bundles form a generator of Dperf(Y), we get for F∈Db(Y˜):   F∈WCN(ϱ)⟺F∈⊥(ϱ+(Dperf(Y)))⟺F∈⊥Qϱ*L∀L∈PicY⟺F∈CY(ϱ*L)∀L∈PicY.□ Remark 5.15 Following the proof of Proposition 5.14, we see that, on the right-hand side of (5.3), it is sufficient to take the intersection over all powers of a given ample line bundle. In our cyclic setup, if S consists of isolated points, we even have WCN(ϱ)=CY(OY˜) so that the weakly crepant neighbourhood is computed by a Calabi–Yau neighbourhood of a single object. The same should hold in general if Y has isolated singularities. 6. Stability conditions for Kummer threefolds Let A be an abelian variety of dimension g. Consider the action of G=μ2 by ±1. Then the fixed point set A[2] consists of the 4g two-torsion points. Consider the quotient A¯ (the singular Kummer variety) of A by G, and the blow-up K(A) (the Kummer resolution) of A¯ in A[2]. This setup satisfies Condition 3.1, with m=2 and n=g and we get Corollary 6.1 The functor Ψ:DGb(A)→Db(K(A))is fully faithful, and  Db(K(A))=⟨Db(𝗉𝗍),…,Db(𝗉𝗍)︸(g−2)4gtimes,Ψ(DGb(A))⟩. To explore a potentially useful consequence of this result, we need to recall that a Bridgeland stability condition on a reasonable C-linear triangulated category D consists of the heart A of a bounded t-structure in D and a function from the numerical Grothendieck group of D to the complex numbers satisfying some axioms, see [15]. Corollary 6.2 There exists a Bridgeland stability condition on Db(K(A)), for an abelian threefold A. Proof To begin with, by [11, Corollary 10.3], there is a stability condition on DGb(A). Denote by A⊂DGb(A) the corresponding heart; it is a tilt of the standard heart [11, Section 2]. For a two-torsion point x∈A[2], we set Ex≔Oϱ−1(π(x))(−1). Then, since g=dimA=3, the semi-orthogonal decomposition of Corollary 6.1 is given by   Db(K(A))=⟨{Ex}x∈A[2],DGb(A)⟩. (6.1) Next, we want to show that, for every x∈A[2], there exists an integer i such that Hom≤i(Ex,Ψ(F))=0 for all F∈A⊂DGb(A). Indeed, the cohomology of any complex in the heart of the stability condition on DGb(A), as constructed in [11, Corollary 10.3], is concentrated in an interval of length three. The functor Ψ has cohomological amplitude at most 3, since q*G:CohG(A)→Coh(K(A)) is an exact functor of abelian categories, and every sheaf on A has a locally free resolution of length dimA=3. This implies that the cohomology of any complex in Ψ(DGb(A)) is contained in a fixed interval of length 6. This proves the above claim. Using [18, Proposition 3.5(b)], this then implies that ⟨Ex,Ψ(DGb(A))⟩ has a stability condition; compare the argument in [10, Corollary 3.8]. We can proceed to show that, for x≠y∈A[2], there exists an integer i such that Hom≤i(Ey,⟨Ex,Ψ(DGb(A))⟩)=0 and so there is a stability condition on ⟨Ey,Ex,Ψ(DGb(A))⟩. After 43 steps we have constructed a stability condition on Db(K(A)); compare (6.1).□ Acknowledgements It is a pleasure to thank Tarig Abdelgadir, Martin Kalck, Sönke Rollenske and Evgeny Shinder for comments and discussions. We are grateful to the anonymous referee for very careful inspection. References 1 R. Abuaf, Categorical crepant resolutions for quotient singularities, Math. Z.  282 ( 2016), 679– 689. Google Scholar CrossRef Search ADS   2 D. Arinkin and A. Căldăraru, When is the self-intersection of a subvariety a fibration? Adv. Math.  231 ( 2012), 815– 842. Google Scholar CrossRef Search ADS   3 D. Arinkin, A. Căldăraru and M. 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Derived categories of resolutions of cyclic quotient singularities

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Abstract For a cyclic group G acting on a smooth variety X with only one character occurring in the G-equivariant decomposition of the normal bundle of the fixed point locus, we study the derived categories of the orbifold [X/G] and the blow-up resolution Y˜→X/G. Some results generalize known facts about X=An with diagonal G-action, while other results are new also in this basic case. In particular, if the codimension of the fixed point locus equals |G|, we study the induced tensor products under the equivalence Db(Y˜)≅Db([X/G]) and give a ‘flop–flop = twist’ type formula. We also introduce candidates for general constructions of categorical crepant resolutions inside the derived category of a given geometric resolution of singularities and test these candidates on cyclic quotient singularities. 1. Introduction For geometric, homological and other reasons, it has become commonplace to study the bounded derived category of a variety. One of the many intriguing aspects are connections, some of them conjectured, some of them proven, to birational geometry. One expected phenomenon concerns a birational correspondence   of smooth varieties. Then we should have A fully faithful embedding Db(X)↪Db(X′), if q*KX≤p*KX′. A fully faithful embedding Db(X′)↪Db(X), if q*KX≥p*KX′. An equivalence Db(X′)≅Db(X), in the flop case q*KX=p*KX′.This is proven in many instances, see [12, 14, 28, 38]. Another very interesting aspect of derived categories is their occurrence in the context of the McKay correspondence. Here, one of the key expectations is that the derived category of a crepant resolution Y˜→X/G of a Gorenstein quotient variety is derived equivalent to the corresponding quotient orbifold: Db(Y˜)≅Db([X/G])=DGb(X). In [9], this expectation is proven in many cases under the additional assumption that Y˜≅HilbG(X) is the fine moduli space of G-clusters on X. It is enlightening to view the derived McKay correspondence as an orbifold version of the conjecture on derived categories under birational correspondences described above; for more information on this point of view, see [29, Section 2], where the conjecture is called the DK-Hypothesis. Indeed, if we denote the universal family of G-clusters by Z⊂Y˜×X, we have the following diagram of birational morphisms of orbifolds:   (1.1)Since the pullback of the canonical sheaf of X/G under π is the canonical sheaf of [X/G], the condition that ϱ is a crepant resolution amounts to saying that (1.1) is a flop of orbifolds. In many situations, a crepant resolution of X/G does not exist. However, given a resolution ϱ:Y˜→X, the DK-Hypothesis still predicts the behaviour of the categories Db(Y˜) and DGb(X) if ϱ*KX/G≥KY˜ or ϱ*KX/G≤KY˜. Another related idea is that even though a crepant resolution does not exist in general, there should always be a categorical crepant resolution of Db(X/G), see [35]. The hope is to find such a categorical resolution as an admissible subcategory of the derived category Db(Y˜) of a geometric resolution. Besides dimensions 2 and 3, one of the most studied testing grounds for the above, and related, ideas is the isolated quotient singularity An/μm. Here, the cyclic group μm of order m acts on the affine space by multiplication with a primitive mth root of unity ζ. In this paper, we consider the following straight-forward generalization of this set-up. Namely, let X be a quasi-projective smooth complex variety acted upon by the finite cyclic group μm. We assume that only 1 and μm occur as the isotropy groups of the action and write S≔𝖥𝗂𝗑(μm)⊂X for the fixed point locus. Fix a generator g of μm and assume that g acts on the normal bundle N≔NS/X by multiplication with some fixed primitive mth root of unity ζ. Then the blow-up Y˜→X/μm with centre S is a resolution of singularities; see Section 3 for further details. There are four particular cases we have in mind: X=An with the diagonal action of any μm. X=Z2, where Z is a smooth projective variety of arbitrary dimension, and μ2=S2 acts by permuting the factors. Then Y˜≅Z[2], the Hilbert scheme of two points. X is an abelian variety, μ2 acts by ±1. In this case, Y˜ is known as the Kummer resolution. X→Y=X/μm is a cyclic covering of a smooth variety Y, branched over a divisor. Here, n=1 and X˜=X, Y˜=Y. This case has been studied in [32]. First, we prove the following result in Section 3.1. This is probably well known to experts, but we could not find it in the literature. Write G≔μm. Proposition A (=Proposition 3.2) The resolution obtained by blowing up the fixed point locus in X/Gis isomorphic to the G-Hilbert scheme: Y˜≅HilbG(X). We set n≔codim(S↪X) and find the following dichotomy, in accordance with the DK-Hypothesis. We keep the notation from diagram (1.1). In particular, for n=m, we obtain new instances of BKR-style derived equivalences between orbifold and resolution. Theorem B (=Theorem4.1) The functor Φ≔p*q*:Db(Y˜)→DGb(X)is fully faithful for m≥nand an equivalence for m=n. For m>n, there is a semi-orthogonal decomposition of DGb(X)consisting of Φ(Db(Y˜))and m−npieces equivalent to Db(S). The functor Ψ≔q*p*:DGb(X)→Db(Y˜)is fully faithful for n≥mand an equivalence for n=m. For n>m, there is a semi-orthogonal decomposition of Db(Y˜)consisting of Ψ(DGb(X))and n−mpieces equivalent to Db(S). For a more exact statement with an explicit description of the embeddings of the Db(S) components into Db(Y˜) and DGb(X), see Section 4. In particular, for m>n, the push-forward a*:Db(S)→DGb(X) along the embedding a:S↪X of the fixed point divisor is fully faithful. In the basic affine case (a), the result of the theorem is also stated in [29, Example 4] and there are related results in the more general toroidal case in [29, Section 3]. Proofs, in the basic case, are given in [1, Section 4] for n≥m and in [27] for n=2. If n=1, the quotient is already smooth and we have Y˜=X/G — here the semi-orthogonal decomposition categorifies the natural decomposition of the orbifold cohomology; compare [43]. The n=1 case is also proven in [36, Theorem 3.3.2]. We study the case m=n, where Φ and Ψ are equivalences, in more detail. On both sides of the equivalence, we have distinguished line bundles. The line bundle OY˜(Z) on Y˜, corresponding to the exceptional divisor, admits an mth root L. On [X/G], there are twists of the trivial line bundle by the group characters OX⊗χi. For i=−m+1,…,−1,0, we have Ψ(OX⊗χi)≅Li. Furthermore, we see that the functors Db(S)→Db(Y˜) and Db(S)→DGb(X), which define fully faithful embeddings in the n>m and m<n cases, respectively, become spherical for m=n, and hence induce twist autoequivalences; see Section 2.8 for details on spherical functors and twists. We show that the tensor products by the distinguished line bundles correspond to the spherical twists under the equivalences Ψ and Φ. In particular, one part of Theorem 4.26 is the following formula. Theorem C There is an isomorphism Ψ−1(Ψ(̲)⊗L−1)≅𝖳a*−1(̲⊗χ−1)of autoequivalences of DGb(X)where the inverse spherical twist 𝖳a*−1is defined by the exact triangle of functors  𝖳a*−1→id→a*(a*(̲)G)→. The tensor powers of the line bundle L form a strong generator of Db(Y˜), thus Theorem C, at least theoretically, completely describes the tensor product   ̲⊗^̲≔Ψ−1(Ψ(̲)⊗Ψ(̲)):DGb(X)×DGb(X)→DGb(X)induced by Ψ on DGb(X). There is related unpublished work on induced tensor products under the McKay correspondence in dimensions 2 and 3 by T. Abdelgadir, A. Craw, J. Karmazyn, and A. King. In Corollary 4.27, we also get a formula which can be seen as a stacky instance of the ‘flop–flop = twist’ principle as discussed in [4]. In Section 5, we introduce a general candidate for a weakly crepant categorical resolution (see [35] or Section 5.1 for this notion), namely the weakly crepant neighbourhood WCN(ϱ)⊂Db(Y˜), inside the derived category of a given resolution ϱ:Y˜→Y of a rational Gorenstein variety Y. The idea is pretty simple: by Grothendieck duality, there is a canonical section s:OY˜→Oϱ of the relative dualizing sheaf, and this induces a morphism of Fourier–Mukai transforms t≔ϱ*(̲⊗s):ϱ*→ϱ!. Set ϱ+≔cone(t) and WCN(ϱ)≔ker(ϱ+). Then, by the very construction, we have ϱ*∣WCN(ϱ)≅ϱ!∣WCN(ϱ) which amounts to the notion of categorical weak crepancy. There is one remaining condition needed to ensure that WCN(ϱ) is a categorical weakly crepant resolution: whether it is actually a smooth category; this holds as soon as it is an admissible subcategory of Db(Y˜) which means that its inclusion has adjoints. We prove that, in the Gorenstein case m∣n of our set-up of cyclic quotients, WCN(ϱ)⊂Db(Y˜) is an admissible subcategory; see Theorem 5.4. In Section 5.4, we observe that there are various weakly crepant resolutions inside Db(Y˜). However, a strongly crepant categorical resolution inside Db(Y˜) is unique, as we show in Proposition 5.8. Our concept of weakly crepant neighbourhoods was motivated by the idea that some non-CY objects possess ‘CY neighbourhoods’ (a construction akin to the spherical subcategories of spherelike objects in [25]), that is full subcategories in which they become Calabi–Yau. This relationship is explained in Section 5.5. In the final Section 6, we construct Bridgeland stability conditions on Kummer three-folds as an application of our results, see Corollary 6.2. Conventions. We work over the complex numbers. All functors are assumed to be derived. We write Hi(E) for the ith cohomology object of a complex E∈Db(Z) and H*(E) for the complex ⊕iHi(Z,E)[−i]. If a functor Φ has a left/right adjoint, they are denoted ΦL, ΦR. There are a number of spaces, maps and functors repeatedly used in this text. For the convenience of the reader, we collect our notation as the following:   G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα    G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα    G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα    G=μm=⟨g⟩actsonsmoothX  S=𝖥𝗂𝗑(G)⊂X,n=dim(X)−dim(S)  N=NS/Xwithg∣N=ζ·idN  χ:G→C*,χ(g)=ζ−1  LY˜∈Pic(Y˜)withLX˜m=OY˜(Z)  LX˜=OX˜(Z)∈PicG(X˜)withtrivial  actiononLX˜∣Z=OZ(Z)=Oν(−1)  Φ≔p*◦q*◦𝗍𝗋𝗂𝗏  Ψ≔(−)G◦q*◦p*  Θβ≔i*(ν*(̲)⊗Oν(β))  Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα  2. Preliminaries 2.1. Fourier–Mukai transforms and kernels Recall that given an object E in Db(Z×Z′), where Z and Z′ are smooth and projective, we get an exact functor Db(Z)→Db(Z′), F↦pZ′*(E⊗pZ*F). Such a functor, denoted by 𝖥𝖬E, is called a Fourier–Mukai transform (or FM transform) and E is its kernel. See [26] for a thorough introduction to FM transforms. For example, if Δ:Z→Z×Z is the diagonal map and L is in Pic(Z), then 𝖥𝖬Δ*L(F)=F⊗L. In particular, 𝖥𝖬OΔ is the identity functor. Convention. We will write 𝖬L for the functor 𝖥𝖬Δ*L. The calculus of FM transforms is, of course, not restricted to smooth and projective varieties. Note that f* maps Db(Z) to Db(Z′) as soon as f:Z→Z′ is proper. In order to be able to control the tensor product and pullbacks, one can restrict to perfect complexes. Recall that a complex of sheaves on a quasi-projective variety Z is called perfect if it is locally quasi-isomorphic to a bounded complex of locally free sheaves. The triangulated category of perfect complexes on Z is denoted by Dperf(Z). It is a full subcategory of Db(Z). These two categories coincide if and only if Z is smooth. We will sometimes take cones of morphisms between FM transforms. Of course, one needs to make sure that these cones actually exist. Luckily, if one works with FM transforms, this is not a problem, because the maps between the functors come from the underlying kernels and everything works out, even for (reasonable) schemes which are not necessarily smooth and projective, see [5]. 2.2. Group actions and derived categories Let G be a finite group acting on a smooth variety X. Recall that a G-equivariant coherent sheaf is a pair (F,λg), where F∈Coh(X) and λg:F→∼g*F are isomorphisms satisfying a cocycle condition. The category of G-equivariant coherent sheaves on X is denoted by CohG(X). It is an abelian category. The equivariant-derived category, denoted by DGb(X), is defined as Db(CohG(X)), see, for example [42] for details. Recall that for every subgroup G′⊂G, the restriction functor 𝖱𝖾𝗌:DGb(X)→DG′b(X) has the induction functor 𝖨𝗇𝖽:DG′b(X)→DGb(X) as a left and right adjoint (see for example [42, Section 1.4]). It is given for F∈Db(Z) by   𝖨𝗇𝖽(F)=⨁[g]∈G′⧹Gg*F (2.1)with the G-linearization given by the G′-linearization of F together with appropriate permutations of the summands. If G acts trivially on X, there is also the functor 𝗍𝗋𝗂𝗏:Db(X)→DGb(X) which equips an object with the trivial G-linearization. Its left and right adjoint is the functor (̲)G:DGb(X)→Db(X) of invariants. Given an equivariant morphism f:X→X′ between varieties endowed with G-actions, there are equivariant pushforward and pullback functors, see, for example [42, Section 1.3] for details. We will sometimes write f*G for (̲)G◦f*. It is also well known that the category DGb(X) has a tensor product and the usual formulas, for example the adjunction formula, hold in the equivariant setting. Finally, we need to recall that a character κ of G acts on the equivariant category by twisting the linearization isomorphisms with κ. If F∈DGb(X), we will write F⊗κ for this operation. We will tacitly use that twisting by characters commutes with the equivariant pushforward and pullback functors along G-equivariant maps. 2.3. Semi-orthogonal decompositions References for the following facts are, for example, [12, 13]. Let T be a Hom-finite triangulated category. A semi-orthogonal decomposition of T is a sequence of full triangulated subcategories A1,…,Am such that (a) if Ai∈Ai and Aj∈Aj, then Hom(Ai,Aj[l])=0 for i>j and all l, and (b) the Ai generate T, that is, the smallest triangulated subcategory of T containing all the Ai is already T. We write T=⟨A1,…,Am⟩. If m=2, these conditions boil down to the existence of a functorial exact triangle A2→T→A1 for any object T∈T. A subcategory A of T is right admissible if the embedding functor ι has a right adjoint ιR, left admissible if ι has a left adjoint ιL, and admissible if it is left and right admissible. Given any triangulated subcategory A of T, the full subcategory A⊥⊆T consists of objects T such that Hom(A,T[k])=0 for all A∈A and all k∈Z. If A is right admissible, then T=⟨A⊥,A⟩ is a semi-orthogonal decomposition. Similarly, T=⟨A,⊥A⟩ is a semi-orthogonal decomposition if A is left admissible, where ⊥A is defined in the obvious way. Examples typically arise from so-called exceptional objects. Recall that an object E∈Db(Z) (or any C-linear triangulated category) is called exceptional if Hom(E,E)=C and Hom(E,E[k])=0 for all k≠0. The smallest triangulated subcategory containing E is then equivalent to Db(Spec(C)) and this category, by abuse of notation again denoted by E, is admissible, leading to a semi-orthogonal decomposition Db(Z)=⟨E⊥,E⟩. A sequence of objects E1,…,En is called an exceptional collection if Db(Z)=⟨(E1,…,En)⊥,E1,…,En⟩ and all Ei are exceptional. The collection is called full if (E1…,En)⊥=0. Note that any fully faithful FM transform Φ:Db(X)→Db(X′) gives a semi-orthogonal decomposition Db(X′)=⟨Φ(Db(X))⊥,Φ(Db(X))⟩, because any FM transform has a right and a left adjoint, see [26, Proposition 5.9]. 2.4. Dual semi-orthogonal decompositions Let T be a triangulated category together with a semi-orthogonal decomposition T=⟨A1,…,An⟩ such that all Ai are admissible. Then there is the left-dual semi-orthogonal decomposition T=⟨Bn,…,B1⟩ given by Bi≔⟨A1,…,Ai−1,Ai+1,…,An⟩⊥. There is also a right-dual decomposition, but we will always use the left-dual and refer to it simply as the dual semi-orthogonal decomposition. We summarize the properties of the dual semi-orthogonal decomposition needed later on in the following: Lemma 2.1 Let T=⟨A1,…,An⟩be a semi-orthogonal decomposition with dual semi-orthogonal decomposition T=⟨Bn,…,B1⟩. ⟨A1,…,Ar⟩=⟨Br,…,B1⟩and ⟨A1,…,Ar⟩⊥=⟨Bn,…,Br+1⟩for 1≤r≤n. If ⟨A1,…,An⟩is given by an exceptional collection that is, Ai=⟨Ei⟩, then its dual is also given by an exceptional collection Bi=⟨Fi⟩such that Hom*(Ei,Fj)=δijC[0]. Proof Part (i) is [21, Proposition 2.7(i)]. Part (ii) is then clear.□ An important classical example is the following: Lemma 2.2 There are dual semi-orthogonal decompositions  Db(Pn−1)=⟨O,O(1),…,O(n−1)⟩,Db(Pn−1)=⟨Ωn−1(n−1)[n−1],…,Ω1(1)[1],O⟩. Proof The fact that both sequences are indeed full goes back to Beilinson, see [26, Section 8.3] for an account. The fact that they are dual is classical and follows by a direct computation, for instance using [16, Lemma 2.5].□ The following relative version is the example of dual semi-orthogonal decompositions which we will need throughout the text. Lemma 2.3 Let ν:Z→Sbe a Pn−1-bundle. There is the semi-orthogonal decomposition  Db(Z)=⟨ν*Db(S),ν*Db(S)⊗Oν(1),…,ν*Db(S)⊗Oν(n−1)⟩whose dual decomposition is given by  Db(Z)=⟨ν*Db(S)⊗Ωνn−1(n−1),…,ν*Db(S)⊗Ων1(1),ν*Db(S)⟩. Proof Part (i) is [40, Theorem 2.6]. Part (ii) follows from Lemma 2.2.□ The following consequence will be used in Section 4.5. Corollary 2.4 If m<n, there is the equality of subcategories of Db(Z)  ⟨ν*Db(S)⊗Oν(m−n),…,ν*Db(S)⊗Oν(−1)⟩=⟨ν*Db(S)⊗Ωνn−1(n−1),…,ν*Db(S)⊗Ωνm(m)⟩. Proof Applying Lemma 2.1(i) to the dual decompositions of Lemma 2.3 gives the equalities   ⟨ν*Db(S)⊗Ωνn−1(n−1),…,ν*Db(S)⊗Ωνm(m)⟩=⟨ν*Db(S),…,ν*Db(S)⊗Oν(m−1)⟩⊥=⟨ν*Db(S)⊗Oν(m−n),…,ν*Db(S)⊗Oν(−1)⟩.□ 2.5. Linear functors and linear semi-orthogonal decompositions Let T be a tensor triangulated category, that is a triangulated category with a compatible symmetric monoidal structure. Moreover, let X be a triangulated module category over T, that is there is an exact functor π*:T→X and a tensor product ⊗:T×X→X, that is an assignment π*(A)⊗E functorial in A∈T and E∈X. We will take T=DGperf(Y) for some variety Y with an action by a finite group G. Note that DGperf(Y) has a (derived) tensor product, and it is compatible with G-linearizations. For X, we have several cases in mind. If X is a smooth G-variety X with a G-equivariant morphism π:X→Y, then we take X=DGb(X)=DGperf(X); this is a tensor-triangulated category itself and π* preserves these structures. If Λ is a finitely generated OY-algebra, then let X=Db(Λ) be the bounded derived category of finitely generated right Λ-modules with π*(A)=A⊗OYΛ and π*(A)⊗E=A⊗OYΛ⊗ΛE=A⊗OYE∈X. Note that if Λ is not commutative, then X is not a tensor category. We say that a full triangulated subcategory A⊂X is T-linear (since in our cases we have T=Dperf(Y), we will also speak of Y-linearity) if   π*(A)⊗E∈AforallA∈TandE∈A.We say that a semi-orthogonal decomposition X=⟨A1,…,An⟩ is T-linear, if all the Ai are T-linear subcategories. We call a class of objects S⊂X (left/right) spanning over T if π*T⊗S is a (left/right) spanning class of X in the non-relative sense. Recall that a subset C⊂X is generating if X=⟨C⟩ is the smallest triangulated category closed under direct summands containing C. The subset C⊂X is called generating over T if C⊗π*T generates Db(X). Let X′ be a further tensor triangulated category together with a tensor-triangulated functor π′*:T→X′. We say that an exact functor Φ:X→X′ is T-linear if there are functorial isomorphisms   Φ(π*(A)⊗E)≅π′*(A)⊗Φ(E)forallA∈TandE∈X.The verification of the following lemma is straight-forward. Lemma 2.5 If Φ:X→X′is T-linear, then Φ(X)is a T-linear subcategory of X′. Let A⊂Db(Y)be a T-linear (left/right) admissible subcategory. Then the essential image of Ais Db(Y)if and only if Acontains a (left/right) spanning class over T. For the following, we consider the case that X=Db(X) for some smooth variety X together with a proper morphism π:X→Y. Lemma 2.6 Let A,B⊂Db(X)be Y-linear full subcategories. Then  A⊂B⊥⟺π*Hom(B,A)=0∀A∈A,B∈B. Proof The direction ⟸ follows immediately from Hom*(B,A)≅Γ(Y,π*Hom(B,A)); recall that all our functors are the derived versions. Conversely, assume that there are A∈A and B∈B such that π*Hom(B,A)≠0. Since Dperf(Y) spans D(QCoh(Y)), this implies that there is an E∈Dperf(Y) such that   0≠Hom*(E,π*Hom(B,A))≅Γ(Y,π*Hom(B,A)⊗E∨)≅Γ(Y,π*(Hom(B,A)⊗π*E∨))≅Γ(Y,π*Hom(B⊗π*E,A))≅Hom*(B⊗π*E,A).By the Y-linearity, we have B⊗π*E∈B and hence A⊂B⊥.□ 2.6. Relative Fourier–Mukai transforms Let π:X→Y and π′:X′→Y be proper morphisms of varieties with X and X′ being smooth. We denote the closed embedding of the fibre product into the product by i:X×YX′↪X×X′. We call Φ:Db(X)→Db(X′) a relative FM transform if Φ=𝖥𝖬ι* for some object ∈Db(X×YX). It is a standard computation that a relative FM transform is linear over Y, with respect to the pullbacks π* and π′*. Furthermore, we have Φ≅p*(q*(̲)⊗), where p and q are the projections of the fibre diagram   (2.2) The right adjoint of Φ is given by ΦR≔q*(p!(̲)⊗∨):Db(X′)→Db(X). We also have the following slightly stronger statement which one could call relative adjointness. Lemma 2.7 For E∈Db(X)and F∈Db(X′), there are functorial isomorphisms  π*′Hom(Φ(E),F)≅π*Hom(E,ΦR(F)). Proof This follows by Grothendieck duality, commutativity of (2.2) and projection formula:   π*′Hom(Φ(E),F)≅π*′Hom(p*(q*E⊗),F)≅π*′p*Hom(q*E⊗,p!F)≅π*q*Hom(q*E,p!F⊗∨)≅π*Hom(E,q*(p!F⊗∨))≅π*Hom(E,ΦR(F)).□ For E,F∈Db(X), using the isomorphism of the previous lemma, we can construct a natural morphism Φ˜:π*Hom(E,F)→π*′Hom(Φ(E),Φ(F)) as the composition   Φ˜=Φ˜(E,F):π*Hom(E,F)→π*′Hom(E,ΦRΦ(F))≅π*′Hom(Φ(E),Φ(F)), (2.3)where the first morphism is induced by the unit of adjunction F→ΦRΦ(F). Note that taking global sections gives back the functor Φ on morphisms, that is Φ=Γ(Y,Φ˜) as maps   Hom*(E,F)≅Γ(Y,π*Hom(E,F))→Γ(Y,π*′Hom(Φ(E),Φ(F)))≅Hom*(Φ(E),Φ(F)).More generally, Φ induces functors for open subsets U⊆Y,   ΦU:Db(W)→Db(W′),whereW=π−1(U)⊆XandW′=π′−1(U)⊆X′,given by restricting the FM kernel ι* to W×W′ and we have ΦU=Γ(U,Φ˜). From this, we see that Φ˜ is compatible with composition which means that the following diagram, for E,F,G∈Db(X), commutes   (2.4) 2.7. Relative tilting bundles Let π:X→Y be a proper morphism of varieties and let X be smooth. Later on, X and Y will have G-actions, and Db(X) will be replaced by DGb(X). We say that V∈Coh(X) is a relative tilting sheaf if ΛV≔Λ≔π*Hom(V,V) is cohomologically concentrated in degree 0 and V is a spanning class over Y. For a more general theory of relative tilting bundles, see [8]. Note that Λ is a finitely generated OY-algebra. We denote the bounded derived category of coherent right modules over Λ by Db(Λ). It is a triangulated module category over Dperf(Y) via π*A=A⊗OYΛ, and Λ is a relative generator. In particular, for A∈Db(X) and M∈Db(Λ), the tensor product A⊗M is over the base OY. The functor π*Hom(V,̲):Coh(X)→Coh(Y) factorizes over Coh(Λ). Since it is left exact, we can consider its right-derived functor π*Hom(V,̲):Db(X)→Db(Λ). This yields a relative tilting equivalence: Proposition 2.8 Let V∈Db(X)be a relative tilting sheaf over Y. Then Vgenerates Db(X)over Y, and the following functor defines a Y-linear exact equivalence:   tV≔π*Hom(V,̲):Db(X)→∼Db(Λ). Proof The Y-linearity of tV is due to the projection formula   tV(π*A⊗E)=π*(π*A⊗Hom(V,E))≅A⊗π*Hom(V,E)=A⊗tV(E).Consider the restricted functor tV′:V≔⟨V⊗π*Dperf(Y)⟩→Db(Λ). We show that tV′ is fully faithful, using the adjunctions π*⊣π* and ̲⊗OYΛ⊣𝖥𝗈𝗋, where 𝖥𝗈𝗋:Db(Λ)→Db(Y) is scalar restriction, the projection formula, and the Y-linearity of tV′:   HomOX(π*A⊗V,π*B⊗V)≅HomOX(π*A,π*B⊗Hom(V,V))≅HomOY(A,π*(π*B⊗Hom(V,V)))≅HomOY(A,B⊗Λ)≅HomΛ(A⊗Λ,B⊗Λ)≅HomΛ(tV′(π*A⊗V),tV′(π*B⊗V)).Since objects of type π*A⊗V generate V, this shows that tV′ is fully faithful. We have tV′(V)=Λ. Since Λ is a relative generator, hence a relative spanning class, of Db(Λ), we get an equivalence V≅Db(Λ), see Lemma 2.5. We now claim that the inclusion V↪Db(X) has a right adjoint, namely   tV′−1tV:Db(X)→Db(Λ)→V.For this, take A∈Dperf(Y), F∈Db(X) and compute   HomOX(π*A⊗V,F)≅HomOY(A,π*Hom(V,F))≅HomΛ(A⊗Λ,tV(F))≅HomV(tV′−1(A⊗Λ),tV′−1tV(F))≅HomV(π*A⊗V,tV′−1tV(F)),where we use the projection formula, the adjunction Λ⊗OY̲⊣𝖥𝗈𝗋, the fact that tV′−1 is an equivalence, hence fully faithful, and the Y-linearity of tV′−1. Since the right-admissible Y-linear subcategory V⊂Db(X) contains the relative spanning class V, we get V=Db(X) by Lemma 2.5. This shows that V is a relative generator and that tV=tV′ is an equivalence.□ Let π′:X′→Y be a second proper morphism and let Φ:Db(X)→∼Db(X′) be a relative FM transform. Lemma 2.9 If  Φ˜Λ≔Φ˜(V,V):ΛV=π*Hom(V,V)→π*′Hom(Φ(V),Φ(V))=ΛΦ(V)is an isomorphism, then the following diagram of functors commutes:   (2.5) Proof We first show that Φ˜(V,E):tV(E)→tΦ(V)(Φ(E)) is an isomorphism in Db(Y) for every E∈D(X). Assume first that there is an exact triangle π*A⊗V→E→π*B⊗V for some A,B∈Dperf(Y) and consider the induced morphism of triangles   The outer vertical arrows are isomorphisms because they decompose as   π*Hom(V,V⊗π*A)→∼π*Hom(V,V)⊗A→Φ˜Λ∼π*′Hom(Φ(V),Φ(V))⊗A→∼π*′Hom(Φ(V),Φ(V)⊗π′*A)→∼π*′Hom(Φ(V),Φ(V⊗π*A)).Therefore, the middle vertical arrow is an isomorphism as well. Since V is a relative generator, we can show that Φ˜(V,E) is an isomorphism for arbitrary E∈Db(X) by repeating the above argument. Using the commutativity of (2.4) with E plugged in for G and V plugged in for all other arguments, we see that Φ˜(V,E) induces an ΛΦ(V)-linear isomorphism π*Hom(V,E)⊗ΛVΛΦ(V)→∼π*′Hom(Φ(V),Φ(E)).□ Lemma 2.10 The functor Φis fully faithful if and only if Φ˜Λ:ΛV→ΛΦ(V)is an isomorphism. Proof If Φ is fully faithful, the unit id→ΦRΦ is an isomorphism. Hence, Φ˜Λ is an isomorphism, see (2.3). Conversely, let Φ˜Λ be an equivalence. By Lemma 2.9, we get a commutative diagram   In this diagram, the horizontal functors are tilting equivalences. The right-hand vertical functor is an equivalence, too, by assumption on Φ˜Λ. Hence, Φ:Db(X)→⟨Φ(V)⟩ is an equivalence, which implies that Φ:Db(X)→Db(X′) is fully faithful.□ Lemma 2.11 Let V∈Db(X)be a relative tilting sheaf, Φ:Db(X)→∼Db(X)a relative FM autoequivalence, and ν:V→∼Φ(V)an isomorphism such that  Φ˜Λ=ν◦̲◦ν−1:π*Hom(V,V)→π*Hom(Φ(V),Φ(V)),that is ΦU(φ)◦ν=ν◦φfor all open subsets U⊂Yand φ∈ΛV(U). Then there exists an isomorphism of functors id→∼Φrestricting to ν. Proof We claim that, under our assumptions, the following diagram of functors commutes:   (2.6) We construct a natural isomorphism η:tΦV→∼tV⊗ΛVΛΦV as follows. For E∈Db(X), there is a natural OY-linear isomorphism π*Hom(Φ(V),E)→∼π*Hom(V,E)⊗ΛVΛΦ(V) given by f↦fν⊗1; the inverse map is g⊗1↦gν−1. This map is linear over ΛΦV because, for a local section λ∈π*Hom(Φ(V),Φ(V)), we have by our assumption, setting φ=Φ˜−1(λ):   η(fλ)=fλν⊗1=fνΦ˜−1(λ)⊗1=fν⊗λ=(fν⊗1)λ.Comparing the diagrams (2.6) and (2.5) shows that Φ≅id.□ Corollary 2.12 Let V∈Db(X)be a relative tilting sheaf, Φ1,Φ2:Db(X)→∼Db(X′)relative FM equivalences, and ν:Φ1(V)→∼Φ2(V)an isomorphism such that Φ2,U(φ)◦ν=ν◦Φ1,U(φ)for all φ∈ΛV(U)and U⊂Yopen. Then there exists a isomorphism of functors Φ1→∼Φ2restricting to ν. Moreover, if V=L1⊕⋯⊕Lkdecomposes as a direct sum, then the above condition is satisfied by specifying isomorphisms νi:Φ1(Li)→∼Φ2(Li)inducing functor isomorphisms Φ1,U→∼Φ2,Uon the full finite subcategory {L1∣U,…,Lk∣U}of Db(π−1(U)). Remark 2.13 All the results of this subsection remain valid in an equivariant setting, where a finite group G acts on X and π:X→Y is G-invariant. Then the correct sheaf of OY-algebras is ΛV=π*GHom(V,V). 2.8. Spherical functors An exact functor φ:C→D between triangulated categories is called spherical if it admits both adjoints, if the cone endofunctor F[1]≔cone(idC→φRφ) is an autoequivalence of C, and if the canonical functor morphism φR→FφL[1] is an isomorphism. A spherical functor is called split if the triangle defining F is split. The proper framework for dealing with functorial cones are dg-categories; the triangulated categories in this article are of geometric nature, and we can use Fourier–Mukai transforms. See [6] for proofs in great generality. Given a spherical functor φ:C→D, the cone of the natural transformation 𝖳=𝖳φ≔cone(φφR→idD) is called the twist around φ; it is an autoequivalence of D. The following lemma follows immediately from the definition, since an equivalence has its inverse functor as both left and right adjoint. Lemma 2.14 Let φ:C→Dbe a spherical functor and let δ:D→D′be an equivalence. Then δ◦φ:C→D′is a spherical functor with associated twist functor 𝖳δφ=δ𝖳φδ−1. 3. The geometric setup Let X be a smooth quasi-projective variety together with an action of a finite group G. Let S≔fix(G) be the locus of fixed points. Then S⊂X is a closed subset, which is automatically smooth since, locally in the analytic topology, the action can be linearized by Cartan’s lemma, see [17, Lemma 2]. Also note that X/G has rational singularities, like any quotient singularity over C [31]. Condition 3.1 We make strong assumptions on the group action: G≅μm is a cyclic group. Fix a generator g∈G. Only the trivial isotropic groups 1 and μm occur. The generator g acts on the normal bundle N≔NS/X by multiplication with some fixed primitive mth root of unity ζ. Condition (ii) obviously holds if m is prime. Condition (iii) can be rephrased: there is a splitting TX∣S=TS⊕NS/X because TS is the subsheaf of G-invariants of TX∣S and we work over characteristic 0. By (iii), this is even the splitting into the eigenbundles corresponding to the eigenvalues 1 and ζ. We denote by χ:G→C* the character with χ(g)=ζ−1. Hence, we can reformulate (iii) by saying that G acts on N via χ−1. From these assumptions, we deduce the following commutative diagram:   (3.1)where a, b, i, and j are closed embeddings and π is the quotient morphism. The G-action on X lifts to a G-action on X˜. Since, by assumption, G acts diagonally on N, it acts trivially on the exceptional divisor Z=P(N). In particular, the fixed point locus of the G-action on X˜ is a divisor. Hence, the quotient variety Y˜ is again smooth and the quotient morphism q is flat due to the Chevalley–Shephard–Todd theorem. Since the composition π◦p is G-invariant, it induces the morphism ϱ:Y˜→Y which is easily seen to be birational, hence a resolution of singularities. The preimage ϱ−1(S) of the singular locus is a divisor in Y˜. Hence, by the universal property of the blow-up, we get a morphism Y˜→BlSY which is easily seen to be an isomorphism. 3.1. The resolution as a moduli space of G-clusters The result of this section might be of independent interest. Let X be a smooth quasi-projective variety and G a finite group acting on X. A G-cluster on X is a closed zero-dimensional G-invariant subscheme W⊂X such that the G-representation H0(W,OW) is isomorphic to the regular representation of G. There is a fine moduli space HilbG(X) of G-clusters, called the G-Hilbert scheme. It is equipped with the equivariant Hilbert–Chow morphism τ:HilbG(X)→X/G,W↦𝗌𝗎𝗉𝗉(W), mapping G-clusters to their underlying G-orbits. Proposition 3.2 Let Gbe a finite cyclic group acting on Xsuch that all isotropy groups are either 1 or G, and such that Gacts on the normal bundle N𝖥𝗂𝗑(G)/Xby scalars which means that Condition3.1is satisfied. Then there is an isomorphism  φ:Y˜→≅HilbG(X)withτ◦φ=ϱ. Proof We use the notation from (3.1). One can identify X˜ with the reduced fibre product (Y˜×YX)red which gives a canonical embedding X˜⊂Y˜×X. Under this embedding, the generic fibre of q is a reduced free G-orbit of the action on X. In particular, it is a G-cluster. By the flatness of q, every fibre is a G-cluster and we get the classifying morphism φ:Y˜→HilbG(X) which is easily seen to satisfy τ◦φ=ϱ. Let s∈S and z∈Z with ν(z)∈s. Let ℓ⊂N(s) be the line corresponding to z. Then, one can check that the tangent space of the G-cluster q−1(i(z))⊂X is exactly ℓ. Hence, the G-clusters in the family X˜ are all different so that the classifying morphism φ is injective. For the bijectivity of φ, it is only left to show that the G-orbits supported on a given fixed point s∈S are parametrized by P(N(s)). Let ξ⊂X be such a G-cluster. In particular, ξ is a length m=∣G∣ subscheme concentrated in s, and hence can be identified with an ideal I⊂OX,s/mX,sm of codimension m. By Cartan’s lemma, the G-action on X can be linearized in an analytic neighbourhood of s. Hence, there is an G-equivariant isomorphism   OX,s/mX,sm≅C[x1,…,xk,y1,…,yn]/(x1,…,xk,y1,…,yn)m≕R,where G acts trivially on the xi and by multiplication by ζ−1 on the yi. Furthermore, n=rankNS/X and k=rankTS=dimX−n. By assumption, O(ξ) is the regular μm-representation. In other words,   O(ξ)≅R/I≅χ0⊕χ⊕⋯⊕χm−1, (3.2)where χ is the character given by multiplication by ζ−1. In particular, R/I has a one-dimensional subspace of invariants. It follows that every xi is congruent to a constant polynomial modulo I. Hence, we can make an identification O(ξ)≅R′/J where J is a G-invariant ideal in R′=C[y1,…,yk]/nm where n=(y1,…,yn). The decomposition of the G-representation R′ into eigenspaces is exactly the decomposition into the spaces of homogeneous polynomials. Hence, an ideal J⊂R′ is G-invariant if and only if it is homogeneous. Furthermore, (3.2) implies that   dimC(ni/(J∩ni+ni+1))=1foralli=0,…,m−1,which means that ξ is curvilinear. In summary, ξ can be identified with a homogeneous curvilinear ideal J in R′. The choice of such a J corresponds to a point in P((n/n2)∨)≅P(N(s)), see [22, Remark 2.1.7]. Hence, φ is a bijection and we only need to show that HilbG(X) is smooth. The smoothness in points representing free orbits is clear since the G-Hilbert–Chow morphism is an isomorphism on the locus of these points. So it is sufficient to show that   HomDGb(X)1(Oξ,Oξ)=dimX=n+kfor a G-cluster ξ supported on a fixed point. Following the above arguments, we have   HomDGb(X)*(Oξ,Oξ)≅HomDGb(Ak×An)*(Oξ′,Oξ′),where G acts trivially on Ak and by multiplication by ζ on An. Furthermore, by a transformation of coordinates, we may assume that   ξ′=V(x1,…,xk,y1m,y2,…,yn)⊂Ak×An.We have Oξ′≅O0⊠Oη, where   η=V(y1m,y2,…,yn)⊂An.By Künneth formula, we get   HomDGb(Ak×An)*(Oξ′,Oξ′)≅HomDb(Ak)*(O0,O0)⊗HomDGb(An)*(Oη,Oη)≅∧*(Ck)⊗HomDGb(An)*(Oη,Oη).Furthermore, HomDGb(An)0(Oη,Oη)≅H0(Oη)G≅C. Hence, it is sufficient to show that HomDGb(An)1(Oη,Oη)≅Cn. Note that η is contained in the line ℓ=V(y2,…,yn). On ℓ, we have the Koszul resolution   0→Oℓ→·y1mOℓ→Oη→0.Using this, we compute   HomDb(ℓ)*(Oη,Oη)≅Oη[0]⊕Oη[−1].Note that the normal bundle of ℓ, as an equivariant bundle, is given by Nℓ/An≅(Oℓ⊗χ−1)⊕n−1. By [2, Theorem 1.4], we have   HomDb(An)*(Oη,Oη)≅HomDb(ℓ)*(Oη,Oη⊗∧*Nℓ/An)≅HomDb(ℓ)*(Oη,Oη)⊗∧*((Oℓ⊗χ−1)⊕n−1).Evaluating in degree 1 gives   HomDb(An)1(Oη,Oη)≅Oη⊕(Oη⊗χ−1)⊕n−1.Since, as a G-representation, Oη≅χ0⊕χ1⊕…⊕χm−1, we get an n-dimensional space of invariants   HomDGb(An)1(Oη,Oη)≅HomDb(An)1(Oη,Oη)G≅Cn.□ The following lemma is needed later in Section 4.4, but its proof fits better into this section. Lemma 3.3 Assume that m=∣G∣≥n=codim(S↪X). Let ξ1,ξ2⊂Xbe two different G-clusters supported on the same point s∈S. Then HomDGb(X)*(Oξ1,Oξ2)=0. Proof By the same arguments as in the proof of the previous proposition we can reduce to the claim that   HomDGb(An)*(Oη1,Oη2)=0,where η1=V(y1m,y2,…,yn) and η2=V(y1,y2m,y3,…,yn). Set ℓ1=V(y2,…,yn), ℓ2=V(y1,y3,…,yn), E=⟨ℓ1,ℓ2⟩=V(y3,…,yn) and consider the diagram of closed embeddings   where Nt≅(OE⊗χ−1)⊕n−2. By [33, Lemma 3.3] (alternatively, one may consult [23] or [3] for more general results on derived intersection theory), we get   HomDb(An)*(Oη1,Oη2)=HomDb(An)*(ι1*Oη1,ι2*Oη2)≅HomDb(ℓ2)*(ι2*ι1*Oη1,Oη2)≅HomDb(ℓ2)*(u*v*Oη1,Oη2)⊗∧*Nt∣ℓ2≅HomDb(ℓ2)*(u*v*Oη1,Oη2)⊗∧*(Oℓ2⊗χ−1)⊕n−2. (3.3)We consider the Koszul resolution 0→Oℓ1→y1mOℓ1→Oη1→0 of Oη1. Note that this is an equivariant resolution when we consider Oℓ1 equipped with the canonical linearization since y1m is a G-invariant function. Applying u*v*, we get an equivariant isomorphism   u*v*Oη1≅O0⊕O0[1]. (3.4)Similarly, we have the equivariant Koszul resolution 0→Oℓ⊗χ→·yOℓ→O0→0 of O0, where we set ℓ≔ℓ2 and y≔y2. Applying Hom(̲,Oη2) to the resolution, we get   0→C[y]/ym⊗→·yC[y]/ym⊗χ−1→0and taking cohomology yields   HomDb(ℓ2)*(O0,Oη2)≅C⟨ym−1⟩[0]⊕C⟨1⟩⊗χ−1[−1]≅O0⊗χ−1[0]⊕O0⊗χ−1[−1]. (3.5) Plugging (3.4) and (3.5) into (3.3) gives   HomDb(An)*(Oη1,Oη2)≅(O0⊗χ−1[0]⊕O0⊕2⊗χ−1[−1]⊕O0⊗χ−1[−2])⊗∧*(χ−1)⊕n−2.The irreducible representations occurring are χ−1,χ−2,…,χ−(n−1), hence the invariants vanish (recall that m≥n).□ 4. Proof of the main result In this section, we will study the derived categories Db(Y˜) and DGb(X) in the setup described in the previous section. In particular, we will prove Theorems B and C. We set n=codim(S↪X) and m=∣G∣, in other words G=μm. We consider, for α∈Z/mZ and β∈Z, the exact functors   Φ≔p*◦q*◦𝗍𝗋𝗂𝗏:Db(Y˜)→DGb(X)Ψ≔(−)G◦q*◦p*:DGb(X)→Db(Y˜)Θβ≔i*(ν*(̲)⊗Oν(β)):Db(S)→Db(Y˜)Ξα≔(a*◦𝗍𝗋𝗂𝗏)⊗χα:Db(S)→DGb(X).With this notation, the precise version of Theorem B is Theorem 4.1 The functor Φis fully faithful for m≥nand an equivalence for m=n. For m>n, all the functors Ξαare fully faithful and there is a semi-orthogonal decomposition  DGb(X)=⟨Ξn−m(Db(S)),…,Ξ−1(Db(S)),Φ(Db(Y˜))⟩. The functor Ψis fully faithful for n≥mand an equivalence for n=m. For n>m, all the functors Θβare fully faithful and there is a semi-orthogonal decomposition  Db(Y˜)=⟨Θm−n(Db(S)),…,Θ−1(Db(S)),Ψ(DGb(X))⟩. Remark 4.2 We will see later in Lemma 4.14 that KY˜≤ϱ*KY for m≥n and KY˜≥ϱ*KY for n≥m. Hence, Theorem 4.1 is in accordance with the DK-Hypothesis as described in the introduction. For the proof, we first need some more preparations. 4.1. Generators and linearity Lemma 4.3 The bundle V≔OX⊗C[G]=OX⊗(χ0⊕⋯⊕χm−1)is a relative tilting sheaf for DGb(X)over Dperf(Y). Proof If L∈Pic(Y)⊂Dperf(Y) is an ample line bundle, then so is π*(L). Hence, Db(X) has a generator of the form E≔π*(OY⊕L⊕⋯⊕L⊗k) for some k≫0; see [41]. In particular, E is a spanning class of Db(X). Using the adjunction 𝖱𝖾𝗌⊣𝖨𝗇𝖽⊣𝖱𝖾𝗌, it follows that 𝖨𝗇𝖽(E)≅E⊕E⊗χ⊕⋯⊕E⊗χm−1 is a spanning class of DGb(X). Hence, V=𝖨𝗇𝖽OX is a relative spanning class of DGb(X) over Dperf(Y). Since V is a vector bundle, so is Hom(V,V)=V∨⊗V. The map π is finite, hence π* is exact (does not need to be derived). Finally, taking G-invariants is exact because we work in characteristic 0. Altogether, π*GHom(V,V) is a sheaf concentrated in degree 0.□ Lemma 4.4 The functors Φand Ψ, and for all α,β∈Zthe subcategories  Ξα(Db(S))=a*(Db(S))⊗χα⊂DGb(X)andΘβ(Db(S))=i*ν*Db(S)⊗OY˜(β)⊂Db(Y˜)are Y-linear for π*𝗍𝗋𝗂𝗏:Dperf(Y)→DGb(X)and ϱ*:Dperf(Y)→Db(Y˜), respectively. Proof We first show that Φ is Y-linear. Recall that in our setup this means   Φ(ϱ*(E)⊗F)≅π*𝗍𝗋𝗂𝗏(E)⊗Φ(F)for any E∈Dperf(Y) and F∈Db(Y˜). But this holds, since   π*𝗍𝗋𝗂𝗏(E)⊗Φ(F)≅π*𝗍𝗋𝗂𝗏(E)⊗p*q*𝗍𝗋𝗂𝗏(F)≅p*(p*π*𝗍𝗋𝗂𝗏(E)⊗q*𝗍𝗋𝗂𝗏(F))≅p*(q*ϱ*𝗍𝗋𝗂𝗏(E)⊗q*𝗍𝗋𝗂𝗏(F))≅p*q*𝗍𝗋𝗂𝗏(ϱ*(E)⊗F).The proof that Ψ is Y-linear is similar and is left to the reader. The Y-linearity of the image categories follows from Lemma 2.5(i).□ Lemma 4.5 The set of sheaves S≔{OY˜}∪{is*Ωr(r)∣s∈S,r=0,…,n−1}forms a spanning class of Db(Y˜)over Y, where is:Pn−1≅ϱ−1(s)↪Y˜denotes the fibre embedding. Proof We need to show that Sˆ≔ϱ*Dperf(Y)⊗S is a spanning class of Db(Y˜). Let y˜∈Y˜⧹Z. Then y=ϱ(y˜) is a smooth point of Y. Hence, Oy∈Dperf(Y) and Oy˜∈ϱ*Dperf(Y)=ϱ*Dperf(Y)⊗OY˜⊂Sˆ. Thus, an object E∈Db(Y˜) with 𝗌𝗎𝗉𝗉E∩(Y˜⧹Z)≠∅ satisfies Hom*(E,Sˆ)≠0≠Hom*(Sˆ,E); see [26, Lemma 3.29]. Let now 0≠E∈Db(Y˜) with 𝗌𝗎𝗉𝗉E⊂Z. Then there exists s∈S such that is*E≠0≠is!E; see again [26, Lemma 3.29]. Since the Ωr(r) form a spanning class of Pn−1, we get by adjunction Hom*(E,S)≠0≠Hom*(S,E).□ 4.2. On the equivariant blow-up Recall that the blow-up morphism q:X˜→X is G-equivariant. Let LX˜∈PicG(X˜) (we will sometimes simply write L instead of LX˜) be the equivariant line bundle OX˜(Z) equipped with the unique linearization whose restriction to Z gives the trivial action on OZ(Z)≅Oν(−1). We consider a point z∈Z with ν(z)=s corresponding to a line ℓ⊂NS/X(s). Then the normal space NZ/X˜(z) can be equivariantly identified with ℓ. It follows by Condition 3.1 that NZ/X˜≅(LX˜⊗χ−1)∣Z as an equivariant bundle. Hence, in CohG(X˜), there is the exact sequence   0→LX˜−1⊗χ→OX˜→OZ→0, (4.1)where both OX˜ and OZ are equipped with the canonical linearization, which is the one given by the trivial action over Z. Lemma 4.6 For ℓ=0,…,n−1we have p*LX˜ℓ=OX⊗χℓ. Proof We have p*OX˜≅OX, both, OX˜ and OX, equipped with the canonical linearizations. Hence, the assertion is true for ℓ=0. By induction, we may assume that p*LX˜ℓ−1≅OX⊗χℓ−1. We tensor (4.1) by LX˜ℓ to get   0→LX˜ℓ−1⊗χ→LX˜ℓ→Oν(−ℓ)→0.Since 0≤ℓ≤n−1, we have p*Oν(−ℓ)=0. Hence, we get an isomorphism   p*(LX˜ℓ)≅p*(LX˜ℓ−1⊗χ)≅p*(LX˜ℓ−1)⊗χ≅OX⊗χℓ−1⊗χ≅OX⊗χℓ.□ Lemma 4.7 The smooth blow-up p:X˜→Xhas G-linearized relative dualizing sheaf  ωp≅LX˜n−1⊗χ1−n∈PicG(X˜). Proof The non-equivariant relative dualizing sheaf of the blow-up is ωp≅OX˜((n−1)Z). Since p is G-equivariant, ωp has a unique linearization such that p!=p*(̲)⊗ωp:DGb(X)→DGb(X˜) is the right-adjoint of p*:DGb(X˜)→DGb(X). We now compute this linearization of ωp. As the equivariant pull-back p* is fully faithful, p!:DGb(X)→DGb(X˜) is fully faithful, too. Hence, adjunction gives an isomorphism of equivariant sheaves, p*ωp≅p*p!OX≅OX. The claim now follows from Lemma 4.6.□ We denote by is:Pn−1≅ϱ−1(s)↪Y˜ the embedding of the fibre of ϱ and by js:Pn−1≅p−1(s)↪X˜ the embedding of the fibre of p over s∈S. Lemma 4.8 For s∈Sand r=0,…,n−1, the cohomoloy sheaves of p*Os∈DGb(X˜)are  H−r(p*Os)≅js*(Ωr(r)⊗χr). Proof It is well known that, for the underlying non-equivariant sheaves, we have H−r(p*Os)≅js*Ωr(r); see [26, Proposition 11.12]. Since the sheaves Ωr(r) are simple, that is End(Ωr(r))=C, we have H−r(p*Os)≅js*(Ωr(r)⊗χαr) for some αr∈Z/mZ. So we only need to show αr=r. Let r∈{0,…,n−1}. We have p!L−r≅p*(L−r+n−1⊗χ1−n) by Lemma 4.7. Since −r+n−1∈{0,…,n−1}, Lemma 4.6 gives p!L−r≅OX⊗χ−r. By adjunction,   C[0]≅HomDGb(X)*(OX⊗χ−r,Os⊗χ−r)≅HomDGb(X˜)*(L−r,p*Os⊗χ−r).By Lemma 2.3, for r≠v, we have   HomDb(X˜)*(OX˜(−rZ),js*Ωv(v))≅HomDb(Pn−1)*(O(r),Ωv(v))=0.Using the spectral sequence in DGb(X˜)  E2u,v=Homu(L−r,Hv(p*Os⊗χ−r))⇒Eu+v=Homu+v(L−r,p*Os⊗χ−r),it follows that   C[0]≅HomDGb(X˜)*(L−r,p*Os⊗χ−r)≅HomDGb(X˜)*(L−r,H−r(p*Os)⊗χ−r)[r]≅(HomDb(Pn−1)*(O(r),Ωr(r))⊗χαr⊗χ−r)G[r]≅(C[−r]⊗χαr−r)G[r],where the last isomorphism is again due to Lemma 2.3. Comparing the first and the last term of the above chain of isomorphisms, we get C≅(χαr−r)G which implies αr=r.□ Corollary 4.9 Let n≥mand ℓ∈{0,…,m−1}. Let λ≥0be the largest integer such that ℓ+λm≤n−1. Then  H*(Ψ(Os⊗χ−ℓ))≅is*(⨁t=0λΩℓ+tm(ℓ+tm)[ℓ+tm]). Proof Since the (non-derived) functor q*G:CohG(X˜)→Coh(Y˜) is exact, we have   H−r(Ψ(Os⊗χ−ℓ))0≅q*G(H−r(p*Os)⊗χ−ℓ)and the claim follows from Lemma 4.8.□ 4.3. On the cyclic cover The morphism q:X˜→Y˜=X˜/G is a cyclic cover branched over the divisor Z. This geometric situation and the derived categories involved are studied in great detail in [32]. However, we will only need the following basic facts, all of which can be found in [32, Section 4.1]. Lemma 4.10 The sheaf of invariants q*G(OX˜⊗χ−1)is a line bundle which we denote LY˜−1∈Pic(Y˜). LY˜m≅OY˜(Z). q*G(OX˜⊗χα)≅LY˜αfor α∈{−m+1,…,0}. q*◦𝗍𝗋𝗂𝗏:Db(Y˜)↪DGb(X˜)is fully faithful, due to q*G(OX˜)≅OY˜. q*(𝗍𝗋𝗂𝗏(LY˜))≅LX˜are isomorphic G-equivariant line bundles. In particular, LY˜∣Z≅LX˜∣Z≅Oν(−1). Corollary 4.11 Ψ(OX⊗χα)≅LY˜αfor α∈{−m+1,…,0}. Lemma 4.12 The relative dualizing sheaf of q:X˜→Y˜=X˜/Gis ωq≅OX˜((m−1)Z). Proof Since the G-action on W≔X˜⧹Z is free, we have ωq∣W≅OW. Hence, ωq≅OX˜(αZ) for some α∈Z. We have Hom(OZ,OX˜)≅OZ(Z)[−1]≅j*Oν(−1)[−1], and hence   i*Oν(−1)[−1]≅q*j*Oν(−1)[−1]≅q*Hom(OZ,OX˜)≅q*Hom(OZ,q*OY˜)≅q*Hom(OZ,q!LY˜−α)byLemma4.10(v)≅Hom(q*OZ,LY˜−α)byGrothendieckduality≅OZ(Z)⊗LY˜−α[−1]≅i*Oν(−m+α)[−1]byLemma4.10(ii)+(vi),and thus we conclude α=m−1.□ Remark 4.13 As an equivariant bundle, we have ωq≅LX˜m−1⊗χ, but we will not use this. Corollary 4.14 We have ωY˜∣Z≅Oν(m−n). Proof We have ωX˜∣Z≅Oν(−n+1); compare Lemma 4.7. Furthermore, ωY˜∣Z≅(q*ωY˜)∣Z. Hence,   Oν(1−m)≅4.12ωq∣Z≅ωX˜∣Z⊗ωY˜∣Z∨≅Oν(1−n)⊗ωY˜∣Z∨.□ 4.4. The case m≥n. Throughout this subsection, let m≥n Proposition 4.15 (i) If m>n, then the functor Ξαis fully faithful for any α∈Z/mZ. (ii) Let m−n≥2and α≠β∈Z/mZ. Then  ΞβRΞα=0⟺α−β∈{n−m+1¯,n−m+2¯,…,−1¯}. Proof Recall that Ξβ=(a*◦𝗍𝗋𝗂𝗏(̲))⊗χβ:Db(S)→DGb(X). Hence, the right-adjoint of Ξβ is given by ΞβR≅(a!(̲)⊗χ−β)G. By [2, Theorem 1.4 and Section 1.20],   ΞβRΞα≅(a!a*(̲)⊗χα−β)G≅((̲)⊗∧*N⊗χα−β)G≅(̲)⊗(∧*N⊗χα−β)G,where by Condition 3.1, the G-action on ∧ℓN is given by χ−ℓ. We see that (∧*N)G≅∧0N[0]≅OS[0];here we use that m>n. This shows that, in the case α=β, we have ΞαRΞα≅id which proves (i). Furthermore, since the characters occurring in ∧*N are χ0, χ−1,…, χ−n, we obtain (ii) from   ΞβRΞα≠0⟺(∧*N⊗χα−β)G≠0⟺0¯∈{α−β¯,α−β−1¯,…,α−β−n¯},thatisΞβRΞα=0⟺α−β∈{n+1¯,…,m−1¯}={n−m+1¯,n−m+2¯,…,−1¯}.□ Corollary 4.16 For m>n, there is a semi-orthogonal decomposition  DGb(X)=⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S)),A⟩,where A=⊥⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S))⟩. Proposition 4.17 The functor Φ=p*q*𝗍𝗋𝗂𝗏:Db(Y˜)→DGb(X)is fully faithful. Proof By [26, Proposition 7.1], we only need to show for x,y∈Y˜ that   HomDGb(X)i(Φ(Ox),Φ(Oy))={Cifx=yandi=00ifx≠yori∉[0,dimX].By Proposition 3.2, Φ(Ox)=Oξ for some G-cluster ξ. Hence,   HomDGb(X)0(Φ(Ox),Φ(Ox))≅H0(Oξ)G≅C.Furthermore, since Φ(Ox) is a sheaf, the complex Hom*(Φ(Ox),Φ(Ox)) is concentrated in degrees 0,…,dim(X). It remains to prove the orthogonality for x≠y. If ϱ(x)≠ϱ(y), the corresponding G-clusters are supported on different orbits. Hence, their structure sheaves are orthogonal. If ϱ(x)=ϱ(y), but x≠y, the orthogonality was shown in Lemma 3.3.□ Lemma 4.18 The functor Φfactors through A. Proof By Corollary 4.16, this statement is equivalent to ΦRΞα=0 for α∈{n−m,…,−1}, where ΦR:DGb(X)→Db(Y˜) is the right adjoint of Φ. Since the composition ΦRΞα is a Fourier–Mukai transform, it is sufficient to test the vanishing on skyscraper sheaves of points, see [34, Section 2.2]. So we have to prove that   ΦRΞα(Os)≅ΦR(Os⊗χα)=0for every s∈S and every α∈{n−m,…,−1}. We have ΦR≅q*Gp!; recall that q*G stands for (̲)G◦q*. By Lemma 4.7 together with Lemma 4.8, we have   H−r(p!Os)≅is*(Ωr(r+1−n)⊗χr+1−n),where the non-vanishing cohomologies occur for r∈{0,…,n−1}. Thus, the linearizations of the cohomologies of p!(Os⊗χα) are given by the characters χγ for γ∈{α+1−n,…,α}. We see that, for α∈{n−m,…,−1}, the trivial character does not occur in H*(p!Os⊗χα). This implies that q*p!(Os⊗χα) has vanishing G-invariants.□ We denote by B⊂DGb(X) the full subcategory generated by the admissible subcategories Ξα(Db(S)) for α∈{n−m,…,−1} and Φ(Db(Y˜)). By the above, these admissible subcategories actually form a semi-orthogonal decomposition   B=⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S)),Φ(Db(Y˜))⟩. Proposition 4.19 We have the (essential) equalities B=DGb(X)and Φ(Db(Y˜))=A. For the proof, we need the following: Lemma 4.20 We have p*LX˜r⊗χ−λ∈Bfor r∈Zand λ∈{0,…,m−n}. Proof By Lemma 4.10, LX˜r≅q*(𝗍𝗋𝗂𝗏(LY˜r)). Hence,   p*(LX˜r)≅p*q*(𝗍𝗋𝗂𝗏(LY˜r))=Φ(LY˜r)∈Φ(Db(Y˜))⊂Bwhich proves the assertion for λ=0. We now proceed by induction over λ. Tensoring (4.1) by LX˜r⊗χ−λ and applying p*, we get the exact triangle   p*LX˜r−1⊗χ−(λ−1)→p*LX˜r⊗χ−λ→p*j*OZ(−r)⊗χ−λ→, (4.2)where OZ(−r) carries the trivial G-action. The first term of the triangle is an object of B by induction. Furthermore, by diagram (3.1), we have p*j*OZ(−r)≅a*ν*OZ(−r). Hence, the third term of (4.2) is an object of a*DGb(S)⊗χ−λ=Ξ−λ(Db(S))⊂B. Thus, also the middle term is an object of B which gives the assertion.□ Proof of Proposition 4.19 The second assertion follows from the first one since, if B=DGb(X) holds, both, Φ(Db(Y˜)) and A, are given by the left-orthogonal complement of   ⟨Ξn−m(Db(S)),Ξn−m+1(Db(S)),…,Ξ−1(Db(S))⟩in DGb(X). The subcategories Ξα(Db(S)) and Φ(Db(Y˜)) of DGb(X) are Y-linear by Lemma 4.4. Hence, for the equality B=DGb(X) it suffices to show that   OX⊗χℓ∈B=⟨Db(S)⊗χn−m,…,Db(S)⊗χ−1,Φ(Db(Y˜))⟩for every ℓ∈Z/mZ; see Lemma 4.3. Combining Lemmas 4.10 and 4.6, we see that   Φ(LY˜ℓ)≅p*(q*𝗍𝗋𝗂𝗏(LY˜)ℓ)≅OX⊗χℓforℓ=0,…,n−1.In particular, OX⊗χℓ∈Φ(Db(Y˜))⊂B for ℓ=0,…,n−1. Setting r=0 in the previous lemma, we find that also OX⊗χℓ for ℓ=n−m,…,−1 is an object of B.□ Combining the results of this subsection gives Theorem 4.1(i). 4.5. The case n≥m. Throughout this subsection, let n≥m Proposition 4.21 Let n>m. Then the functors Θβ:Db(S)→Db(Y˜)are fully faithful for every β∈Zand there is a semi-orthogonal decomposition  Db(Y˜)=⟨C(m−n),C(m−n+1),…,C(−1),D⟩,where C(ℓ)≔Θℓ(Db(S))=i*ν*Db(S)⊗OY˜(ℓ)and  D={E∈Db(Y˜)∣i*E∈⊥⟨ν*Db(S)⊗OY˜(m−n),…,ν*Db(S)⊗OY˜(−1)⟩}={E∈Db(Y˜)∣i*E∈⟨ν*Db(S),…,ν*Db(S)⊗OY˜(m−1)⟩}. Proof This follows from [35, Theorem 1]. However, for convenience, we provide a proof for our special case. By construction, ΘβR≅ν*𝖬Oν(−β)i!. We start with the standard exact triangle of functors id→i!i*→𝖬OZ(Z)[−1]→ (see for example [26, Corollary 11.4]). By Lemma 4.10, OZ(Z)≅Oν(−m), and thus the above triangle induces for any α,β∈Z  ν*𝖬Oν(α−β)ν*→ΘβRΘα→ν*𝖬Oν(α−β−m)ν*→.By projection formula, we can rewrite this as   (̲)⊗ν*Oν(α−β)→ΘβRΘα→(̲)⊗ν*Oν(α−β−m)→.Now, ν*Oν≅OS and ν*Oν(γ)=0 for γ∈{−n+1,…,−1}. Hence, ΘβRΘβ≅id and ΘβRΘα=0 if α−β∈{m−n+1,…,−1}. Therefore, we get a semi-orthogonal decomposition   Db(Y˜)=⟨C(m−n),C(m−n+1),…,C(−1),D⟩.The description of the left-orthogonal D follows by the adjunction i*⊣i*.□ Lemma 4.22 The functor Ψ:DGb(X)→Db(Y˜)factors through D. Proof By Lemma 4.3, the equivariant bundles OX,OX⊗χ,…OX⊗χm−1 generate DGb(X) over Dperf(Y), and therefore so do the bundles OX⊗χ−m+1,…,OX⊗χ−1,OX obtained by twisting with χ1−m. Hence, it is sufficient to prove that Ψ(OX⊗χα)∈D for α∈{−m+1,…,0} as Ψ and D are Y-linear, see Lemma 4.4. Indeed, by Lemma 4.10 we have i*LY˜α=LY˜∣Zα≅Oν(−α), hence   Ψ(OX⊗χα)≅4.11q*G(OX⊗χα)≅LY˜α∈Dforα∈{−m+1,…,0}.□ Proposition 4.23 The functor Ψ:DGb(X)→Db(Y˜)is fully faithful. Proof We first observe that V≔OX⊗C[G]=OX⊗(χ0⊕⋯⊕χm−1) is a relative tilting bundle for DGb(X) over Dperf(Y); see Lemma 4.3. For the fully faithfulness, we follow Lemma 2.10. So we need to show that Ψ induces an isomorphism ΛV=π*GHom(V,V)→∼ϱ*Hom(Ψ(V),Ψ(V)). In turn, it suffices to consider the direct summands of V. Thus, let α,β∈{−m+1,…,0} and compute   π*GHom*(OX⊗χα,OX⊗χβ)≅π*GHom*(OX⊗χα+n−1⊗χ1−n,OX⊗χβ)≅4.6π*GHom*(p*LX˜α+n−1⊗χ1−n,OX⊗χβ)≅4.7π*GHom*(p*(LX˜α⊗ωp),OX⊗χβ)≅π*Gp*HomX*(LX˜α⊗ωp,p!OX⊗χβ)≅π*Gp*HomX˜*(LX˜α,p*OX⊗χβ)≅4.10ϱ*q*GHomX˜*(q*q*G(OX˜⊗χα),OX˜⊗χβ)≅ϱ*HomY˜*(q*G(OX˜⊗χα),q*G(OX˜⊗χβ))=ϱ*HomY˜*(Ψ(OX˜⊗χα),Ψ(OX˜⊗χβ)).□ We denote by E⊂Db(Y˜) the full subcategory generated by the admissible subcategories Ψ(DGb(X)) and Θℓ(Db(S))=i*ν*Db(S)⊗OY˜(ℓ) for ℓ∈{m−n,…,−1}. By the above, these admissible subcategories actually form a semi-orthogonal decomposition   E=⟨Θm−n(Db(S)),…,Θ−1(Db(S)),Ψ(DGb(X))⟩⊆Db(Y˜). Proposition 4.24 We have the (essential) equalities E=Db(Y˜)and Ψ(DGb(X))=D. Proof Analogously to Proposition 4.19, it is sufficient to prove the equality E=Db(Y˜). As E is constructed from images of fully faithful FM transforms (which have both adjoints), it is admissible in Db(Y˜). Therefore, it suffices to show that E contains a spanning class for Db(Y˜). Moreover, because all functors and categories involved are Y-linear, it suffices to prove that the relative spanning class S of Lemma 4.5 is contained in E. We already know that OY˜≅Ψ(OX)∈Ψ(DGb(X))⊂E. By Corollary 2.4, we get for s∈S and r∈{m,…,n−1}  is*Ωr(r)∈⟨Θm−n(Db(S)),…,Θ−1(Db(S))⟩⊂E.By Corollary 4.9, we have, for ℓ∈{0,…,m−1}, an exact triangle   E→Ψ(Os⊗χ−ℓ)→is*Ωℓ(ℓ)[ℓ]→,where E is an object in the triangulated category spanned by is*Ωr(r) for r∈{m,…,n−1}. In particular, the first two terms of the exact triangle are objects in E. Hence, also is*Ωℓ(ℓ)∈E for ℓ∈{0,…,m−1}.□ Combining the results of this subsection gives Theorem 4.1(ii). 4.6. The case m=n: spherical twists and induced tensor products Throughout this subsection, m=n, so that both functors Φ and Ψ are equivalences. We will show that the functors Θβ and Ξα, which were fully faithful in the cases n>m and m>n, respectively, are now spherical. Furthermore, the spherical twists along these functors allow to describe the transfer of the tensor structure from one side of the derived McKay correspondence to the other. We set Θ≔Θ0 and Ξ≔Ξ0. Proposition 4.25 For every α∈Z/mZ, the functor Ξα:Db(S)→DGb(X)is a split spherical functor with cotwist 𝖬ωS/X[−n]. Proof Since Ξα≅𝖬χαΞ, it is sufficient to prove the assertion for α=0, see Lemma 2.14. Following the proof of Proposition 4.15, we have ΞRΞ≅(̲)⊗(∧*N)G, where G acts on ∧ℓN by χ−ℓ. From rankN=n=m=ordχ, we get   (∧*N)G≅OS[0]⊕detN[−n]≅OS[0]⊕ωS/X[−n].Hence, ΞRΞ≅id⊕C with C≔𝖬ωS/X[−n]. Moreover, ΞR≅CΞL follows from a!≅Ca*.□ We introduce autoequivalences 𝖬L:Db(Y˜)→Db(Y˜) and 𝖬χ:DGb(X)→DGb(X) given by the tensor products with the line bundle LY˜ and the character χ, respectively. Theorem 4.26 There are the following relations between functors: Ψ−1≅𝖬χΦ𝖬Ln−1; ΨΞ≅Θ, in particular, the functors Θβare spherical too; 𝖳Θ≅Ψ𝖳ΞΨ−1; Ψ−1𝖬LΨ≅𝖬χ𝖳Ξand Ψ−1𝖬L−1Ψ≅𝖳Ξ−1𝖬χ−1. Proof In the verification of (i), we use ωp≅LX˜n−1⊗χ, from Lemma 4.7 and m=n:   Ψ−1≅ΨL≅p!q*≅4.7p*𝖬LX˜n−1⊗χq*≅4.10𝖬χp*q*𝖬Ln−1≅𝖬χΦ𝖬Ln−1.For (ii), first note that, since the G-action on Z⊂X˜ is trivial, we have   Θ≅i*ν*≅q*j*ν*≅q*Gj*ν*𝗍𝗋𝗂𝗏.Hence, the base change morphism ϑ:p*a*→j*ν* induces a morphism of functors   ϑˆ:ΨΞ≅q*Gp*a*𝗍𝗋𝗂𝗏→q*Gj*ν*𝗍𝗋𝗂𝗏≅Θwhich in turn is induced by a morphism between the Fourier–Mukai kernels, see [34, Section 2.4]. Hence, it is sufficient to show that ϑ induces an isomorphism ΨΞ(Os)≅Θ(Os) for every s∈S, see [34, Section 2.2]. The morphism ϑ induces an isomorphism on degree zero cohomology L0p*a*(Os)≅Op−1(a(s))≅j*L0ν*(Os). But there are no cohomologies in non-zero degrees for j*ν* since ν is flat and j a closed embedding. Furthermore, the non-zero cohomologies of p*a* vanish after taking invariants, see Corollary 4.9. Hence, ϑˆ(Os) is indeed an isomorphism. The second assertion of (ii) and (iii) is direct consequences of Proposition 4.25 and the formula ΨΞ≅Θ, see Lemma 2.14. For (iv), it is sufficient to prove the second relation, and we employ Corollary 2.12 with Li=OX⊗χi, see also Lemma 4.3. Recall that 𝖳Ξ−1=cone(id→ΞΞL)[−1], and ΞL≅(̲)Ga*. For 1≠α∈Z/nZ, we get   ΞL𝖬χ−1(OX⊗χα)≅(OS⊗χα−1)G≅0.Hence, 𝖳Ξ−1𝖬χ−1(OX⊗χα)≅OX⊗χα−1. We have ΞL(OX)=OS. Therefore, ΞΞL(OX)≅a*OS and 𝖳Ξ−1(OX)≅IS. In summary,   𝖳Ξ−1𝖬χ−1(OX⊗χα)≅{OX⊗χα−1forα≠1,ISforα=1.On the other hand, for α∈{−n+1,…,0}, we have Ψ(OX⊗χα)≅Lα, see Corollary 4.11. Hence, we have   Ψ−1𝖬L−1Ψ(OX⊗χα)≅OX⊗χα−1forα∈{−n+2,…,0}.For α=−n+1, we use (i) to get   Ψ−1𝖬L−1Ψ(OX⊗χ1−n)≅Ψ−1(LY˜−n)≅𝖬χΦ(LY˜−1)≅4.10p*(LX˜−1⊗χ)≅IS,where we get the last isomorphism by applying p* to the exact sequence (4.1). Therefore, for every α∈Z/nZ, we obtain isomorphisms   κα:F1(Lα)≔𝖳Ξ−1𝖬χ−1(OX⊗χα)→∼F2(Lα)≔Ψ−1𝖬L−1Ψ(OX⊗χα).Finally, we have to check that the isomorphisms κα can be chosen in such a way that they form an isomorphism of functors κ:F1,V∣{L0,…,Ln−1}→∼F2,V∣{L0,…,Ln−1} over every open set V⊂Y. Let U≔Y⧹S⊂Y the open complement of the singular locus. We claim that F1,U≅𝖬χ−1≅F2,U. This is clear for F2=𝖳Ξ−1𝖬χ−1. Furthermore, the map p:X˜→X is an isomorphism and q:X˜→Y˜ is a free quotient when restricted to W≔π−1(U). Since also LX˜=q*G(OX˜⊗χ), we get ΨU≅𝖬χ−1∣U. Hence, over W, the κi∣W can be chosen functorially. By the above computations, each κi∣W is given by a section of the trivial line bundle. As S has codimension at least 2 in X, the sections κi∣W over W uniquely extend to sections κi over X. The commutativity of the diagrams relevant for the functoriality now follows from the commutativity of the diagrams restricted to the dense subset W.□ The relations of Theorem 4.26 allow to transfer structures between Db(Y˜) and DGb(X). For example, we can deduce the formula Ψ𝖬χ−1Ψ−1≅𝖳Θ𝖬L−1. Since OX⊗χα for α∈{−(n−1),…,0} form a relative generator of DGb(X), their images Lα under Ψ do as well. Hence, at least theoretically, our formulas give a complete description of the tensor products induced by Ψ (and also Φ) on both sides. Note that Φ and Ψ are both equivalences, but not inverse to each other. Hence, they induce non-trivial autoequivalences ΨΦ∈Aut(DGb(X)) and ΦΨ∈Aut(Db(Y˜)). Considering the setup of the McKay correspondence as a flop of orbifolds as in diagram (1.1), it makes sense to call them flop–flop autoequivalences. These kinds of autoequivalences were widely studied for flops of varieties, see [4, 7, 19, 20, 45]. The general picture seems to be that the flop–flop autoequivalences can be expressed via spherical and P-twists induced by functors naturally associated to the centres of the flops. This picture is called the ‘flop–flop = twist’ principle, see [4]. The following can be seen as the first instance of an orbifold ‘flop–flop = twist’ principle which we expect to hold in greater generality. Corollary 4.27 ΨΦ≅𝖳Θ𝖬L−n≅𝖳Θ𝖬OY˜(−Z). Remark 4.28 Let us assume m=n=2 so that χ−1=χ. Then, for every k∈N, we get   Φ(L−k)≅ISk⊗χk, (4.3)where ISk denotes the k th power of the ideal sheaf of the fixed point locus. Indeed,   Φ(L−k)≅4.26(i)𝖬χΨ−1(L−k−1)≅𝖬χ(Ψ−1𝖬L−1Ψ)k(L−1)≅4.11𝖬χ(Ψ−1𝖬L−1Ψ)k(O⊗χ)≅4.26(iv)(𝖬χ𝖳Ξ−1)k(OX)≅ISk⊗χk.The last isomorphism follows inductively using the short exact sequences   0→ISk+1→ISk→ISk/ISk+1→0and the fact that the natural action of μ2 on ISk/ISk+1 is given by χk. Let now S be a surface and X=S2 with μ2 acting by permutation of the factors. Then Y˜=S[2] is the Hilbert scheme of two points and LY˜ is the square root of the boundary divisor Z parametrizing double points. For a vector bundle F on S of rank r, we have   detF[2]≅LY˜−r⊗DdetF,where F[2] denotes the tautological rank 2r bundle induced by F and, for L∈PicS, we put DL≔ϱ*π*(L⊠L)G∈PicS[2]. Hence, by the OY-linearity of Φ, formula (4.3) recovers the n=2 case of [44, Theorem 1.8]. 5. Categorical resolutions 5.1. General definitions Recall from [35] that a categorical resolution of a triangulated category T is a smooth triangulated category T˜ together with a pair of functors P*:T˜→T and P*:Tperf→T˜ such that P* is left adjoint to P* on Tperf and the natural morphism of functors idTperf→P*P* is an isomorphism. Here, Tperf is the triangulated category of perfect objects in T. Moreover, a categorical resolution (T˜,P*,P*) is weakly crepant if the functor P* is also right adjoint to P* on Tperf. For the notion of smoothness of a triangulated category, see for example [30]. For us, it is sufficient to notice that every admissible subcategory of Db(Z) for some smooth variety Z is smooth. In fact, we will always consider categorical resolutions of Db(Y), for some variety Y with rational Gorenstein singularities, inside Db(Y˜) for some fixed (geometric) resolution of singularities ϱ:Y˜→Y. By this, we mean an admissible subcategory T˜⊂Db(Y˜) such that ϱ*:Dperf(Y)→Db(Y˜) factorizes through T˜. By Grothendieck duality, we get a canonical isomorphism OY≅ϱ*OY˜≅ϱ*ωϱ. This induces a global section s of ωϱ, unique up to a global unit (that is scalar multiplication by an element of OY(Y)×), and hence a morphism of functors   t≔ϱ*(̲⊗s):ϱ*→ϱ!.Since this morphism can be found between the corresponding Fourier–Mukai kernels, we may define the cone of functors ϱ+≔cone(t):Db(Y˜)→Db(Y). Definition 5.1 The weakly crepant neighbourhood of Yinside Db(Y˜) is the full triangulated subcategory   WCN(ϱ)≔ker(ϱ+)⊂Db(Y˜). Proposition 5.2 If WCN(ϱ) is a smooth category (which is the case if it is an admissible subcategory of Db(Y˜)), it is a categorical weakly crepant resolution of singularities. Proof By adjunction formula, tϱ*:ϱ*ϱ*→ϱ!ϱ* is an isomorphism. Hence, ϱ+ϱ*=0 and ϱ*:Dperf(Y)→Db(Y˜) factors through WCN(ϱ). By definition, ϱ! is the left adjoint to ϱ*. Since ϱ* and ϱ! agree on WCN(ϱ), we also have the adjunction ϱ*⊣ϱ* on WCN(ϱ).□ Remark 5.3 We think of WCN(ϱ) as the biggest weakly crepant categorical resolution inside the derived category Db(Y˜) of a given geometric resolution ϱ:Y˜→Y. The only thing that prevents us from turning this intuition into a statement is the possibility that, for a given weakly crepant resolution T⊂Db(Y˜), there might be an isomorphism ϱ*∣T≅ϱ!∣T which is not the restriction of t (up to scalars). 5.2. The weakly crepant neighbourhood in the cyclic setup In the case of the resolution of the cyclic quotient singularities discussed in the earlier sections, WCN(ϱ) is indeed a categorical resolution by the following result. We use the notation of Section 3; recall G=μm. Theorem 5.4 Let Y=X/G, ϱ:Y˜→Yand i:Z=ϱ−1(S)↪Y˜be as in Section3. Assume m∣n=codim(S↪X)and n>m. Then there is a semi-orthogonal decomposition  WCN(ϱ)=⟨i*(E),Ψ(Dμmb(X))⟩,where  E=⟨A(−m+1),A(−m+2)…,A(−1),A⊗Ωn−m−1(n−m−1),A⊗Ωn−m−2(n−m−2),…,A⊗Ωm(m)⟩with A≔ν*Db(S)and A(i)≔A⊗O(i); the A⊗Ωi(i)parts of the decomposition do not occur for n=2m. In particular, WCN(ϱ)is an admissible subcategory of Db(Y˜). Proof We first want to show that Ψ(Dμmb(X))⊂WCN(ϱ). For this, by Lemma 4.3, it is sufficient to show that LY˜a=Ψ(OX⊗χa)∈WCN(ϱ) for every a∈{−m+1,…,0}. The equivariant derived category Dμmb(X) is a strongly (hence also weakly) crepant categorical resolution of the singularities of Y via the functors π*,π*μm, see [1, Theorem 1.0.2]. Since Ψ◦π*≅ϱ* (see Lemma 4.4), C≔Ψ(Dμmb(X)) is a crepant resolution via the functors ϱ*,ϱ*. Hence, ϱ*LY˜a≅ϱ!LY˜a for a∈{−m+1,…,0} and it is only left to show that this isomorphism is induced by t. Again by the Y-linearity of Ψ, we have ϱ*LY˜a≅π*(OX⊗χa)μm which is a reflexive sheaf on the normal variety Y (this follows for example by [24, Corollary 1.7]). By construction, t induces an isomorphism over Y⧹S. Since the codimension of S is at least 2, t:ϱ*LY˜a→ϱ!LY˜a is an isomorphism of reflexive sheaves over all of Y, see [24, Proposition 1.6]. By Theorem 4.1(ii), we have Db(Y˜)≅⟨B,C⟩ with   B≅i*⟨A(m−n),…,A(−1)⟩≅i*(⟨A,A(1),…,A(m−1)⟩⊥).We have ϱ*B=0. It follows that WCN(ϱ)=⟨B∩ker(ϱ!),C⟩. Indeed, consider an object A∈Db(Y˜). It fits into an exact triangle C→A→B→ with C∈C and B∈B. From the morphism of triangles   we see that t(A) is an isomorphism if and only if ϱ!B=0. It is left to compute B∩ker(ϱ!). Let F∈Db(Z) and B=i*F. By Lemma 4.14,   ϱ!B≅ϱ!i*F≅b*ν*(F⊗Oν(m−n)).We see that B∈kerϱ! if and only if ν*(F⊗Oν(m−n))=0 if and only if F∈ν*Db(S)(n−m)⊥. Hence, B∩kerϱ!=i*(F⊥) with   F=⟨A,A(1),…,A(m−1),A(n−m)⟩⊂Db(Z).Carrying out the appropriate mutations within the semi-orthogonal decomposition   Db(Z)=⟨A(−m+1),A(−m+2),…,A(n−m−1),A(n−m)⟩,we see that F⊥=E; compare Lemma 2.3. Since E⊂⟨A(m−n),…,A(−1)⟩ is an admissible subcategory, we find that i*:E→Db(Y˜) is fully faithful and has adjoints. Hence, WCN(ϱ)⊂Db(Y˜) is admissible.□ Remark 5.5 We have Db(Y˜)=⟨i*(A⊗Ωn−1(n−m)),WCN(ϱ)⟩. In other words, we can achieve categorical weak crepancy by dropping only one Db(S) part of the semi-orthogonal decomposition of Db(Y˜). 5.3. The discrepant category and some speculation Let Y be a variety with rational Gorenstein singularities and ϱ:Y˜→Y a resolution of singularities. Then, ϱ is a crepant resolution if and only if Db(Y˜)=WCN(ϱ); compare [1, Proposition 2.0.10]. We define the discrepant category of the resolution as the Verdier quotient   disc(ϱ)≔Db(Y˜)/WCN(ϱ).By [39, Remark 2.1.10], since WCN(ϱ) is a kernel, and hence a thick subcategory, we have disc(ϱ)=0 if and only if Db(Y˜)=WCN(ϱ). Therefore, we can regard disc(ϱ) as a categorical measure of the discrepancy of the resolution ϱ:Y˜→Y. In our cyclic quotient setup, where Y˜≅HilbG(X) is the simple blow-up resolution, we have disc(ϱ)≅Db(S) by Remark 5.5 and [37, Lemma A.8]. Hence, in this case, disc(ϱ) is the smallest non-zero category that one could expect (this is most obvious in the case that S is a point). This agrees with the intuition that the blow-up resolution is minimal in some way. Question 1 Given a variety Y with rational Gorenstein singularities, is there a resolution ϱ:Y˜→Y of minimal categorical discrepancy in the sense that, for every other resolution ϱ′:Y˜′→Y, there is a fully faithful embedding disc(ϱ)↪disc(ϱ′)? Often, in the case of a quotient singularity, a good candidate for a resolution of minimal categorical discrepancy should be the G-Hilbert scheme. At least, we can see that disc(ϱ) grows if we further blow up the resolution away from the exceptional locus. Proposition 5.6 Let ϱ:Y˜→Ybe a resolution of singularities and let f:Y˜′→Y˜be the blow-up in a smooth centre C⊂Y˜which is disjoint from the exceptional locus of ϱ. Set ϱ′≔ϱf:Y˜′→Y. Then there is a semi-orthogonal decomposition  disc(ϱ′)=⟨Db(C),disc(ϱ)⟩. We first need the following general. Lemma 5.7 Let Dbe a triangulated category, C⊂Da triangulated subcategory, and D=⟨A,B⟩a semi-orthogonal decomposition so that the right-adjoint iB!of the inclusion iB:B↪Dsatisfies iB!(C)⊂B∩C. Then there is a semi-orthogonal decomposition  D/C≅⟨A/(A∩C),B/(B∩C)⟩. Proof For every object D∈D, we have an exact triangle   iB!D→D→iA*D→, (5.1)where iA* is the left-adjoint to the embedding iA:A→D. Considering an object C∈C shows that our assumption iB!(C)⊂B∩C implies iA*(C)⊂A∩C. Let C∈C and A∈A. Then, using the long exact Hom-sequence associated to the triangle (5.1), we see that every morphism C→A factors as C→ιA*C→A. Hence, the embedding iA descends to a fully faithful embedding i¯A:A/(A∩C)→D/C, by [37, Proposition B.2] (set W=A, V=A∩C and use (ff2) of loc. cit.). Similarly, we get an induced fully faithful embedding i¯B:B/(B∩C)→D/C (use (ff2)op instead of (ff2)). Now let us show that HomD/C(B/(B∩C),A/(A∩C))=0. For B∈B and A∈A, a morphism B→A in D/C is represented by a roof   B←βD→αA,where β:D→B is a morphism in D with cone(β)∈C and α:D→A is any morphism in D, see [39, Definition 2.1.11]. Put C≔cone(β)[−1]∈C. We apply the triangle of functors iA*→id→iB!→ (formally, iA* has to be replaced by iAiA* and iB! by iBiB!) to the triangle of objects C→D→B→ and obtain the diagram   where we have used iB!B=B and iA*B=0. Now iB!C∈C∩B by assumption. The left column thus forces iA*C≅iA*D∈C. We get that coneβγ∈C since coneβ,coneγ∈C, see [39, Lemma 1.5.6]. Therefore, we get another roof representing the same morphism in D/C, replacing D by iB!D:   B←βγiB!D→αγA.However, iB!D∈B and HomD(B,A)=0, so the morphism is 0 in D/C. Finally, we need to show that A/(A∩C) and B/(B∩C) generate D/C, but this is clear, because A and B generate D.□ Proof of Proposition 5.6 We have a semi-orthogonal decomposition Db(Y˜′)=⟨A,B⟩ with B=f*Db(Y˜) and   A=⟨ι*(g*Db(C)⊗Og(−c+1)),…,ι*(g*Db(C)⊗Og(−1))⟩.Here, c=codim(C↪Y˜) and g and ι are the Pn−1-bundle projection and the inclusion of the exceptional divisor of the blow-up f:Y˜′→Y˜. Let U≔Y˜⧹C. For F∈Db(Y˜′), we have (f*F)∣U≅(f!F)∣U. Hence, if F∈WCN(ϱ′), we must have (f*F)∣U∈WCN(ϱ∣U). Since ϱ is an isomorphism in a neighbourhood of C, an object E∈Db(Y˜) is contained in WCN(ϱ) if and only if its restriction E∣U is contained in WCN(ϱ∣U). In summary,   f*F∈WCN(ϱ)foreveryF∈WCN(ϱ′).We have f*OY˜′≅OY˜≅f*ωf. By the projection formula, it follows that f*∣B≅f!∣B. Hence, we have B∩WCN(ϱ′)=f*WCN(ϱ) and B/(WCN(ϱ′)∩B)≅disc(ϱ). Now, we can apply Lemma 5.7 with C=WCN(ϱ′) to get a semi-orthogonal decomposition   disc(ϱ′)=⟨A/(WCN(ϱ′)∩A),disc(ϱ)⟩.We have f*(A)=0, hence ϱ*′(A)=0. Accordingly,   WCN(ϱ′)∩A=ker(ϱ!′)∩A=ker(f!)∩A.The second equality is due to the fact that all objects of f!A are supported on C, where ϱ is an isomorphism. Now, in analogy to the computations of the proof of Theorem 5.4 and Remark 5.5, we get a semi-orthogonal decomposition   A=⟨ι*(g*Db(C)⊗Ωc−1(c−1)),ker(f!)∩A⟩.Hence, A/(WCN(ϱ′)∩A)≅ι*(g*Db(C)⊗Ωc−1(c−1))≅Db(C).□ 5.4. (Non-)unicity of categorical crepant resolutions Let Y˜→Y be a resolution of rational Gorenstein singularities and let D⊂Dperf(Y˜) be an admissible subcategory which is a weakly crepant resolution, that is ϱ*:Dperf(Y)→Db(Y˜) factors through D and ϱ*∣D≅ϱ!∣D. Then every admissible subcategory D′⊂D with the property that ϱ*:Dperf(Y)→Db(Y˜) factors through D′ is a weakly crepant resolution, too. In particular, in our setup of cyclic quotients, there is a tower of weakly crepant resolutions of length n−m given by successively dropping the Db(S) parts of the semi-orthogonal decomposition of WCN(ϱ). We see that weakly crepant categorical resolutions are not unique, even if we fix the ambient derived category Db(Y˜) of a geometric resolution Y˜→Y. In contrast, strongly crepant categorical resolutions are expected to be unique up to equivalence, see [35, Conjecture 4.10]. A strongly crepant categorical resolution of Db(Y) is a module category over Db(Y) with trivial relative Serre functor, see [35, Section 3]. For an admissible subcategory D⊂Db(Y˜) of the derived category of a geometric resolution of singularities ϱ:Y˜→Y this condition means that D is Y-linear and there are functorial isomorphisms   ϱ*Hom(A,B)∨≅ϱ*Hom(B,A) (5.2)for A,B∈D. In our cyclic setup, Ψ(DG(X))⊂Db(Y˜) is a strongly crepant categorical resolution, see [35, Theorem 1] or [1, Theorem 10.2]. We require strongly crepant categorical resolutions to be indecomposable, which means that they do not decompose into direct sums of triangulated categories or, in other words, they do not admit both-sided orthogonal decompositions. Under this additional assumption, we can prove that strongly crepant categorical resolutions are unique if we fix the ambient derived category of a geometric resolution. Proposition 5.8 Let Y˜→Ybe a resolution of Gorenstein singularities and D,D′⊂Db(Y˜)admissible indecomposable strongly crepant subcategories. Then D=D′. Proof The intersection D∩D′ is again an admissible Y-linear subcategory of Db(Y˜) containing ϱ*(Dperf(Y)). Furthermore, condition (5.2) is satisfied for every pair of objects of D∩D′; so the intersection is again a strongly crepant resolution. Hence, we can assume D′⊂D. Let A be the right-orthogonal complement of D′ in D, so that we have a semi-orthogonal decomposition D=⟨A,D′⟩. By Lemma 2.6, this means that ϱ*Hom(D,A)=0 for A∈A and D∈D′. But then, by (5.2), we also get ϱ*Hom(A,D)=0 so that D=A⊕D′.□ 5.5. Connection to Calabi–Yau neighbourhoods In [25], spherelike objects and their spherical subcategories were introduced and studied. The paper hinted at a role of these notions for birationality questions of Calabi–Yau varieties. One of the starting points for our project was to consider Calabi–Yau neighbourhoods (a generalization of spherical subcategories) as candidates for categorical crepant resolutions of Calabi–Yau quotient varieties. In this subsection, we describe the connection to the weakly crepant resolutions considered above. We recall some abstract homological notions. Let T be a Hom-finite C-linear triangulated category and E∈T an object. We say that 𝖲E∈T is a Serre dual object for E if the functors Hom*(E,−) and Hom*(−,𝖲E)∨ are isomorphic. By the Yoneda lemma, 𝖲E is then uniquely determined. Fix an integer d. We call the object E a d-Calabi–Yau object, if E[d] is a Serre dual of E, d-spherelike if Hom*(E,E)=C⊕C[−d], and d-spherical if E is d-spherelike and a d-Calabi–Yau object.Note a smooth compact variety X of dimension d is a strict Calabi–Yau variety precisely if the structure sheaf OX is a d-spherical object of Db(X). In [25], the authors show that if E is a d-spherelike object, there exists a unique maximal triangulated subcategory of T in which E becomes d-spherical. In the following, we will imitate this construction for a larger class of objects. Definition 5.9 Let E∈T be an object in a triangulated category having a Serre dual 𝖲E. We call E a d-selfdual object if Hom(E,E[d])≅C, that is by Serre duality there is a morphism w:E→ω(E)≔𝖲E[−d] unique up to scalars, and the induced map w*:Hom*(E,E)→∼Hom*(E,ω(E)) is an isomorphism.In particular, a d-selfdual object satisfies Hom*(E,E)≅Hom(E,E)∨[−d], hence the name. Remark 5.10 If an object is d-spherelike, then it is d-selfdual; compare [25, Lemma 4.2]. For a d-selfdual object E, there is a triangle E→wω(E)→QE→E[1] induced by w. By our assumption, we get Hom*(E,QE)=0. Thus, following an idea suggested by Martin Kalck after discussing [25, Section 7] with Michael Wemyss, we propose the following: Definition 5.11 The Calabi–Yau neighbourhood of a d-selfdual object E∈T is the full triangulated subcategory   CY(E)≔⊥QE⊆T. Proposition 5.12 If E∈Tis a d-selfdual object then E∈CY(E)is a d-Calabi-Yau object. Proof If T∈CY(E), apply Hom*(T,−) to the triangle E→ω(E)→QE.□ Using the same proof as for [25, Theorem 4.6], we see that the Calabi–Yau neighbourhood is the maximal subcategory of T in which a d-selfdual object E becomes d-Calabi–Yau. Proposition 5.13 If U⊂Tis a full triangulated subcategory and E∈Uis d-Calabi–Yau, then U⊂CY(E). Proposition 5.14 Let Ybe a projective variety with rational Gorenstein singularities and trivial canonical bundle of dimension d=dimYand consider a resolution of singularities ϱ:Y˜→Y. Then, for every line bundle L∈PicY, the pull-back ϱ*L∈Db(Y˜)is d-selfdual. Furthermore, we have  WCN(ϱ)=⋂L∈PicYCY(ϱ*L). (5.3) Proof Note that, by our assumption that ωY is trivial, we have ωY˜≅ωϱ. Hence, by Grothendieck duality, there is a morphism wL:ϱ*L→ϱ*L⊗ωY˜ unique up to scalar multiplication, namely wL=idϱ*L⊗s, where s is the non-zero section of ωY˜≅ωϱ; compare the previous Section 5.1. Furthermore, wL*:Hom*(ϱ*L,ϱ*L)→Hom*(ϱ*L,ϱ*L⊗ωY˜) is an isomorphism, still by Grothendieck duality, which means that ϱ*L is d-selfdual. Recall that WCN(ϱ)=ker(ϱ+), where ϱ+ is defined as the cone   ϱ*→tϱ!→ϱ+→.By adjunction, we get WCN(ϱ)=⊥(ϱ+(Dperf(Y))), where ϱ+=ϱ+R is given by the triangle   ϱ+→ϱ*→tRϱ!→.Note that tR=(̲)⊗s. Hence, tR(L)=wL:ϱ*L→ϱ*L⊗ωY˜ and ϱ+(L)=Qϱ*L[−1]; compare Definition 5.11. Since the line bundles form a generator of Dperf(Y), we get for F∈Db(Y˜):   F∈WCN(ϱ)⟺F∈⊥(ϱ+(Dperf(Y)))⟺F∈⊥Qϱ*L∀L∈PicY⟺F∈CY(ϱ*L)∀L∈PicY.□ Remark 5.15 Following the proof of Proposition 5.14, we see that, on the right-hand side of (5.3), it is sufficient to take the intersection over all powers of a given ample line bundle. In our cyclic setup, if S consists of isolated points, we even have WCN(ϱ)=CY(OY˜) so that the weakly crepant neighbourhood is computed by a Calabi–Yau neighbourhood of a single object. The same should hold in general if Y has isolated singularities. 6. Stability conditions for Kummer threefolds Let A be an abelian variety of dimension g. Consider the action of G=μ2 by ±1. Then the fixed point set A[2] consists of the 4g two-torsion points. Consider the quotient A¯ (the singular Kummer variety) of A by G, and the blow-up K(A) (the Kummer resolution) of A¯ in A[2]. This setup satisfies Condition 3.1, with m=2 and n=g and we get Corollary 6.1 The functor Ψ:DGb(A)→Db(K(A))is fully faithful, and  Db(K(A))=⟨Db(𝗉𝗍),…,Db(𝗉𝗍)︸(g−2)4gtimes,Ψ(DGb(A))⟩. To explore a potentially useful consequence of this result, we need to recall that a Bridgeland stability condition on a reasonable C-linear triangulated category D consists of the heart A of a bounded t-structure in D and a function from the numerical Grothendieck group of D to the complex numbers satisfying some axioms, see [15]. Corollary 6.2 There exists a Bridgeland stability condition on Db(K(A)), for an abelian threefold A. Proof To begin with, by [11, Corollary 10.3], there is a stability condition on DGb(A). Denote by A⊂DGb(A) the corresponding heart; it is a tilt of the standard heart [11, Section 2]. For a two-torsion point x∈A[2], we set Ex≔Oϱ−1(π(x))(−1). Then, since g=dimA=3, the semi-orthogonal decomposition of Corollary 6.1 is given by   Db(K(A))=⟨{Ex}x∈A[2],DGb(A)⟩. (6.1) Next, we want to show that, for every x∈A[2], there exists an integer i such that Hom≤i(Ex,Ψ(F))=0 for all F∈A⊂DGb(A). Indeed, the cohomology of any complex in the heart of the stability condition on DGb(A), as constructed in [11, Corollary 10.3], is concentrated in an interval of length three. The functor Ψ has cohomological amplitude at most 3, since q*G:CohG(A)→Coh(K(A)) is an exact functor of abelian categories, and every sheaf on A has a locally free resolution of length dimA=3. This implies that the cohomology of any complex in Ψ(DGb(A)) is contained in a fixed interval of length 6. This proves the above claim. 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