AR-Components of domestic finite group schemes: McKay-Quivers and Ramification

AR-Components of domestic finite group schemes: McKay-Quivers and Ramification Abstract For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander–Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of SL(2). Moreover, for a normal subgroup scheme N of a finite group scheme G, we show that there is a connection between the ramification indices of the restriction morphism ℙ(VN)→ℙ(VG) between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander–Reiten quivers. 1. Introduction The Auslander–Reiten quiver of a self-injective finite-dimensional algebra is a powerful tool for understanding its representation theory via combinatorical invariants. In this work, we are mainly interested in the Auslander–Reiten components of a group algebra kG of domestic representation type for a finite group scheme G over an algebraically closed field k. We say that an algebra A has tame representation type if it possesses infinitely many isomorphism classes of indecomposable modules, and if in each dimension almost all isomorphism classes of indecomposable modules occur in only finitely many one-parameter families. If additionally the number of one-parameter families is uniformly bounded, we say that A has domestic representation type. The group algebra kG≔k[G]* of a finite group scheme G is the dual of its coordinate ring. We say that G is domestic if its group algebra is domestic. In [11], Farnsteiner classified the domestic finite group schemes over an algebraically closed field of characteristic p>2. Any such group scheme can be associated to a so-called amalgamated polyhedral group scheme. Moreover, the non-simple blocks of an amalgamated polyhedral group scheme are Morita-equivalent to a trivial extension of a radical square zero tame hereditary algebra. In this way, the components of the Auslander–Reiten quiver of these group schemes are classified abstractly. Our goal is to describe these components in a direct way by using tensor products, McKay-quivers and ramification indices of certain morphisms. We will start by describing the Euclidean components. For this purpose, we show how to extend certain almost split sequences from a normal subgroup scheme N⊆G to almost split sequences of G if the group scheme G/N is linearly reductive. Moreover, for a simple G/N-module S, we will show that the tensor functor −⊗kS sends these extended almost split sequences to almost split sequences. In Section 2, we will use these results to show that for any amalgamated polyhedral group scheme G, there is a finite linearly reductive subgroup scheme G˜⊆SL(2) such that the Euclidean components of Γs(G) can be explicitly described by the McKay-quiver ϒL(1)(G˜). Of great importance for the proofs of these results is the fact that the category of G-modules is closed under taking tensor products of G-modules. This comes into play in the definition of the McKay-quiver, for the construction of new almost split sequences and in the description of the Euclidean components. Thanks to this property, we are also able to introduce geometric invariants for the representation theory of G. If G is any finite group scheme, one can endow the even cohomology ring H•(G,k) with the structure of a commutative graded k-algebra. Thanks to the Friedlander–Suslin Theorem [13], this algebra is finitely generated. Therefore, the maximal ideal spectrum VG of H•(G,k) is an affine variety. As H•(G,k) is graded, we can also consider its projectivized variety P(VG). Now let us again assume that N⊆G is a normal subgroup scheme such that G/N is linearly reductive. Then the ramification indices of the restriction morphism P(VN)→P(VG) will give upper bounds for ranks of the corresponding tubes in the Auslander–Reiten quiver. Here a tube Z/(r)[A∞] of rank r can be regarded as a quiver which is arranged on an infinite tube with circumference r. Moreover, if G is an amalgamated polyhedral group scheme and N=G1 is its first Frobenius kernel, the ranks are equal to the corresponding ramification indices. Altogether we will prove the following: Theorem Let Gbe an amalgamated polyhedral group scheme and Θa component of the stable Auslander–Reiten quiver Γs(G). Then the following hold: If Θ is Euclidean, then there is a component Q of the separated quiver ϒL(1)(G˜)sand a concrete isomorphism Θ≅Z[Q]. Let Θ be a tube and eΘthe ramification index of the restriction morphism P(VG1)→P(VG)at the corresponding point xΘ. Then Θ≅Z/(eΘ)[A∞]. There seems to be a connection to a result of Crawley–Boevey [4], which states that a finite-dimensional tame algebra has only finitely many non-homogeneous tubes. On the other side, the restriction morphism P(VN)→P(VG) is finite and has constant ramification on an open dense subset of P(VN). Therefore, there are only finitely many exceptional ramification points. In our situation, all but finitely many points will be unramified and a tube can only be non-homogeneous, if it belongs to the image of a ramification point. 2. Induction of almost split sequences Let k be an algebraically closed field. Given a finite group scheme G with normal subgroup scheme N, we want to investigate conditions under which the induction functor indNG:modN→modG sends an almost split exact sequence to a direct sum of almost split exact sequences. We start by giving an overview of the functorial approach to almost split sequences. For further details, we refer the reader to [1, IV.6]. Let A be a finite-dimensional k-algebra. Denote by FunopA and FunA the categories of contravariant and covariant k-linear functors from modA to modk. A functor F in FunopA is finitely generated if the functor F is isomorphic to a quotient of HomA(−,M) for some M∈modA. A functor F in FunopA is finitely presented if there is an exact sequence HomA(−,M)→HomA(−,N)→F→0 of functors in FunopA for some M,N∈modA. The full subcategory of FunopA consisting of the finitely presented functors will be denoted by mmodA. Up to isomorphism, the finitely generated projective functors in FunopA are exactly the functors of the form HomA(−,M). Such a functor is indecomposable if and only if the A-module M is indecomposable. For A-modules, M and N we define the radical of HomA(M,N) as RadA(M,N)≔{φ∈HomA(M,N)∣φisnotanisomorphism}. Then RadA(−,M) is a subfunctor of HomA(−,M) and we define the functor SM≔HomA(−,M)/RadA(−,M). Up to isomorphism, the simple functors in FunopA are exactly the functors of the form SM with an indecomposable A-module M. The projective cover of SM is HomA(−,M). Let N be an indecomposable A-module. An A-module homomorphism g:M→N is (minimal) right almost split if and only if the induced exact sequence HomA(−,M)→HomA(−,N)→SN→0 of functors in FunopA is a (minimal) projective presentation of SN. A functor F:modA→modB induces a functor F:mmodA→mmodB via F(HomA(−,M))=HomB(−,F(M)). There are dual notions and results for left almost split morphisms and functors in FunA. In [24, 3.8], Reiten and Riedtmann used this functorial approach to show that for a skew group algebra A∗G, with a finite group G, the induction functor ind1G:modA→modA∗G and the restriction functor res1G:modA∗G→modA send almost split sequences to direct sums of almost split sequences if the group order of G is invertible in k. Their proof relied on the following properties of the involved functors: (A) (i) There is a split monomorphism of functors idmodA→res1Gind1G. (ii) There is a split epimorphism of functors ind1Gres1G→idmodA*G. (B) (ind1G,res1G) and (res1G,ind1G) are adjoint pairs of functors. (C̃) There is a finite group G acting on modA such that for every A-module M, there is a decomposition res1Gind1GM=⊕g∈GMg and if φ:M→N is A-linear, then res1Gind1G(φ)=(g.φ)g∈G:⊕g∈GMg→⊕g∈GNg. In [24, 3.5], it was shown that these properties also hold for the induced functors ind1G:mmodA→mmodA∗G and res1G:mmodA*G→mmodA and that they imply the following property: (C) ind1G:mmodA→mmodA∗G and res1G:mmodA∗G→mmodA preserve semisimple objects and projective covers. We want to apply the ideas of the proof in the context of group algebras of finite group schemes. If G is a finite group scheme, then the dual of its coordinate ring kG≔(k[G])* is called the group algebra of G. In this situation, we will not always have analogous results for the induction and restriction functor. For example, in [10, 3.1.4], it was already shown that for the restriction functor, this is possible if and only if the ending term of the almost split sequence fulfils a certain regularity property. Let N be a normal subgroup scheme of G. The N-modules which will be of our interest are restrictions of G-modules. To obtain an analog of property (C̃), we will use the following result: Lemma 2.1 Let Gbe a finite group scheme and Nbe a normal subgroup scheme of G. Let M be a G-module. Then there is a G-linear isomorphism ψM:indNGresNGM→M⊗kk(G/N)which is natural in M. In particular, resNGindNGM≅Mn, where n=dimkk(G/N). Proof This follows directly from the tensor identity. Alternatively, exactly as in the proof of [16, 5.1] one can show that the map ψM:indNGM→M⊗kk(G/N),a⊗m↦∑(a)a(1)m⊗π(a(2)) is an isomorphism of G-modules, where π:kG→k(G/N) is the canonical projection. A direct computation shows that it is natural in M.□ Proposition 2.2 Let Gbe a finite group scheme and Nbe a normal subgroup scheme such that G/Nis linearly reductive. Let X,Yand E be N-modules such that every indecomposable direct summand of the modules X,Yand E is the restriction of a G-module, and E:0→X⟶φE⟶ψY→0is an almost split exact sequence of N-modules.Then indNGEis a direct sum of almost split exact sequences. Proof Denote by C the full subcategory of modN consisting of direct sums of indecomposable N-modules which are restrictions of G-modules. Due to 2.1, the properties (A)(i) and (C̃) hold in C if G is a group acting trivially on C. As G/N is linearly reductive, the k(G/N)-Galois extension kG:kN is separable by [6, 3.15]. This yields property (A)(ii). Thanks to [18, 1.7(5)], the ring extension kG:kN is a free Frobenius extension of first kind, i.e. kG is a finitely generated free kN-module, and there is a (kG,kN)-bimodule isomorphism kG→HomN(kG,kN). Hence, by [21, 2.1], the induction and coinduction functors are equivalent, so that property (B) holds. As ψ is minimal right almost split, the exact sequence HomN(−,E)→HomN(−,Y)→SY→0 is a minimal projective presentation of SY. We can now apply the arguments of the proof of [24, 3.6] by keeping the following things in mind: The simple functors which occur in our situation are all of the form SV with V being an indecomposable module in C. The projective covers which occur in our situation are all of the form HomN(−,V) with V∈C. Therefore, the functor indNGSY is semisimple and the exact sequence indNGHomN(−,E)→indNGHomN(−,Y)→indNGSY→0 is a minimal projective presentation of indNGSY≅SindNGY. Hence, indNGψ is a direct sum of minimal right almost split homomorphisms. Dually, one can show that indNGφ is a direct sum of minimal left almost split homomorphisms.□ Let G be a group scheme and X and M be G-modules. We define the k-vector spaces RadG2(X,M)≔{α∣∃Z∈ModG,φ∈RadG(X,Z),ψ∈RadG(Z,M):α=ψ◦φ} and IrrG(X,M)≔RadG(X,M)/RadG2(X,M). Lemma 2.3 Let Gbe a reduced group scheme, Nbe a normal subgroup scheme of Gand X and M be G-modules. Then RadN(X,M)and RadN2(X,M)are G-submodules of HomN(X,M). Proof As G is reduced, we only need to prove that RadN(X,M) and RadN2(X,M) are G(k)-submodules of HomN(X,M) (see [15, Remark after 2.8]). Let g∈G(k) and φ∈RadN(X,M). Assume that g.φ is an isomorphism with inverse ψ. Then g−1.ψ is an inverse of φ, a contradiction. Let g∈G(k) and φ∈RadN2(X,M). Then there is an N-module Z and homomorphisms α∈RadN(X,Z), β∈RadN(Z,M) such that φ=β◦α. Denote by Zg the N-module Z with action twisted by g−1, i.e. h·z=hg−1z for h∈kN and z∈Zg. As X and M are G-modules, we can define the N-linear maps α˜:X→Zg,x↦α(g−1x)andβ˜:Zg→M,z↦gβ(z). Then we obtain g.φ=β˜◦α˜. If α˜ is an isomorphism with inverse γ:Zg→X, then the N-linear map γ˜:Z→X,z→g−1γ(z) is an inverse of α, a contradiction. In the same way, β˜ is not an isomorphism. Hence g.φ=β˜◦α˜∈RadN2(X,M).□ Proposition 2.4 Let Gbe a finite subgroup scheme of a reduced group scheme Hand N⊆Ga normal subgroup scheme of Hsuch that G/Nis linearly reductive. Let X and M be indecomposable G-modules such that there is an almost split exact sequence E:0→τN(M)⟶Xn⟶M→0of N-modules. Then there is a short exact sequence E˜:0→N⟶X⊗kIrrN(X,M)⟶M→0of G-modules such that resNGE˜=E. Proof Consider the map ψ˜:X⊗kHomN(X,M)→M,x⊗f↦f(x) of G-modules. As G/N is linearly reductive, the short exact sequence 0→RadN2(X,M)⟶RadN(X,M)⟶IrrN(X,M)→0 of G-modules splits. This yields a decomposition RadN(X,M)≅RadN2(X,M)⊕IrrN(X,M). Hence, there results a G-linear map ψ:X⊗kIrrN(X,M)→M by restricting ψ˜ to X⊗kIrrN(X,M). Let (fi)1≤i≤n:Xn→M be the surjection given by E. Since E is almost split, the maps (fi)1≤i≤n form a k-basis of IrrN(X,M) (cf. [1, IV.4.2]). Therefore, the restriction resNG(ψ) equals (fi)1≤i≤n.□ In 2.2, we have seen that if we apply the induction functor to certain almost split sequences, they will be the direct sum of almost split sequences. Our next result enables us to describe this decomposition under a certain regularity condition. Let M be a multiplicative group scheme, i.e. the coordinate ring of M is isomorphic to the group algebra kX(M) of its character group. Then we obtain for any M-module M a weight space decomposition M=⊕λ∈X(M)Mλ with Mλ≔{m∈M∣hm=λ(h)mforallh∈kM}. Let G be a finite group scheme and N⊆G0 a normal subgroup scheme of G such that G0/N is linearly reductive. By Nagata’s Theorem [5, IV,§3,3.6], an infinitesimal group scheme is linearly reductive if and only if it is multiplicative. Let M be a G0-module. For any λ∈X(G0/N), we obtain a G0-module M⊗kkλ, the tensor product of M with the one-dimensional G0/N-module defined by λ. This defines an action of X(G0/N) (here we identify X(G0/N) with a subgroup of X(G0) via the canonical inclusion X(G0/N)↪X(G0) which is induced by the canonical projection G0↠G0/N) on the isomorphism classes of G0-modules and we define the stabilizer of a G0-module as X(G0/N)M≔{λ∈X(G0/N)∣M⊗kkλ≅M}. A G0-module M is called G0/N-regular if its stabilizer is trivial. Proposition 2.5 In the situation of2.4, assume that M and N are G0/N-regular. Let S be a simple G/N-module. Then the short exact sequence E˜⊗kSis almost split. Proof Let k(G/N)=⊕i=1nSi be the decomposition into simple G/N-modules. For any G-module U, let ψU:indNGU→U⊗kk(G/N) be the natural isomorphism given by 2.1. We obtain the following commutative diagram with exact rows: As all vertical arrows are isomorphisms, the two exact sequences indNGE and ⊕i=1nE˜⊗kSi are equivalent. Thanks to 2.2, the exact sequence indNGE is a direct sum of almost split sequences. By [16, 5.1], the G-modules M⊗kS and N⊗kS are indecomposable. Moreover, the sequence E˜⊗kS does not split. Otherwise, the sequence resNG(E˜⊗kS) would split and, therefore, also E. Hence, the sequence E˜⊗kS is equivalent to an indecomposable direct summand of indNGE. As the Krull–Schmidt theorem holds in the category of short exact sequences of finite-dimensional G-modules (as the space of morphisms between those sequences is finite-dimensional, cf. [17]), it follows that E˜⊗kSi is almost split.□ 3. The McKay and Auslander–Reiten quiver of finite domestic group schemes Let k be an algebraically closed field of characteristic p>2. The group algebra kSL(2)1 is isomorphic to the restricted universal enveloping algebra U0(sl(2)) of the restricted Lie algebra sl(2). There are one-to-one correspondences between the representations of SL(2)1, U0(sl(2)) and sl(2). The indecomposable representations of the restricted Lie algebra sl(2) were classified by Premet in [23]. For d∈N0, we consider the (d+1)-dimensional Weyl module V(d) of highest weight d. Weyl modules are rational SL(2)-modules which are obtained by twisting the 2-dimensional standard module with the Cartan involution ( x↦−xtr) and taking its dth symmetric power. For d≤p−1, we obtain in this way exactly the simple U0(sl(2))-modules L(0),…,L(p−1). Let A be a self-injective k-algebra. We denote by Γs(A) the stable Auslander–Reiten quiver of A. The vertices of this valued quiver are the isomorphism classes of non-projective indecomposable A-modules and the arrows correspond to the irreducible morphisms between these modules. Moreover, we have an automorphism τA of Γs(A), called the Auslander–Reiten translation. For a self-injective algebra, the Auslander–Reiten translation is the composite ν◦Ω2, where ν denotes the Nakayama functor of modA and Ω the Heller shift of modA. For further details, we refer to [1, 2]. Let Q be a quiver. We denote by Z[Q] the translation quiver with underlying set Z×Q, arrows (n,x)→(n,y) and (n+1,y)→(n,x) for any arrow x→y in Q and translation τ:Z[Q]→Z[Q] given by τ(n,x)=(n+1,x). Let Θ⊆Γs(A) be a connected component of the stable Auslander–Reiten quiver of A. Thanks to the Struktursatz of Riedtmann [25], there is an isomorphism of stable translation quivers Θ≅Z[TΘ]/Π, where TΘ denotes a directed tree and Π is an admissible subgroup of Aut(Z[TΘ]). The underlying undirected tree T¯Θ is called the tree class of Θ. The Auslander–Reiten quiver of each block of kSL(2)1 consists of two components of type Z[A˜1,1] and infinitely many homogeneous tubes Z[A∞]/(τ). Thanks to [10, 4.1], each of the p−1 Euclidean components Θ(i) contains exactly one simple SL(2)1-module L(i) with 0≤i≤p−2. This component is then given by Θ(i)={Ω2n(L(i)),Ω2n+1(L(p−2−i))∣n∈Z} with almost split sequences 0→Ω2n+2(L(i))⟶Ω2n+1(L(p−2−i))⊕Ω2n+1(L(p−2−i))⟶Ω2n(L(i))→0. In the following, it will be convenient to have a common notation for the Weyl modules and their duals. We set V(n,i)≔{V(np+i)ifn≥0V(−np+i)*ifn≤0 for n∈Z and 0≤i≤p−1. Lemma 3.1 Let Gbe a finite subgroup scheme of SL(2)with SL(2)1⊆Gsuch that G/SL(2)1is linearly reductive. Let 0≤i≤p−2and n∈Z. Then ΩG2n(L(i))≅V(2n,i)and ΩG2n+1(L(i))≅V(2n+1,p−2−i).In particular, Θ(i)={V(n,i)∣n∈Z}. Proof By [27, 7.1.2], there is for n≤0 a short exact sequence 0→V(n,i)⟶P(i)⊗k(V(n)*)[1]⟶V(n−1,p−2−i)→0 of SL(2)-modules, where (V(n)*)[1] denotes the first Frobenius twist of V(n)* (cf. [15, I.9.10]). As P(i) is a projective SL(2)-module, we obtain that the G-module P(i)⊗k(V(n)*)[1] is projective. Hence, there is a projective G-module P with ΩG(V(n−1,p−2−i))⊕P≅V(n,i). Since V(n,i) is a non-projective indecomposable G-module, we obtain ΩG(V(n−1,p−2−i))≅V(n,i). Dualizing the above sequence yields ΩG(V(n,i))=V(n+1,p−2−i) for n≥0 in the same way.□ Lemma 3.2 For all n∈Zand 0≤i≤p−2the SL(2)-modules L(1)[1]and IrrSL(2)1(V(n+1,i),V(n,i))are isomorphic. Proof As in the proof of 3.1, there is for all n∈Z a short exact sequence 0→V(n+1,i)⟶P⟶V(n,p−2−i)→0 of SL(2)-modules with P being projective over SL(2)1. Applying the functor HomSL(2)1(−,V(n,i)) to this sequence yields the following exact sequence of SL(2)-modules: HomSL(2)1(P,V(n,i))→αHomSL(2)1(V(n+1,i),V(n,i))→βExtSL(2)11(V(n,p−2−i),V(n,i))→ExtSL(2)11(P,V(n,i)). As P is a projective SL(2)1-module, we obtain that ExtSL(2)11(P,V(n,i))=(0). Hence the map β is surjective. By [7, 2.4], the SL(2)-modules L(1)[1] and ExtSL(2)11(V(n,p−2−i),V(n,i)) are isomorphic. As imα=kerβ is contained in RadSL(2)12(V(n+1,i),V(n,i)), we obtain a surjective morphism L(1)[1]→IrrSL(2)1(V(n+1,i),V(n,i)) of SL(2)-modules. Since both modules are 2-dimensional, this map is an isomorphism.□ Let H be a finite linearly reductive group scheme, S1,…,Sn a complete set of pairwise non-isomorphic simple H-modules and L be an H-module. For each 1≤j≤n, there are aij≥0 such that L⊗kSj≅⊕i=1naijSi. The McKay quiver ϒL(H) of H relative to L is the quiver with underlying set {S1,…Sn} and aij arrows from Si to Sj. Let G be a domestic finite group scheme. We denote by Glr the largest linearly reductive normal subgroup scheme of G. Let Z be the center of the group scheme SL(2). Thanks to [11, 4.3.2], there is a finite linearly reductive subgroup scheme G˜ of SL(2) with Z⊆G˜ such that G/Glr is isomorphic to (SL(2)1G˜)/Z. Group schemes which arise in this fashion are also called amalgamated polyhedral group schemes. By the above, any domestic finite group scheme can be associated to one of these group schemes. If G is an amalgamated polyhedral group scheme, we set Gˆ≔SL(2)1G˜. For any Euclidean diagram (A˜n)n∈N,(D˜n)n≥4 and (E˜n)6≤n≤8, we will denote in the same way the quiver where each edge •−• is replaced by a pair of arrows •⇆•. As shown in the proof of [8, 7.2.3], the McKay quiver ϒL(1)[1](Gˆ/Gˆ1) is isomorphic to one of the quivers A˜2npr−1−1,D˜npr−1+2,E˜6,E˜7,E˜8, where r is the height of Gˆ0 and (n,p)=1. For any quiver Q, we denote by Qs its separated quiver. If {1,…n} is the vertex set of Q, then Qs has 2n vertices {1,…n,1′,…n′} and arrows i→j′ if and only if i→j is an arrow in Q. The separated quiver of one of the quivers A˜2npr−1−1,D˜npr−1+2,E˜6,E˜7,E˜8 is the union of 2 quivers with the same underlying graph as the original quiver and each vertex is either a source or a sink. Theorem 3.3 Let Gbe an amalgamated polyhedral group scheme and Θ a component of Γs(G)containing a G-module of complexity 2. Let Q be a connected component of ϒL(1)[1](Gˆ/Gˆ1)s. Then Θ is isomorphic to Z[Q]. Proof Thanks to [8, 7.3.2], all the non-simple blocks of kG are Morita equivalent to the principal block B0(G) of kG. Additionally, by [8, 1.1], the block B0(G) is isomorphic to the block B0(Gˆ). Therefore, it suffices to prove this result for G≔Gˆ. In view of [12, 5.6] and [9, 3.1], all modules belonging to the component Θ have complexity 2. Let S1,…,Sm be the simple G/G1-modules and M∈Θ. Due to [16, 7.4], there are 0≤l≤p−2, n≥0 and 1≤j≤m such that M≅V(n,l)⊗kSj. Thanks to [8, 7.4.1], the group algebra kG is symmetric. Therefore, the Auslander–Reiten translation τG equals ΩG2 (see [3, 4.12.8]). Applying 2.4 and 2.5 to the almost split exact sequence 0→V(n+2,l)⟶V(n+1,l)⊕V(n+1,l)⟶V(n,l)→0 of SL(2)1-modules yields the almost split exact sequence 0→τG(V(n,l))⟶V(n+1,l)⊗kIrrSL(2)1(V(n+1,l),V(n,l))⟶V(n,l)→0 of G-modules. Due to 3.1, we have τG(V(n,l))≅ΩG2(V(n,l))≅V(n+2,l). In conjunction with 3.2 and 2.2, we now obtain the almost split exact sequence 0→V(n+2,l)⊗kSj⟶V(n+1,l)⊗kL(1)[1]⊗kSj⟶V(n,l)⊗kSj→0. Hence, τG(V(n,l)⊗kSj)≅V(n+2,l)⊗kSj. Moreover, due to the decomposition L(1)[1]⊗kSj≅⊗i=1maijSi, there are aij arrows V(n+1,l)⊗kSi→V(n,l)⊗kSj and aij arrows V(n+2,l)⊗kSj→V(n+1,l)⊗kSi in Θ. Without loss, we can now assume that n=0, so that V(0,l)⊗kSj belongs to Θ. Denote by {1,…,m} the vertex set of ϒL(1)[1](G/G1) and by {1,…,m,1′,…,m′} the vertex set of its separated quiver. Let Q be the connected component of ϒL(1)[1](G/G1)s which contains j′. If N is another module in Θ, then there are μ∈Z, 0≤l˜≤p−2 and t∈{1,…,m} with N≅V(μ,l˜)⊗kSt. By the above, M and N can only lie in the same component if l=l˜. If μ=2ν is even, then τG−ν(N)≅V(0,l)⊗kSt. As M and N are in the same component, there is a path V(0,l)⊗lSj←V(1,l)⊗lSi1→V(0,l)⊗lSi2←…←V(1,l)⊗lSir→V(0,l)⊗lSt in Θ. This gives rise to a path j′←i1→i2′←…←ir→t′ in the separated quiver ϒL(1)[1](G/G1)s. Consequently, t′∈Q. Similarly, if μ is odd, we obtain t∈Q. Moreover, for each arrow i→t′ in Q and μ∈Z, we have arrows φi,t′,μ:V(μ+1,l)⊗kSi→V(μ,l)⊗kStand φt′,i,μ+1:V(μ+2,l)⊗kSt→V(μ+1,l)⊗kSi in Θ. Now let ψ:Z[Q]→Θ be the morphism of stable translation quivers given by ψ(ν,t)=V(2ν+1,l)⊗kStforeachν∈Zandt∈{1,…,m} ψ(ν,t′)=V(2ν,l)⊗kStforeachν∈Zandt′∈{1′,…,m′} ψ((ν,i)→(ν,t′))=φi,t′,2ν ψ((ν+1,t′)→(ν,i))=φt′,i,2ν+1. One now easily checks that this is an isomorphism.□ 4. Quotients of support varieties and ramification The goal of this section is to describe a geometric connection between the tubes in the Auslander–Reiten quiver of a finite group scheme G and the corresponding tubes in the Auslander–Reiten quiver of a normal subgroup scheme N of G. We will see that the support variety of N is a geometric quotient of the support variety of G. The geometric connection will then be given via the ramification indices of the quotient morphism. Let k be an algebraically closed field. We say that X is a variety, if it is a separated reduced prevariety over k and we will identify it with its associated separated reduced k-scheme of finite type (cf. [14, 15]). A point x∈X is always supposed to be closed and, therefore, also to be k-rational, as k is algebraically closed. Let x∈X, R be a commutative k-algebra and ιR:k→R be the canonical inclusion. Then we denote by xR≔X(ιR)(x) the image of x in X(R). An action of a group scheme on a variety is always supposed to be an action via morphisms of schemes. Definition 4.1 Let H be a group scheme acting on a variety X. A pair (Y,q) consisting of a variety Y and an H-invariant morphism q:X→Y is called categorical quotient of X by the action of H, if for every H-invariant morphism q′:X→Y′ of varieties, there is a unique morphism α:Y→Y′ such that q′=α◦q. A pair (Y,q) consisting of a variety Y and an H-invariant morphism q:X→Y of varieties is called geometric quotient of X by the action of H, if the underlying topological space of Y is the quotient of the underlying topological space of X by the action of the group H(k) and q:X→Y is an H-invariant morphism of schemes such that the induced homomorphism of sheafs OY→q*(OX)H is an isomorphism. If x∈X is a point, then the stabilizer Hx is the subgroup scheme of H given by Hx(R)={g∈H(R)∣g.xR=xR} for every commutative k-algebra R. Thanks to [22, 12.1], there is for any finite group scheme H and any quasi-projective variety X and up to isomorphism uniquely determined geometric quotient which will be denoted by X/H. Moreover, the quotient morphism q:X→X/H is finite, i.e. there exists an open affine covering X/H=⋃i∈IVi such that q−1(Vi) is affine and the ring homomorphism k[Vi]→k[q−1(Vi)] is finite for all i∈I. Example 4.2 Let H be a finite group scheme and A be a finitely generated commutative k-algebra. Then the set X≔MaxspecA of maximal ideals of A is an affine variety. An action of H on X gives A the structure of a kH-module algebra over the Hopf algebra kH. This means that A is an H-module such that h.(ab)=∑(h)(h(1).a)(h(2).b) and h.1=ε(h)1 where ε:kH→k is the counit of kH. Then the set AH≔{a∈A∣h.a=ε(h)aforallh∈kH} of H-invariants is a subalgebra of A. As H is finite, the subalgebra is also finitely generated and X/H=MaxspecAH. Let A=⊕n≥0An be additionally graded with A0=k and A+=⊕n>0An be its irrelevant ideal. Then the set X=ProjA of maximal homogeneous ideals which do not contain A+ is a projective variety. If H acts on X, then X/H=ProjAH. Let X be a variety with structure sheaf OX. For a point x∈X, we denote by OX,x the local ring at the point x. We say that a point x∈X is simple, if its local ring OX,x is regular. If R is a commutative local ring, we will denote by Rˆ its completion at its unique maximal ideal. Let V be a k-vector space of dimension n with basis b1,…,bn. Let t1,…,tn be the corresponding dual basis of V*. We define the ring of polynomial functions of V as k[V]≔k[t1,…,tn]. Its completion k[[t1,…,tn]] at the maximal ideal (t1,…,tn) will be denoted by k[[V]]. For future reference, we will recall the following facts: Remark 4.3 Let X be an n-dimensional variety and x∈X. A point x∈X is simple if and only if OˆX,x≅k[[x1,…,xn]]. Let H be a finite group scheme acting on X and assume that there is a geometric quotient (Y,q) of this action. Then OˆY,q(x)≅(OˆX,x)Hx [20, Exercise 4.5(ii)]. Let H be a linearly reductive group scheme acting on k[[x1,…,xn]] via algebra automorphisms. Then there is an n-dimensional H-module V such that there is an H-equivariant isomorphism k[[V]]≅k[[x1,…,xn]] of k-algebras (cf. [26, Proof of 1.8]). Definition 4.4 Let f:X→Y be a finite morphism, x∈X and y=f(x). Let my be the maximal ideal of the local ring OY,y Then ex(f)≔dimkOX,x/myOX,x is called the ramification index of f at x. As the morphism f is finite, the induced homomorphism OY,y→OX,x endows OX,x with the structure of a finitely generated OY,y-module. Therefore, the number ex(f) is finite. For n∈N, let μ(n) be the finite group scheme with μ(n)(R)={x∈R∣xn=1} for every commutative k-algebra R. For a finite group scheme H, we denote by ∣H∣≔dimkkH its order. Lemma 4.5 Let Hbe a finite linearly reductive group scheme which acts faithfully on a 1-dimensional irreducible quasi-projective variety X. Denote by q:X→X/Hthe quotient morphism. If x∈Xis a simple point, then Hx≅μ(n)for some n∈Nand ex(q)=∣Hx∣. Proof Since x is a simple point and X is 1-dimensional, (i) yields an Hx-equivariant isomorphism OˆX,x≅k[[T]]. By (iii), we can assume that the one-dimensional k-vector space ⟨T⟩k is an Hx-module. As X is irreducible, the field of fractions of OX,x is the function field k(X) of X. If K is the kernel of the action of Hx on OX,x, then it also acts trivially on k(X) and therefore also on X. As H acts faithfully on X, it follows that K is trivial. Therefore, ⟨T⟩k is a faithful Hx-module and we can assume Hx=μ(n)⊆GL1, where n=∣Hx∣. As Hx acts via algebra automorphisms, this yields k[[T]]Hx=k[[Tn]]. By (ii), we have OˆX/H,q(x)≅OˆX,xHx. As a result, we obtain ex(q)=dimkOˆX,x/mq(x)OˆX,x=dimkk[T]/(Tn)=n=∣Hx∣. □ Let k be an algebraically closed field of characteristic p>0 and G be a finite group scheme. Denote by VG the cohomological support variety of G. By definition VG=MaxspecH•(G,k) is the spectrum of maximal ideals of the even cohomology ring H•(G,k) of G (in characteristic 2 one takes instead the whole cohomology ring). The projectivization of the cohomological support variety will be denoted by P(VG). For every finite-dimensional G-module M, there is a natural homomorphism ΦM:H•(G,k)→ExtG*(M,M) of graded k-algebras. The cohomological support variety of M is then defined as the subvariety VG(M)=Maxspec(H•(G,k)/kerΦM) of VG. For a subgroup scheme H of G let ι*,H:P(VH)→P(VG) be the morphism which is induced by the canonical inclusion ι:kH→kG. If N is a normal subgroup scheme of G, then G/N acts via automorphisms of graded algebras on H•(N,k) and, therefore, on P(VN). Proposition 4.6 Let Gbe a finite group scheme and Nbe a normal subgroup scheme of Gsuch that G/Nis linearly reductive. Then (P(VG),ι*,N)is a geometric quotient for the action of G/Non P(VN). Proof Since G/N is linearly reductive, the Lyndon–Hochschild–Serre spectral sequence [15, I.6.6(3)] yields an isomorphism ι•:H•(G,k)→H•(N,k)G/N. Therefore, (P(VG),ι*,N) is a geometric quotient for the action of G/N on P(VN).□ Let H be a group. A k-algebra A is called strongly H-graded if it admits a decomposition A=⊕g∈HAg such that AgAh=Agh for all g,h∈H. If U⊆H is a subgroup, the subalgebra AU≔⊕g∈UAg is a strongly U-graded k-algebra. Let N⊆H be a normal subgroup of H. Then A can be regarded as a strongly H/N-graded k-algebra via AgN≔⊕x∈gNAx for all g∈H. Let M be an A1-module. For g∈H, we denote by Mg the A1-module with the same underlying space M and A1-action twisted by g−1. We will call the subgroup GM≔{g∈H∣Mg≅M} the stabilizer of M. Let G be a finite group scheme, N⊆G be a normal subgroup scheme with G0⊂N and set G≔(G/N)(k). Due to the decomposition G≅G0⋊Gred, the group algebra kG is isomorphic to the skew group algebra kG0*G(k). As G0⊆N, we, therefore, obtain that kG has the structure of G-graded k-algebra with (kG)1=kN. Let M be an N-module. Then the Hopf-subalgebra (kG)GM of kG determines a unique subgroup scheme GM of G with kGM=(kG)GM. The group G acts on the module category modN via equivalences of categories modN→modN,M↦Mg for g∈G. Since these equivalences commute with the Auslander–Reiten translation of Γs(N), each g∈G induces an automorphism tg of the quiver Γs(N). Therefore, G acts on the set of components of Γs(N). For a component Θ, we write Θg=tg(Θ) and let GΘ={g∈G∣Θg=Θ} be the stabilizer of Θ. If M is an N-module which belongs to the component Θ, then GM⊆GΘ. As above, there is a unique subgroup scheme GΘ⊆G with kGΘ=(kG)GΘ. Now let N be an indecomposable non-projective N-module and Ξ the corresponding component in Γs(N). Assume that there is an indecomposable non-projective direct summand M of indNGN and let Θ be the corresponding component in Γs(G). A stable translation quiver with tree class A∞ has for each vertex M only one sectional path to the end of the component [2, (VII)]. The length of this path is called the quasi-length ql(M) of M. A stable translation quiver of the form Z[A∞]/(τn), n≥1, is called a tube of rank n. These components contain for each l≥1 exactly n modules of quasi-length l. Tubes of rank 1 are also called homogeneous tubes and all other tubes are called exceptional tubes. If A and B are G-modules which belong to the same AR-component ϒ, then VG(A)=VG(B) (cf. [9, 3.1]). Therefore, we can define VG(ϒ)≔VG(A) for some G-module A belonging to ϒ. If ϒ is a tube, then ∣P(VG(ϒ))∣=1 (cf. [9, 3.3(3)]). Thanks to [12, 5.6], we have ι*,N−1(P(VG(M)))=P(VN(resNGM)). By [19, 4.5.8], the module resNGM has an indecomposable direct summand which belongs to Ξ. Hence, xΞ∈ι*,N−1(P(VG(M)))=ι*,N−1(xΘ), so that ι*,N(xΞ)=xΘ. Proposition 4.7 Let Ξ be a tube of rank n and Θ be a tube of rank m. Assume that G0⊆Nand set G≔(G/N)(k). Moreover, assume the following G/Nis linearly reductive, i.e. p does not divide the order of G, G acts faithfully on P(VN), the variety P(VN)is one-dimensional and irreducible, xΞis a simple point of P(VN), and all modules belonging to Ξ are GΞstable.Then m≤exΞ(ι*,N)n. Proof By the above, the group algebra kG is a strongly G-graded k-algebra with (kG)1=kN. As G acts faithfully on the one-dimensional irreducible variety P(VN) with simple point xΞ, we obtain due to 4.5 that the stabilizer GxΞ is a cyclic group and we have the equality exΞ(ι*,N)=∣GxΞ∣. Now the assertion follows directly from [16, 6.3(d)].□ Let Z be the center of the group scheme SL(2) and G˜ a finite linearly reductive subgroup scheme of SL(2) with Z⊆G˜. As mentioned in Section 2, every domestic finite group scheme can be associated to one of the amalgamated polyhedral group schemes (SL(2)1G˜)/Z. There are five different possible classes for the choice of G˜ (for details see [8, 3.3]): binary cyclic group scheme, binary dihedral group scheme, binary tetrahedral group scheme, binary octahedral group scheme, and binary icosahedral group scheme. The latter three are always reduced group schemes. Proposition 4.8 Let Gbe an amalgamated polyhedral group scheme, N≔G1its first Frobenius kernel and Θ be a tube. Then Θ has rank exΞ(ι*,N). Proof If G˜ is a non-reduced binary dihedral group scheme, then one obtains this result directly from [16, 7.6] by comparing the numbers. With the same methods as in the proof of [16, 7.6], one can classify the indecomposable modules for the amalgamated cyclic group scheme. Again one obtains this result by comparing numbers. The other cases are either proved in the same way or with the following arguments: We can assume that G˜ is reduced. Let M be an indecomposable G-module which belongs to Θ and U be an indecomposable direct summand of resGΞGM with indGΞGU=M. Denote by Λ the Auslander–Reiten component of Γs(GΞ) which contains U. Then [19, 4.5.10] yields, that indGΞG:Λ→Θ is an isomorphism of stable translation quivers. Now the assertion follows from the cyclic case, as GΞ(k) is a cyclic group.□ Acknowledgement The results of this article are part of my doctoral thesis, which I am currently writing at the University of Kiel. I would like to thank my advisor Rolf Farnsteiner for his continuous support as well as for helpful remarks. Furthermore, I thank the members of my working group for proofreading. References 1 I. Assem , A. Skowronski and D. Simson , Elements of the Representation Theory of Associative Algebras: Volume 1: Techniques of Representation Theory Vol. 65 , Cambridge University Press , Cambridge, 2006 . Google Scholar CrossRef Search ADS 2 M. Auslander , I. Reiten and S. Smalø , Representation Theory of Artin Algebras, Cambridge Studies in Advanced Mathematics , Cambridge University Press , Cambridge, 1997 . 3 D. J. Benson , Representations and Cohomology: Volume 1, Basic Representation Theory of Finite Groups and Associative Algebras Vol. 2 , Cambridge University Press , Cambridge, 1998 . 4 W. W. Crawley-Boevey , On tame algebras and bocses , Proc. Lond. Math. Soc. 3 ( 1988 ), 451 – 483 . Google Scholar CrossRef Search ADS 5 M. Demazure and P. 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Kirchhoff , Classification of indecomposable modules for finite group schemes of domestic representation type, arXiv:1509.05203, 2015 . 17 H. Krause , Krull-Schmidt categories and projective covers, arXiv:1410.2822, 2014 . 18 H. F. Kreimer and M. Takeuchi , Hopf algebras and Galois extensions of an algebra , Indiana Univ. Math. J. 30 ( 1981 ), 675 – 692 . Google Scholar CrossRef Search ADS 19 A. Marcus , Representation Theory of Group Graded Algebras, Nova Science, 1999 . 20 B. Moonen , Abelian varieties, http://www.math.ru.nl/personal/bmoonen/research.html#bookabvar, 2015 . [Unpublished Book; Online; accessed 17-November-2015]. 21 K. Morita , Adjoint pairs of functors and Frobenius extensions , Sci. Rep. Tokyo Kyoiku Daigaku. Sect. A 9 ( 1965 ), 40 – 71 . 22 D. Mumford , C. P. Ramanujam and J. I. Manin , Abelian varieties, Volume 5 of Tata Institute of Fundamental Research studies in Mathematics , Oxford University Press , London, 1974 . 23 A. A. Premet , The Green ring of a simple three-dimensional Lie p-algebra , Soviet Math. 35 ( 1991 ), 56 – 67 . 24 I. Reiten and C. Riedtmann , Skew group algebras in the representation theory of Artin algebras , J. Algebra 92 ( 1985 ), 224 – 282 . Google Scholar CrossRef Search ADS 25 C. Riedtmann , Algebren, Darstellungsköcher, Überlagerungen und zurück , Commentarii Mathematici Helvetici 55 ( 1980 ), 199 – 224 . Google Scholar CrossRef Search ADS 26 M. Satriano , The Chevalley–Shephard–Todd Theorem for finite linearly reductive group schemes , Algebra Number Theory 6 ( 2012 ), 1 – 26 . Google Scholar CrossRef Search ADS 27 S. Xanthopoulos , On a question of Verma about indecomposable representations of algebraic groups and of their Lie algebras. PhD Thesis, Queen Mary, University of London, 1992 . © 2017. Published by Oxford University Press. All rights reserved. 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AR-Components of domestic finite group schemes: McKay-Quivers and Ramification

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Abstract

Abstract For a domestic finite group scheme, we give a direct description of the Euclidean components in its Auslander–Reiten quiver via the McKay-quiver of a finite linearly reductive subgroup scheme of SL(2). Moreover, for a normal subgroup scheme N of a finite group scheme G, we show that there is a connection between the ramification indices of the restriction morphism ℙ(VN)→ℙ(VG) between their projectivized cohomological support varieties and the ranks of the tubes in their Auslander–Reiten quivers. 1. Introduction The Auslander–Reiten quiver of a self-injective finite-dimensional algebra is a powerful tool for understanding its representation theory via combinatorical invariants. In this work, we are mainly interested in the Auslander–Reiten components of a group algebra kG of domestic representation type for a finite group scheme G over an algebraically closed field k. We say that an algebra A has tame representation type if it possesses infinitely many isomorphism classes of indecomposable modules, and if in each dimension almost all isomorphism classes of indecomposable modules occur in only finitely many one-parameter families. If additionally the number of one-parameter families is uniformly bounded, we say that A has domestic representation type. The group algebra kG≔k[G]* of a finite group scheme G is the dual of its coordinate ring. We say that G is domestic if its group algebra is domestic. In [11], Farnsteiner classified the domestic finite group schemes over an algebraically closed field of characteristic p>2. Any such group scheme can be associated to a so-called amalgamated polyhedral group scheme. Moreover, the non-simple blocks of an amalgamated polyhedral group scheme are Morita-equivalent to a trivial extension of a radical square zero tame hereditary algebra. In this way, the components of the Auslander–Reiten quiver of these group schemes are classified abstractly. Our goal is to describe these components in a direct way by using tensor products, McKay-quivers and ramification indices of certain morphisms. We will start by describing the Euclidean components. For this purpose, we show how to extend certain almost split sequences from a normal subgroup scheme N⊆G to almost split sequences of G if the group scheme G/N is linearly reductive. Moreover, for a simple G/N-module S, we will show that the tensor functor −⊗kS sends these extended almost split sequences to almost split sequences. In Section 2, we will use these results to show that for any amalgamated polyhedral group scheme G, there is a finite linearly reductive subgroup scheme G˜⊆SL(2) such that the Euclidean components of Γs(G) can be explicitly described by the McKay-quiver ϒL(1)(G˜). Of great importance for the proofs of these results is the fact that the category of G-modules is closed under taking tensor products of G-modules. This comes into play in the definition of the McKay-quiver, for the construction of new almost split sequences and in the description of the Euclidean components. Thanks to this property, we are also able to introduce geometric invariants for the representation theory of G. If G is any finite group scheme, one can endow the even cohomology ring H•(G,k) with the structure of a commutative graded k-algebra. Thanks to the Friedlander–Suslin Theorem [13], this algebra is finitely generated. Therefore, the maximal ideal spectrum VG of H•(G,k) is an affine variety. As H•(G,k) is graded, we can also consider its projectivized variety P(VG). Now let us again assume that N⊆G is a normal subgroup scheme such that G/N is linearly reductive. Then the ramification indices of the restriction morphism P(VN)→P(VG) will give upper bounds for ranks of the corresponding tubes in the Auslander–Reiten quiver. Here a tube Z/(r)[A∞] of rank r can be regarded as a quiver which is arranged on an infinite tube with circumference r. Moreover, if G is an amalgamated polyhedral group scheme and N=G1 is its first Frobenius kernel, the ranks are equal to the corresponding ramification indices. Altogether we will prove the following: Theorem Let Gbe an amalgamated polyhedral group scheme and Θa component of the stable Auslander–Reiten quiver Γs(G). Then the following hold: If Θ is Euclidean, then there is a component Q of the separated quiver ϒL(1)(G˜)sand a concrete isomorphism Θ≅Z[Q]. Let Θ be a tube and eΘthe ramification index of the restriction morphism P(VG1)→P(VG)at the corresponding point xΘ. Then Θ≅Z/(eΘ)[A∞]. There seems to be a connection to a result of Crawley–Boevey [4], which states that a finite-dimensional tame algebra has only finitely many non-homogeneous tubes. On the other side, the restriction morphism P(VN)→P(VG) is finite and has constant ramification on an open dense subset of P(VN). Therefore, there are only finitely many exceptional ramification points. In our situation, all but finitely many points will be unramified and a tube can only be non-homogeneous, if it belongs to the image of a ramification point. 2. Induction of almost split sequences Let k be an algebraically closed field. Given a finite group scheme G with normal subgroup scheme N, we want to investigate conditions under which the induction functor indNG:modN→modG sends an almost split exact sequence to a direct sum of almost split exact sequences. We start by giving an overview of the functorial approach to almost split sequences. For further details, we refer the reader to [1, IV.6]. Let A be a finite-dimensional k-algebra. Denote by FunopA and FunA the categories of contravariant and covariant k-linear functors from modA to modk. A functor F in FunopA is finitely generated if the functor F is isomorphic to a quotient of HomA(−,M) for some M∈modA. A functor F in FunopA is finitely presented if there is an exact sequence HomA(−,M)→HomA(−,N)→F→0 of functors in FunopA for some M,N∈modA. The full subcategory of FunopA consisting of the finitely presented functors will be denoted by mmodA. Up to isomorphism, the finitely generated projective functors in FunopA are exactly the functors of the form HomA(−,M). Such a functor is indecomposable if and only if the A-module M is indecomposable. For A-modules, M and N we define the radical of HomA(M,N) as RadA(M,N)≔{φ∈HomA(M,N)∣φisnotanisomorphism}. Then RadA(−,M) is a subfunctor of HomA(−,M) and we define the functor SM≔HomA(−,M)/RadA(−,M). Up to isomorphism, the simple functors in FunopA are exactly the functors of the form SM with an indecomposable A-module M. The projective cover of SM is HomA(−,M). Let N be an indecomposable A-module. An A-module homomorphism g:M→N is (minimal) right almost split if and only if the induced exact sequence HomA(−,M)→HomA(−,N)→SN→0 of functors in FunopA is a (minimal) projective presentation of SN. A functor F:modA→modB induces a functor F:mmodA→mmodB via F(HomA(−,M))=HomB(−,F(M)). There are dual notions and results for left almost split morphisms and functors in FunA. In [24, 3.8], Reiten and Riedtmann used this functorial approach to show that for a skew group algebra A∗G, with a finite group G, the induction functor ind1G:modA→modA∗G and the restriction functor res1G:modA∗G→modA send almost split sequences to direct sums of almost split sequences if the group order of G is invertible in k. Their proof relied on the following properties of the involved functors: (A) (i) There is a split monomorphism of functors idmodA→res1Gind1G. (ii) There is a split epimorphism of functors ind1Gres1G→idmodA*G. (B) (ind1G,res1G) and (res1G,ind1G) are adjoint pairs of functors. (C̃) There is a finite group G acting on modA such that for every A-module M, there is a decomposition res1Gind1GM=⊕g∈GMg and if φ:M→N is A-linear, then res1Gind1G(φ)=(g.φ)g∈G:⊕g∈GMg→⊕g∈GNg. In [24, 3.5], it was shown that these properties also hold for the induced functors ind1G:mmodA→mmodA∗G and res1G:mmodA*G→mmodA and that they imply the following property: (C) ind1G:mmodA→mmodA∗G and res1G:mmodA∗G→mmodA preserve semisimple objects and projective covers. We want to apply the ideas of the proof in the context of group algebras of finite group schemes. If G is a finite group scheme, then the dual of its coordinate ring kG≔(k[G])* is called the group algebra of G. In this situation, we will not always have analogous results for the induction and restriction functor. For example, in [10, 3.1.4], it was already shown that for the restriction functor, this is possible if and only if the ending term of the almost split sequence fulfils a certain regularity property. Let N be a normal subgroup scheme of G. The N-modules which will be of our interest are restrictions of G-modules. To obtain an analog of property (C̃), we will use the following result: Lemma 2.1 Let Gbe a finite group scheme and Nbe a normal subgroup scheme of G. Let M be a G-module. Then there is a G-linear isomorphism ψM:indNGresNGM→M⊗kk(G/N)which is natural in M. In particular, resNGindNGM≅Mn, where n=dimkk(G/N). Proof This follows directly from the tensor identity. Alternatively, exactly as in the proof of [16, 5.1] one can show that the map ψM:indNGM→M⊗kk(G/N),a⊗m↦∑(a)a(1)m⊗π(a(2)) is an isomorphism of G-modules, where π:kG→k(G/N) is the canonical projection. A direct computation shows that it is natural in M.□ Proposition 2.2 Let Gbe a finite group scheme and Nbe a normal subgroup scheme such that G/Nis linearly reductive. Let X,Yand E be N-modules such that every indecomposable direct summand of the modules X,Yand E is the restriction of a G-module, and E:0→X⟶φE⟶ψY→0is an almost split exact sequence of N-modules.Then indNGEis a direct sum of almost split exact sequences. Proof Denote by C the full subcategory of modN consisting of direct sums of indecomposable N-modules which are restrictions of G-modules. Due to 2.1, the properties (A)(i) and (C̃) hold in C if G is a group acting trivially on C. As G/N is linearly reductive, the k(G/N)-Galois extension kG:kN is separable by [6, 3.15]. This yields property (A)(ii). Thanks to [18, 1.7(5)], the ring extension kG:kN is a free Frobenius extension of first kind, i.e. kG is a finitely generated free kN-module, and there is a (kG,kN)-bimodule isomorphism kG→HomN(kG,kN). Hence, by [21, 2.1], the induction and coinduction functors are equivalent, so that property (B) holds. As ψ is minimal right almost split, the exact sequence HomN(−,E)→HomN(−,Y)→SY→0 is a minimal projective presentation of SY. We can now apply the arguments of the proof of [24, 3.6] by keeping the following things in mind: The simple functors which occur in our situation are all of the form SV with V being an indecomposable module in C. The projective covers which occur in our situation are all of the form HomN(−,V) with V∈C. Therefore, the functor indNGSY is semisimple and the exact sequence indNGHomN(−,E)→indNGHomN(−,Y)→indNGSY→0 is a minimal projective presentation of indNGSY≅SindNGY. Hence, indNGψ is a direct sum of minimal right almost split homomorphisms. Dually, one can show that indNGφ is a direct sum of minimal left almost split homomorphisms.□ Let G be a group scheme and X and M be G-modules. We define the k-vector spaces RadG2(X,M)≔{α∣∃Z∈ModG,φ∈RadG(X,Z),ψ∈RadG(Z,M):α=ψ◦φ} and IrrG(X,M)≔RadG(X,M)/RadG2(X,M). Lemma 2.3 Let Gbe a reduced group scheme, Nbe a normal subgroup scheme of Gand X and M be G-modules. Then RadN(X,M)and RadN2(X,M)are G-submodules of HomN(X,M). Proof As G is reduced, we only need to prove that RadN(X,M) and RadN2(X,M) are G(k)-submodules of HomN(X,M) (see [15, Remark after 2.8]). Let g∈G(k) and φ∈RadN(X,M). Assume that g.φ is an isomorphism with inverse ψ. Then g−1.ψ is an inverse of φ, a contradiction. Let g∈G(k) and φ∈RadN2(X,M). Then there is an N-module Z and homomorphisms α∈RadN(X,Z), β∈RadN(Z,M) such that φ=β◦α. Denote by Zg the N-module Z with action twisted by g−1, i.e. h·z=hg−1z for h∈kN and z∈Zg. As X and M are G-modules, we can define the N-linear maps α˜:X→Zg,x↦α(g−1x)andβ˜:Zg→M,z↦gβ(z). Then we obtain g.φ=β˜◦α˜. If α˜ is an isomorphism with inverse γ:Zg→X, then the N-linear map γ˜:Z→X,z→g−1γ(z) is an inverse of α, a contradiction. In the same way, β˜ is not an isomorphism. Hence g.φ=β˜◦α˜∈RadN2(X,M).□ Proposition 2.4 Let Gbe a finite subgroup scheme of a reduced group scheme Hand N⊆Ga normal subgroup scheme of Hsuch that G/Nis linearly reductive. Let X and M be indecomposable G-modules such that there is an almost split exact sequence E:0→τN(M)⟶Xn⟶M→0of N-modules. Then there is a short exact sequence E˜:0→N⟶X⊗kIrrN(X,M)⟶M→0of G-modules such that resNGE˜=E. Proof Consider the map ψ˜:X⊗kHomN(X,M)→M,x⊗f↦f(x) of G-modules. As G/N is linearly reductive, the short exact sequence 0→RadN2(X,M)⟶RadN(X,M)⟶IrrN(X,M)→0 of G-modules splits. This yields a decomposition RadN(X,M)≅RadN2(X,M)⊕IrrN(X,M). Hence, there results a G-linear map ψ:X⊗kIrrN(X,M)→M by restricting ψ˜ to X⊗kIrrN(X,M). Let (fi)1≤i≤n:Xn→M be the surjection given by E. Since E is almost split, the maps (fi)1≤i≤n form a k-basis of IrrN(X,M) (cf. [1, IV.4.2]). Therefore, the restriction resNG(ψ) equals (fi)1≤i≤n.□ In 2.2, we have seen that if we apply the induction functor to certain almost split sequences, they will be the direct sum of almost split sequences. Our next result enables us to describe this decomposition under a certain regularity condition. Let M be a multiplicative group scheme, i.e. the coordinate ring of M is isomorphic to the group algebra kX(M) of its character group. Then we obtain for any M-module M a weight space decomposition M=⊕λ∈X(M)Mλ with Mλ≔{m∈M∣hm=λ(h)mforallh∈kM}. Let G be a finite group scheme and N⊆G0 a normal subgroup scheme of G such that G0/N is linearly reductive. By Nagata’s Theorem [5, IV,§3,3.6], an infinitesimal group scheme is linearly reductive if and only if it is multiplicative. Let M be a G0-module. For any λ∈X(G0/N), we obtain a G0-module M⊗kkλ, the tensor product of M with the one-dimensional G0/N-module defined by λ. This defines an action of X(G0/N) (here we identify X(G0/N) with a subgroup of X(G0) via the canonical inclusion X(G0/N)↪X(G0) which is induced by the canonical projection G0↠G0/N) on the isomorphism classes of G0-modules and we define the stabilizer of a G0-module as X(G0/N)M≔{λ∈X(G0/N)∣M⊗kkλ≅M}. A G0-module M is called G0/N-regular if its stabilizer is trivial. Proposition 2.5 In the situation of2.4, assume that M and N are G0/N-regular. Let S be a simple G/N-module. Then the short exact sequence E˜⊗kSis almost split. Proof Let k(G/N)=⊕i=1nSi be the decomposition into simple G/N-modules. For any G-module U, let ψU:indNGU→U⊗kk(G/N) be the natural isomorphism given by 2.1. We obtain the following commutative diagram with exact rows: As all vertical arrows are isomorphisms, the two exact sequences indNGE and ⊕i=1nE˜⊗kSi are equivalent. Thanks to 2.2, the exact sequence indNGE is a direct sum of almost split sequences. By [16, 5.1], the G-modules M⊗kS and N⊗kS are indecomposable. Moreover, the sequence E˜⊗kS does not split. Otherwise, the sequence resNG(E˜⊗kS) would split and, therefore, also E. Hence, the sequence E˜⊗kS is equivalent to an indecomposable direct summand of indNGE. As the Krull–Schmidt theorem holds in the category of short exact sequences of finite-dimensional G-modules (as the space of morphisms between those sequences is finite-dimensional, cf. [17]), it follows that E˜⊗kSi is almost split.□ 3. The McKay and Auslander–Reiten quiver of finite domestic group schemes Let k be an algebraically closed field of characteristic p>2. The group algebra kSL(2)1 is isomorphic to the restricted universal enveloping algebra U0(sl(2)) of the restricted Lie algebra sl(2). There are one-to-one correspondences between the representations of SL(2)1, U0(sl(2)) and sl(2). The indecomposable representations of the restricted Lie algebra sl(2) were classified by Premet in [23]. For d∈N0, we consider the (d+1)-dimensional Weyl module V(d) of highest weight d. Weyl modules are rational SL(2)-modules which are obtained by twisting the 2-dimensional standard module with the Cartan involution ( x↦−xtr) and taking its dth symmetric power. For d≤p−1, we obtain in this way exactly the simple U0(sl(2))-modules L(0),…,L(p−1). Let A be a self-injective k-algebra. We denote by Γs(A) the stable Auslander–Reiten quiver of A. The vertices of this valued quiver are the isomorphism classes of non-projective indecomposable A-modules and the arrows correspond to the irreducible morphisms between these modules. Moreover, we have an automorphism τA of Γs(A), called the Auslander–Reiten translation. For a self-injective algebra, the Auslander–Reiten translation is the composite ν◦Ω2, where ν denotes the Nakayama functor of modA and Ω the Heller shift of modA. For further details, we refer to [1, 2]. Let Q be a quiver. We denote by Z[Q] the translation quiver with underlying set Z×Q, arrows (n,x)→(n,y) and (n+1,y)→(n,x) for any arrow x→y in Q and translation τ:Z[Q]→Z[Q] given by τ(n,x)=(n+1,x). Let Θ⊆Γs(A) be a connected component of the stable Auslander–Reiten quiver of A. Thanks to the Struktursatz of Riedtmann [25], there is an isomorphism of stable translation quivers Θ≅Z[TΘ]/Π, where TΘ denotes a directed tree and Π is an admissible subgroup of Aut(Z[TΘ]). The underlying undirected tree T¯Θ is called the tree class of Θ. The Auslander–Reiten quiver of each block of kSL(2)1 consists of two components of type Z[A˜1,1] and infinitely many homogeneous tubes Z[A∞]/(τ). Thanks to [10, 4.1], each of the p−1 Euclidean components Θ(i) contains exactly one simple SL(2)1-module L(i) with 0≤i≤p−2. This component is then given by Θ(i)={Ω2n(L(i)),Ω2n+1(L(p−2−i))∣n∈Z} with almost split sequences 0→Ω2n+2(L(i))⟶Ω2n+1(L(p−2−i))⊕Ω2n+1(L(p−2−i))⟶Ω2n(L(i))→0. In the following, it will be convenient to have a common notation for the Weyl modules and their duals. We set V(n,i)≔{V(np+i)ifn≥0V(−np+i)*ifn≤0 for n∈Z and 0≤i≤p−1. Lemma 3.1 Let Gbe a finite subgroup scheme of SL(2)with SL(2)1⊆Gsuch that G/SL(2)1is linearly reductive. Let 0≤i≤p−2and n∈Z. Then ΩG2n(L(i))≅V(2n,i)and ΩG2n+1(L(i))≅V(2n+1,p−2−i).In particular, Θ(i)={V(n,i)∣n∈Z}. Proof By [27, 7.1.2], there is for n≤0 a short exact sequence 0→V(n,i)⟶P(i)⊗k(V(n)*)[1]⟶V(n−1,p−2−i)→0 of SL(2)-modules, where (V(n)*)[1] denotes the first Frobenius twist of V(n)* (cf. [15, I.9.10]). As P(i) is a projective SL(2)-module, we obtain that the G-module P(i)⊗k(V(n)*)[1] is projective. Hence, there is a projective G-module P with ΩG(V(n−1,p−2−i))⊕P≅V(n,i). Since V(n,i) is a non-projective indecomposable G-module, we obtain ΩG(V(n−1,p−2−i))≅V(n,i). Dualizing the above sequence yields ΩG(V(n,i))=V(n+1,p−2−i) for n≥0 in the same way.□ Lemma 3.2 For all n∈Zand 0≤i≤p−2the SL(2)-modules L(1)[1]and IrrSL(2)1(V(n+1,i),V(n,i))are isomorphic. Proof As in the proof of 3.1, there is for all n∈Z a short exact sequence 0→V(n+1,i)⟶P⟶V(n,p−2−i)→0 of SL(2)-modules with P being projective over SL(2)1. Applying the functor HomSL(2)1(−,V(n,i)) to this sequence yields the following exact sequence of SL(2)-modules: HomSL(2)1(P,V(n,i))→αHomSL(2)1(V(n+1,i),V(n,i))→βExtSL(2)11(V(n,p−2−i),V(n,i))→ExtSL(2)11(P,V(n,i)). As P is a projective SL(2)1-module, we obtain that ExtSL(2)11(P,V(n,i))=(0). Hence the map β is surjective. By [7, 2.4], the SL(2)-modules L(1)[1] and ExtSL(2)11(V(n,p−2−i),V(n,i)) are isomorphic. As imα=kerβ is contained in RadSL(2)12(V(n+1,i),V(n,i)), we obtain a surjective morphism L(1)[1]→IrrSL(2)1(V(n+1,i),V(n,i)) of SL(2)-modules. Since both modules are 2-dimensional, this map is an isomorphism.□ Let H be a finite linearly reductive group scheme, S1,…,Sn a complete set of pairwise non-isomorphic simple H-modules and L be an H-module. For each 1≤j≤n, there are aij≥0 such that L⊗kSj≅⊕i=1naijSi. The McKay quiver ϒL(H) of H relative to L is the quiver with underlying set {S1,…Sn} and aij arrows from Si to Sj. Let G be a domestic finite group scheme. We denote by Glr the largest linearly reductive normal subgroup scheme of G. Let Z be the center of the group scheme SL(2). Thanks to [11, 4.3.2], there is a finite linearly reductive subgroup scheme G˜ of SL(2) with Z⊆G˜ such that G/Glr is isomorphic to (SL(2)1G˜)/Z. Group schemes which arise in this fashion are also called amalgamated polyhedral group schemes. By the above, any domestic finite group scheme can be associated to one of these group schemes. If G is an amalgamated polyhedral group scheme, we set Gˆ≔SL(2)1G˜. For any Euclidean diagram (A˜n)n∈N,(D˜n)n≥4 and (E˜n)6≤n≤8, we will denote in the same way the quiver where each edge •−• is replaced by a pair of arrows •⇆•. As shown in the proof of [8, 7.2.3], the McKay quiver ϒL(1)[1](Gˆ/Gˆ1) is isomorphic to one of the quivers A˜2npr−1−1,D˜npr−1+2,E˜6,E˜7,E˜8, where r is the height of Gˆ0 and (n,p)=1. For any quiver Q, we denote by Qs its separated quiver. If {1,…n} is the vertex set of Q, then Qs has 2n vertices {1,…n,1′,…n′} and arrows i→j′ if and only if i→j is an arrow in Q. The separated quiver of one of the quivers A˜2npr−1−1,D˜npr−1+2,E˜6,E˜7,E˜8 is the union of 2 quivers with the same underlying graph as the original quiver and each vertex is either a source or a sink. Theorem 3.3 Let Gbe an amalgamated polyhedral group scheme and Θ a component of Γs(G)containing a G-module of complexity 2. Let Q be a connected component of ϒL(1)[1](Gˆ/Gˆ1)s. Then Θ is isomorphic to Z[Q]. Proof Thanks to [8, 7.3.2], all the non-simple blocks of kG are Morita equivalent to the principal block B0(G) of kG. Additionally, by [8, 1.1], the block B0(G) is isomorphic to the block B0(Gˆ). Therefore, it suffices to prove this result for G≔Gˆ. In view of [12, 5.6] and [9, 3.1], all modules belonging to the component Θ have complexity 2. Let S1,…,Sm be the simple G/G1-modules and M∈Θ. Due to [16, 7.4], there are 0≤l≤p−2, n≥0 and 1≤j≤m such that M≅V(n,l)⊗kSj. Thanks to [8, 7.4.1], the group algebra kG is symmetric. Therefore, the Auslander–Reiten translation τG equals ΩG2 (see [3, 4.12.8]). Applying 2.4 and 2.5 to the almost split exact sequence 0→V(n+2,l)⟶V(n+1,l)⊕V(n+1,l)⟶V(n,l)→0 of SL(2)1-modules yields the almost split exact sequence 0→τG(V(n,l))⟶V(n+1,l)⊗kIrrSL(2)1(V(n+1,l),V(n,l))⟶V(n,l)→0 of G-modules. Due to 3.1, we have τG(V(n,l))≅ΩG2(V(n,l))≅V(n+2,l). In conjunction with 3.2 and 2.2, we now obtain the almost split exact sequence 0→V(n+2,l)⊗kSj⟶V(n+1,l)⊗kL(1)[1]⊗kSj⟶V(n,l)⊗kSj→0. Hence, τG(V(n,l)⊗kSj)≅V(n+2,l)⊗kSj. Moreover, due to the decomposition L(1)[1]⊗kSj≅⊗i=1maijSi, there are aij arrows V(n+1,l)⊗kSi→V(n,l)⊗kSj and aij arrows V(n+2,l)⊗kSj→V(n+1,l)⊗kSi in Θ. Without loss, we can now assume that n=0, so that V(0,l)⊗kSj belongs to Θ. Denote by {1,…,m} the vertex set of ϒL(1)[1](G/G1) and by {1,…,m,1′,…,m′} the vertex set of its separated quiver. Let Q be the connected component of ϒL(1)[1](G/G1)s which contains j′. If N is another module in Θ, then there are μ∈Z, 0≤l˜≤p−2 and t∈{1,…,m} with N≅V(μ,l˜)⊗kSt. By the above, M and N can only lie in the same component if l=l˜. If μ=2ν is even, then τG−ν(N)≅V(0,l)⊗kSt. As M and N are in the same component, there is a path V(0,l)⊗lSj←V(1,l)⊗lSi1→V(0,l)⊗lSi2←…←V(1,l)⊗lSir→V(0,l)⊗lSt in Θ. This gives rise to a path j′←i1→i2′←…←ir→t′ in the separated quiver ϒL(1)[1](G/G1)s. Consequently, t′∈Q. Similarly, if μ is odd, we obtain t∈Q. Moreover, for each arrow i→t′ in Q and μ∈Z, we have arrows φi,t′,μ:V(μ+1,l)⊗kSi→V(μ,l)⊗kStand φt′,i,μ+1:V(μ+2,l)⊗kSt→V(μ+1,l)⊗kSi in Θ. Now let ψ:Z[Q]→Θ be the morphism of stable translation quivers given by ψ(ν,t)=V(2ν+1,l)⊗kStforeachν∈Zandt∈{1,…,m} ψ(ν,t′)=V(2ν,l)⊗kStforeachν∈Zandt′∈{1′,…,m′} ψ((ν,i)→(ν,t′))=φi,t′,2ν ψ((ν+1,t′)→(ν,i))=φt′,i,2ν+1. One now easily checks that this is an isomorphism.□ 4. Quotients of support varieties and ramification The goal of this section is to describe a geometric connection between the tubes in the Auslander–Reiten quiver of a finite group scheme G and the corresponding tubes in the Auslander–Reiten quiver of a normal subgroup scheme N of G. We will see that the support variety of N is a geometric quotient of the support variety of G. The geometric connection will then be given via the ramification indices of the quotient morphism. Let k be an algebraically closed field. We say that X is a variety, if it is a separated reduced prevariety over k and we will identify it with its associated separated reduced k-scheme of finite type (cf. [14, 15]). A point x∈X is always supposed to be closed and, therefore, also to be k-rational, as k is algebraically closed. Let x∈X, R be a commutative k-algebra and ιR:k→R be the canonical inclusion. Then we denote by xR≔X(ιR)(x) the image of x in X(R). An action of a group scheme on a variety is always supposed to be an action via morphisms of schemes. Definition 4.1 Let H be a group scheme acting on a variety X. A pair (Y,q) consisting of a variety Y and an H-invariant morphism q:X→Y is called categorical quotient of X by the action of H, if for every H-invariant morphism q′:X→Y′ of varieties, there is a unique morphism α:Y→Y′ such that q′=α◦q. A pair (Y,q) consisting of a variety Y and an H-invariant morphism q:X→Y of varieties is called geometric quotient of X by the action of H, if the underlying topological space of Y is the quotient of the underlying topological space of X by the action of the group H(k) and q:X→Y is an H-invariant morphism of schemes such that the induced homomorphism of sheafs OY→q*(OX)H is an isomorphism. If x∈X is a point, then the stabilizer Hx is the subgroup scheme of H given by Hx(R)={g∈H(R)∣g.xR=xR} for every commutative k-algebra R. Thanks to [22, 12.1], there is for any finite group scheme H and any quasi-projective variety X and up to isomorphism uniquely determined geometric quotient which will be denoted by X/H. Moreover, the quotient morphism q:X→X/H is finite, i.e. there exists an open affine covering X/H=⋃i∈IVi such that q−1(Vi) is affine and the ring homomorphism k[Vi]→k[q−1(Vi)] is finite for all i∈I. Example 4.2 Let H be a finite group scheme and A be a finitely generated commutative k-algebra. Then the set X≔MaxspecA of maximal ideals of A is an affine variety. An action of H on X gives A the structure of a kH-module algebra over the Hopf algebra kH. This means that A is an H-module such that h.(ab)=∑(h)(h(1).a)(h(2).b) and h.1=ε(h)1 where ε:kH→k is the counit of kH. Then the set AH≔{a∈A∣h.a=ε(h)aforallh∈kH} of H-invariants is a subalgebra of A. As H is finite, the subalgebra is also finitely generated and X/H=MaxspecAH. Let A=⊕n≥0An be additionally graded with A0=k and A+=⊕n>0An be its irrelevant ideal. Then the set X=ProjA of maximal homogeneous ideals which do not contain A+ is a projective variety. If H acts on X, then X/H=ProjAH. Let X be a variety with structure sheaf OX. For a point x∈X, we denote by OX,x the local ring at the point x. We say that a point x∈X is simple, if its local ring OX,x is regular. If R is a commutative local ring, we will denote by Rˆ its completion at its unique maximal ideal. Let V be a k-vector space of dimension n with basis b1,…,bn. Let t1,…,tn be the corresponding dual basis of V*. We define the ring of polynomial functions of V as k[V]≔k[t1,…,tn]. Its completion k[[t1,…,tn]] at the maximal ideal (t1,…,tn) will be denoted by k[[V]]. For future reference, we will recall the following facts: Remark 4.3 Let X be an n-dimensional variety and x∈X. A point x∈X is simple if and only if OˆX,x≅k[[x1,…,xn]]. Let H be a finite group scheme acting on X and assume that there is a geometric quotient (Y,q) of this action. Then OˆY,q(x)≅(OˆX,x)Hx [20, Exercise 4.5(ii)]. Let H be a linearly reductive group scheme acting on k[[x1,…,xn]] via algebra automorphisms. Then there is an n-dimensional H-module V such that there is an H-equivariant isomorphism k[[V]]≅k[[x1,…,xn]] of k-algebras (cf. [26, Proof of 1.8]). Definition 4.4 Let f:X→Y be a finite morphism, x∈X and y=f(x). Let my be the maximal ideal of the local ring OY,y Then ex(f)≔dimkOX,x/myOX,x is called the ramification index of f at x. As the morphism f is finite, the induced homomorphism OY,y→OX,x endows OX,x with the structure of a finitely generated OY,y-module. Therefore, the number ex(f) is finite. For n∈N, let μ(n) be the finite group scheme with μ(n)(R)={x∈R∣xn=1} for every commutative k-algebra R. For a finite group scheme H, we denote by ∣H∣≔dimkkH its order. Lemma 4.5 Let Hbe a finite linearly reductive group scheme which acts faithfully on a 1-dimensional irreducible quasi-projective variety X. Denote by q:X→X/Hthe quotient morphism. If x∈Xis a simple point, then Hx≅μ(n)for some n∈Nand ex(q)=∣Hx∣. Proof Since x is a simple point and X is 1-dimensional, (i) yields an Hx-equivariant isomorphism OˆX,x≅k[[T]]. By (iii), we can assume that the one-dimensional k-vector space ⟨T⟩k is an Hx-module. As X is irreducible, the field of fractions of OX,x is the function field k(X) of X. If K is the kernel of the action of Hx on OX,x, then it also acts trivially on k(X) and therefore also on X. As H acts faithfully on X, it follows that K is trivial. Therefore, ⟨T⟩k is a faithful Hx-module and we can assume Hx=μ(n)⊆GL1, where n=∣Hx∣. As Hx acts via algebra automorphisms, this yields k[[T]]Hx=k[[Tn]]. By (ii), we have OˆX/H,q(x)≅OˆX,xHx. As a result, we obtain ex(q)=dimkOˆX,x/mq(x)OˆX,x=dimkk[T]/(Tn)=n=∣Hx∣. □ Let k be an algebraically closed field of characteristic p>0 and G be a finite group scheme. Denote by VG the cohomological support variety of G. By definition VG=MaxspecH•(G,k) is the spectrum of maximal ideals of the even cohomology ring H•(G,k) of G (in characteristic 2 one takes instead the whole cohomology ring). The projectivization of the cohomological support variety will be denoted by P(VG). For every finite-dimensional G-module M, there is a natural homomorphism ΦM:H•(G,k)→ExtG*(M,M) of graded k-algebras. The cohomological support variety of M is then defined as the subvariety VG(M)=Maxspec(H•(G,k)/kerΦM) of VG. For a subgroup scheme H of G let ι*,H:P(VH)→P(VG) be the morphism which is induced by the canonical inclusion ι:kH→kG. If N is a normal subgroup scheme of G, then G/N acts via automorphisms of graded algebras on H•(N,k) and, therefore, on P(VN). Proposition 4.6 Let Gbe a finite group scheme and Nbe a normal subgroup scheme of Gsuch that G/Nis linearly reductive. Then (P(VG),ι*,N)is a geometric quotient for the action of G/Non P(VN). Proof Since G/N is linearly reductive, the Lyndon–Hochschild–Serre spectral sequence [15, I.6.6(3)] yields an isomorphism ι•:H•(G,k)→H•(N,k)G/N. Therefore, (P(VG),ι*,N) is a geometric quotient for the action of G/N on P(VN).□ Let H be a group. A k-algebra A is called strongly H-graded if it admits a decomposition A=⊕g∈HAg such that AgAh=Agh for all g,h∈H. If U⊆H is a subgroup, the subalgebra AU≔⊕g∈UAg is a strongly U-graded k-algebra. Let N⊆H be a normal subgroup of H. Then A can be regarded as a strongly H/N-graded k-algebra via AgN≔⊕x∈gNAx for all g∈H. Let M be an A1-module. For g∈H, we denote by Mg the A1-module with the same underlying space M and A1-action twisted by g−1. We will call the subgroup GM≔{g∈H∣Mg≅M} the stabilizer of M. Let G be a finite group scheme, N⊆G be a normal subgroup scheme with G0⊂N and set G≔(G/N)(k). Due to the decomposition G≅G0⋊Gred, the group algebra kG is isomorphic to the skew group algebra kG0*G(k). As G0⊆N, we, therefore, obtain that kG has the structure of G-graded k-algebra with (kG)1=kN. Let M be an N-module. Then the Hopf-subalgebra (kG)GM of kG determines a unique subgroup scheme GM of G with kGM=(kG)GM. The group G acts on the module category modN via equivalences of categories modN→modN,M↦Mg for g∈G. Since these equivalences commute with the Auslander–Reiten translation of Γs(N), each g∈G induces an automorphism tg of the quiver Γs(N). Therefore, G acts on the set of components of Γs(N). For a component Θ, we write Θg=tg(Θ) and let GΘ={g∈G∣Θg=Θ} be the stabilizer of Θ. If M is an N-module which belongs to the component Θ, then GM⊆GΘ. As above, there is a unique subgroup scheme GΘ⊆G with kGΘ=(kG)GΘ. Now let N be an indecomposable non-projective N-module and Ξ the corresponding component in Γs(N). Assume that there is an indecomposable non-projective direct summand M of indNGN and let Θ be the corresponding component in Γs(G). A stable translation quiver with tree class A∞ has for each vertex M only one sectional path to the end of the component [2, (VII)]. The length of this path is called the quasi-length ql(M) of M. A stable translation quiver of the form Z[A∞]/(τn), n≥1, is called a tube of rank n. These components contain for each l≥1 exactly n modules of quasi-length l. Tubes of rank 1 are also called homogeneous tubes and all other tubes are called exceptional tubes. If A and B are G-modules which belong to the same AR-component ϒ, then VG(A)=VG(B) (cf. [9, 3.1]). Therefore, we can define VG(ϒ)≔VG(A) for some G-module A belonging to ϒ. If ϒ is a tube, then ∣P(VG(ϒ))∣=1 (cf. [9, 3.3(3)]). Thanks to [12, 5.6], we have ι*,N−1(P(VG(M)))=P(VN(resNGM)). By [19, 4.5.8], the module resNGM has an indecomposable direct summand which belongs to Ξ. Hence, xΞ∈ι*,N−1(P(VG(M)))=ι*,N−1(xΘ), so that ι*,N(xΞ)=xΘ. Proposition 4.7 Let Ξ be a tube of rank n and Θ be a tube of rank m. Assume that G0⊆Nand set G≔(G/N)(k). Moreover, assume the following G/Nis linearly reductive, i.e. p does not divide the order of G, G acts faithfully on P(VN), the variety P(VN)is one-dimensional and irreducible, xΞis a simple point of P(VN), and all modules belonging to Ξ are GΞstable.Then m≤exΞ(ι*,N)n. Proof By the above, the group algebra kG is a strongly G-graded k-algebra with (kG)1=kN. As G acts faithfully on the one-dimensional irreducible variety P(VN) with simple point xΞ, we obtain due to 4.5 that the stabilizer GxΞ is a cyclic group and we have the equality exΞ(ι*,N)=∣GxΞ∣. Now the assertion follows directly from [16, 6.3(d)].□ Let Z be the center of the group scheme SL(2) and G˜ a finite linearly reductive subgroup scheme of SL(2) with Z⊆G˜. As mentioned in Section 2, every domestic finite group scheme can be associated to one of the amalgamated polyhedral group schemes (SL(2)1G˜)/Z. There are five different possible classes for the choice of G˜ (for details see [8, 3.3]): binary cyclic group scheme, binary dihedral group scheme, binary tetrahedral group scheme, binary octahedral group scheme, and binary icosahedral group scheme. The latter three are always reduced group schemes. Proposition 4.8 Let Gbe an amalgamated polyhedral group scheme, N≔G1its first Frobenius kernel and Θ be a tube. Then Θ has rank exΞ(ι*,N). Proof If G˜ is a non-reduced binary dihedral group scheme, then one obtains this result directly from [16, 7.6] by comparing the numbers. With the same methods as in the proof of [16, 7.6], one can classify the indecomposable modules for the amalgamated cyclic group scheme. Again one obtains this result by comparing numbers. The other cases are either proved in the same way or with the following arguments: We can assume that G˜ is reduced. Let M be an indecomposable G-module which belongs to Θ and U be an indecomposable direct summand of resGΞGM with indGΞGU=M. Denote by Λ the Auslander–Reiten component of Γs(GΞ) which contains U. Then [19, 4.5.10] yields, that indGΞG:Λ→Θ is an isomorphism of stable translation quivers. Now the assertion follows from the cyclic case, as GΞ(k) is a cyclic group.□ Acknowledgement The results of this article are part of my doctoral thesis, which I am currently writing at the University of Kiel. 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