Endotrivial modules for finite groups schemes II
aa r X i v : . [ m a t h . G R ] A p r ENDOTRIVIAL MODULES FOR FINITE GROUP SCHEMES II
JON F. CARLSON AND DANIEL K. NAKANO
Abstract.
It is well known that if G is a finite group then the group of endotrivialmodules is finitely generated. In this paper we prove that for an arbitrary finite groupscheme G , and for any fixed integer n >
0, there are only finitely many isomorphismclasses of endotrivial modules of dimension n . This provides evidence to support thespeculation that the group of endotrivial modules for a finite group scheme is alwaysfinitely generated. The result also has some applications to questions about lifting andtwisting the structure of endotrivial modules in the case that G is an infinitesimal groupscheme associated to an algebraic group. Introduction
Let S be a finite group scheme over a field k . The endotrivial modules for S form animportant class of modules which, among other things, determine self equivalences of thestable category of S -modules, modulo projective S -modules. In the case that G is thescheme of a finite p -group, there is a complete classification of endotrivial modules [7].This classification has been extended to the group algebras of many other families of finitegroups (cf. [2, 4, 5]).An endotrivial module is an S -module M with the property that Hom k ( M, M ) ∼ = M ⊗ M ∗ ∼ = k ⊕ P (as S -modules) where P is a projective S -module. Two endotrivial modules M and N are equivalent if there exist projective modules P and Q such that M ⊕ P ∼ = N ⊕ Q .The equivalence classes of endotrivial S -modules forms an abelian group T ( S ) under tensorproduct. It was shown by Puig [11] that this group is finitely generated in the case that S is a finite group. This fact also follows from the classification. For arbitrary finite groupschemes, it is an open question as to whether T ( S ) is finitely generated.One of the main ingredients in proving that T ( S ) is finitely generated is a demonstrationthat for any fixed non-negative integer n ≥
0, there are only finitely many endotrivialmodules of dimension equal to n . When S is a unipotent group scheme this was proved bythe authors in [3, Theorem 3.5]. In Section 2, we extend these earlier results by showingthat for arbitrary finite group schemes there are only finitely many endotrivial modulesfor a given dimension. This result will be referred to as the “Finiteness Theorem”.The Finiteness Theorem has some very strong connections with the notion of liftingendotrivial modules to action of H where H is a group scheme containing S as a normalsubgroup scheme. In the case when H is connected, the Finiteness Theorem implies thatevery endotrivial S -module is H -stable (i.e., the twists of an endotrivial module M h are Date : April 4, 2019.1991
Mathematics Subject Classification. all isomorphic to M (as S -module) for all h ∈ H ). Different notions of lifting such as“tensor stability” and “numerical stability” have been investigated recently by Parshalland Scott [10]. In Section 3, we outline these various definitions, and we introduce the newconcept of lifting called “stably lifting” which entails lifting S -modules to H -structures inthe stable module category for S . We connect our new notion of stable lifting with theideas presented by Parshall and Scott.Let G be semisimple algebraic group scheme, B a Borel subgroup with unipotent radical U defined and split over F p , and let k be an algebraically closed field of characterstic p . Let G r , B r , U r denote their r th infinitesimal Frobenius kernels. The existence of the Steinbergmodule shows that every projective G r -module (resp. B r , U r ) lifts to G (resp. B , U ).Furthermore, if M stably lifts then one can use this result to show that all syzygies Ω n ( M )stably lift. In [3, Theorem 5.7, Theorem 6.1], T ( B ) and T ( U ) was completely determinedfor all primes. By using this classification, we prove that all B (resp. U ) endotrivialmodules stably lift to B (resp. U ). We suspect this will also hold for G r . Finally, weexhibit an endotrivial module for B , namely Ω ( k ) when the root system is of type A and p = 2, which does not admit a B -structure.The second author would like to acknowledge Shun-Jen Cheng, Weiqiang Wang andtheir conference organizing team for providing financial support and for efforts in hostinga first rate workshop on representation theory in Taipei during December 2010.2. The finiteness theorem
We begin by introducing the basic definitions which will be used throughout this paper.Let S be a finite group scheme. We will consider the category Mod( S ) of rational S -modules and the stable module category StMod( S ). Since S is a finite group scheme thenotion of projectivity is equivalent to injectivity. For M and N objects in Mod( S ), we saythat [ M ] = [ N ] in StMod( S ) if and only if M ⊕ P ∼ = N ⊕ Q for some projective S -modules, P and Q .Suppose that S is any finite group scheme defined over k . An S -module is an endotrivialmodule provided that, as S -modules,Hom k ( M, M ) ∼ = k ⊕ P for some projective S -module P . Equivalently, an S -module M is endotrivial moduleif [ M ∗ ⊗ M ] = [ k ] in StMod( S ). Note for any S -module M there exists a canonicalisomorphism Hom k ( M, M ) ∼ = M ∗ ⊗ M . We can now define the group T ( S ) of endotrivial S -modules as follows. The objects in T ( S ) are the equivalence classes [ M ] in StMod( S )of endotrivial S -modules. The group operation is given by [ M ] + [ N ] = [ M ⊗ N ]. Theidentity element is the class [ k ] and the inverse of [ M ] is the class [ M ∗ ]. The group T ( S ) isabelian because the associated coproduct used to construct the action of S on the tensorproduct of modules is cocommutative.In this section, we show that the number of endotrivial modules for a finite group scheme H having any particular dimension is finite. The proof follows somewhat the same linesif that in [3] and also [11], and is based on an idea of Dade [8].Suppose that k is an algebraically closed field. In this section it will be more convenientto work with finite-dimensional cocommutative Hopf algebras. For a finite group scheme H , let A = k [ H ] ∗ , the group algebra of H . There is an equivalence of categories between NDOTRIVIAL MODULES FOR FINITE GROUP SCHEMES II 3 H -modules and A -module. Furthermore, the finite dimensional k -algebra A is a cocom-mutative Hopf algebra. As a consequence, projective A -modules are also injective. Let P , . . . , P r be a complete set of representatives of the isomorphism classes of projectiveindecomposable A -modules. Each P i has a simple top S i = P i / Rad( P i ) and a simpleSocle T i = Soc( P i ). The collection { S , . . . , S r } is a complete set of isomorphism classes ofsimple modules, as is the set { T , . . . , T r } . We need not assume here that T i ∼ = S i , thoughthis is often the case.For each i = 1 , . . . , r , we assume that P i is a left ideal in A . That is, we assumethat P i = Ae i for e i a primitive idempotent in A . For each i choose a nonzero element u i ∈ T i ⊆ P i . Then u i has the property that Au i = T i . Note that since u i ∈ P i , we havethat u i = u i e i and that u i P i = { } . Lemma 2.1.
Let i be an integer between 1 and r . Suppose that M is an A -module and that u i M = { } . Then M has a direct summand isomorphic to P i . Moreover, if t i = Dim( u i P i ) is the rank of the operator of left multiplication by u i on P i and a i = Dim( u i M ) /t i , then M ∼ = P a i i ⊕ M ′ where M ′ has no direct summands isomorphic to P i .Proof. Let m ∈ M be an element such that u i m = 0. Then, u i m = u i e i m . Define ψ : P i −→ M by ψ ( ae i ) = ae i m for any a in A . This is well defined since P i = Ae i . Now, ψ ( u i ) = 0 and hence ψ ( T i ) = { } . Therefore, ψ is injective. Because, A is a self-injectivering, the image of ψ is a direct summand of M and hence M ∼ = P i ⊕ N for some submodule N of M . This proves the first statement.The second statement follows by an easy induction beginning with the observation that(as vector spaces) u i M ∼ = u i P i ⊕ u i N. (cid:3) For a positive integer n , let V n denote the variety consisting of all representations of thealgebra A of dimension n . It is defined as follows. Suppose that the collection a , . . . , a t isa chosen set of generators of the algebra A , so that every element of A can be written asa polynomial in (noncommuting) variables a , . . . , a t . We fix this set of generators for theremainder of the discussion in this section. Then we have that A ∼ = k h a , . . . , a t i / J forsome ideal J . A representation of dimension n of A is a homomorphism θ : A −→ M n ( k ),where M n ( k ) is the ring of n × n matrices over k . The representation is completelydetermined by the assignment to each a i of an n × n matrix θ ( a i ) = ( a irs ).For the purposes of defining the variety V n , we consider the polynomial ring R = k [ x irs ]in tn (commuting) variables with 1 ≤ i ≤ t , 1 ≤ r, s ≤ n , and the assignment a i ↔ ( x irs ) ∈ M n ( R )for i = 1 , . . . , t . The ideal J determines an ideal I in the ring R . That is, any relation f ( a , . . . , a t ) in J , when converted into an expression on the matrices using the aboveassignment, defines a collection of relations, one for each r and s in the elements of R .For example, if it were the case that a a = 0 in A , then the polynomial a a would bean element of J , and for each r and s , the polynomial P nu =1 x ru x us would be an elementof I . JON F. CARLSON AND DANIEL K. NAKANO
Lemma 2.2.
Suppose that M is an A -module and P is an indecomposable projective A -module. Let s be a nonnegative integer. Let W be the subset of V n consisting of allrepresentations σ of A having the property that M ⊗ L σ has no submodule isomorphic P s ,where L σ is the A -module affording σ . Then W is a closed set in V n .Proof. By the previous lemma, there is an element u ∈ A such that, if t is the rank of thematrix of u acting on P , then P s is isomorphic to a submodule of M ⊗ L σ if and only ifthe rank of the matrix M u of u on M ⊗ L σ is at least st . Now we observe that u is apolynomial in the (noncommuting) generators of A and hence the entries of the matrix of u on L σ are polynomials in the entries of the matrices of the generators of A on L σ . If wefix a representation of M , then the entries of the matrix of u on M are elements of the basefield k . It follows that the entries of the matrix M u of u on M ⊗ L σ are all polynomialsin the variables of the ring R = k [ x irs ]. Likewise, the determinant of any st × st submatrixof M u is a polynomial in the variables of R . As a consequence, the condition that everysuch determinant is zero (which is the same as saying that the rank of M u is less than st )defines a closed set in V n . (cid:3) Lemma 2.3.
Suppose that M is an endotrivial A -module of dimension n . Let W be theset of all representations σ in V n such that the module L σ afforded by σ is not isomorphicto M ⊗ χ for any one dimensional A -module χ . Then W is a closed set in V n .Proof. Suppose that N is a one dimensional A -module. Because M is endotrivial, we havethat N ⊗ M is endotrivial. Thus( N ⊗ M ) ⊗ ( N ∗ ⊗ M ∗ ) ∼ = k ⊕ r X i =1 P n i i where P , . . . , P r are the indecomposable projective A -modules and n , . . . , n r are non-negative integers. For each i , let W i be the set of all σ ∈ V n with the property that L σ ⊗ N ∗ ⊗ M ∗ does not contain a submodule isomorphic to P n i i . The sets W i are closedby Lemma 2.2. Hence, the set U N = W ∪ · · · ∪ W r is also closed. If σ is not in U N , then L σ ⊗ N ∗ ⊗ M ∗ ∼ = U ⊕ r X i =1 P n i i for some A -module U . But the dimension of U must be one because dim L σ ⊗ N ∗ ⊗ M ∗ =dim( N ⊗ M ) ⊗ ( N ∗ ⊗ M ∗ ). Therefore, L σ ⊗ N ∗ ⊗ M ∗ ⊗ U ∗ ∼ = k ⊕ r X i =1 U ∗ ⊗ P n i i and hence, L σ ∼ = U ⊗ N ⊗ M . Now we claim that W = ∩ N U N where N runs through theone dimensional A -modules. So W is closed. (cid:3) At this point we are ready to prove our main theorem.
Theorem 2.4.
For any natural number n , there is only a finite number of isomorphismclasses of endotrivial modules of dimension n . NDOTRIVIAL MODULES FOR FINITE GROUP SCHEMES II 5
Proof.
Suppose that M is an indecomposable endotrivial module of dimension n . Let U be the subset of V n consisting of representations σ with the property that the underlyingmodule L σ is isomorphic to N ⊗ M for some A -module N of dimension one. Note that A has only finitely many isomorphism classes of dimension one, and hence there are onlyfinitely many isomorphism classes of modules represented in U .By Lemma 2.3, U is an open set in V n . Hence, U , the closure of U is a union ofcomponents in V n . Therefore, the theorem is proved with the observation that V n hasonly finitely many components. (cid:3) Liftings and Stability
Let S be a finite group scheme which is a normal subgroup scheme in a group scheme H . In this section we will describe different notions of when an S -module has a structurethat extends to H .We say that an S -module M lifts to H if M has an H -module structure whose restrictionto S agrees with the (original) S -module structure. This is the strongest form of “lifting”.The weakest form of lifting is the notion of H -stable. Let M be a S -module. For h ∈ H ,one can consider the twisted module M h which is a S ′ = h − Sh -module (cf. [9, I. 2.15]).In particular if h normalizes S then the twisted module M h is an S -module. An S -module M is called H -stable if and only if M h ∼ = M . If the S -module M lifts to H -module, then M is H -stable. The converse statement is not true as we will see in Section 6 (cf. [10,4.2.1]).Following [10, 2.2.2, 2.2.3] we recall the notions of numerical and tensor stablity definedby Parshall and Scott. Definition 3.1. An S -module M is numerically H -stable if there exists an H -module Z such that Z | S ∼ = M ⊕ n . Definition 3.2. An S -module M is tensor H -stable if there exists a finite-dimensional H/S -module Y such that M ⊗ Y is an H -module whose restriction to S coincides with M ⊗ ( Y | S ) . Tensor H -stability is equivalent to numerically H -stable (cf. [10, 2.2.3]). It is clear thatif M lifts to H then M is tensor H -stable and numerically H -stable. Furthermore, tensor H -stable and numerically H -stable imply H -stable.Next we introduce a new concept of lifting which will be relevant for our study ofendotrivial modules. Definition 3.3. An S -module M stably lifts to H if there exists an H -module K suchthat K | S ∼ = M ⊕ P where P is a projective S -module. Observe that in the definition of stable lifting to H , the S -modules K and M representthe same object in StMod( S ), the stable category of all S -modules. If M lifts H then M stably lifts to H . Also, if M is non-projective as an S -module and stably lifts to H thenby using the Krull-Schmidt theorem and the fact that twists of projective S -modules areprojective, it follows that M is H -stable. JON F. CARLSON AND DANIEL K. NAKANO Applications
Suppose that G is a semisimple, simply connected algebraic group, defined and splitover the finite field F p with p elements for a prime p . Let k be the algebraic closure of F p . Let Φ be a root system associated to G with respect to a maximal split torus T . LetΦ + (resp. Φ − ) be the set of positive (resp. negative) roots and ∆ be a base consisting ofsimple roots. Let B be a Borel subgroup containing T corresponding to the negative rootsand let U denote the unipotent radical of B . More generally, if J ⊂ ∆, let L J be the Levisubgroup generated by the root subgroups with roots in ∆, P J the associated (negative)parabolic subgroup and U J its unipotent radical such that P J = L J ⋉ U J .Let H be an affine algebraic group scheme over k and let H r = ker F r . Here F : H → H (1) is the Frobenius map and F r is the r th iteration of the Frobenius map. We note thatthere is a categorical equivalence between modules for the restricted p -Lie algebra Lie( H )of H and H -modules. For each value of r , the group algebra kH r is the distributionalgebra Dist( H r ) (cf. [9]). In general, for the rest of this paper, we use Dist( H r ) to denotethe group algebra of H r .For any group scheme H , let mod( H ) be the category of finite dimensional rational H -modules. This construction can be applied when H = G , B , P J , L J , U , U J and T .Note that the use of T (maximal torus) and T ( − ) (endotrivial group) will be clear fromthe context. Let X := X ( T ) be the integral weight lattice obtained from Φ. The set X has a partial ordering: if λ, µ ∈ X , then λ ≥ µ if and only if λ − µ ∈ P α ∈ Π N α .Let α ∨ = 2 α/ h α, α i is the coroot corresponding to α ∈ Φ. The set of dominant integralweights is defined by X + := X ( T ) + = { λ ∈ X ( T ) : 0 ≤ h λ, α ∨ i for all α ∈ ∆ } . Furthermore, the set of p r -restricted weights is X r ( T ) = { λ ∈ X : 0 ≤ h λ, α ∨ i < p r for all α ∈ ∆ } . Let X ( T r ) be the set of characters of T r which can be identified with the set of onedimensional simple modules for T r .For a reductive algebraic group G , the simple modules are labelled L ( λ ) and the inducedmodules are H ( λ ) = ind GB λ , where λ ∈ X ( T ) + . The Weyl module V ( λ ) is defined as V ( λ ) = H ( − w λ ) ∗ . Let T ( λ ) be the indecomposable tilting module with highest weight λ . We can now apply the Finiteness Theorem to demonstrate, under mild assumptions on H , that every endotrivial module is H -stable. In the following sections we show that theproblem of lifting of endotrivial modules is rather subtle. Theorem 4.1.
Let H be a connected affine algebraic group scheme and S be a finite groupscheme which is a normal subgroup scheme of H . If M is an endotrivial S -module then M is H -stable.Proof. Consider the closed subgroup A = { h ∈ H : M h ∼ = M } in H . According to thefiniteness theorem, this must have finite index in H because there are only finitely manyendotrivial S -modules of any fixed dimension. Therefore, A must contain the connectedcomponent of H . Because H is connected, we have that A = H , which proves the theorem. (cid:3) NDOTRIVIAL MODULES FOR FINITE GROUP SCHEMES II 7
Corollary 4.2.
Let H = G , B , P J , L J , U , U J and T as above, and H r be the r thFrobenius kernel. If M is an endotrivial H r -module, then M is H -stable. Another application of the Finiteness Theorem involves proving that the restriction ofan endotrivial G r -modules to conjugate unipotent radicals of Borel subgroups producessyzygies of the same degree. Theorem 4.3.
Let U and U ′ be the unipotent radicals of the Borel subgroups B and B ′ .If M is an endotrivial G r -module with M | U r ∼ = Ω n U r ( k ) ⊕ ( proj ) and M | U ′ r ∼ = Ω n U ′ r ( k ) ⊕ ( proj ) then n = n .Proof. The Borel subgroups B and B ′ are conjugate by some w ∈ W , and B ′ = B w = w ( B ) w − . According to Corollary 4.2, we have M ∼ = M w as G r -modules. Under thisisomorphism U r is isomorphic to U ′ r and M | U r can be identified with M | U ′ r = M w . Theresult now follows by applying these isomorphisms. (cid:3) Lifting Endotrivial Modules
Let M be an S -module and Ω nS ( M ) ( n = 0 , , , . . . ) be the n th syzygy of M obtainedby taking a projective resolution of M . We will assume that Ω nS ( M ) has no projective S -summands. By taking an injective resolution of M we can define Ω nS ( M ) for n negative.Note that if M is endotrivial over S then Ω nS ( M ) is an endotrivial module for all n ∈ Z .The following result provides conditions on when there are stable liftings for the syzygiesof an H -module M (when considered as an S -module). Proposition 5.1.
Let S be a finite group scheme which is a normal subgroup scheme of H .Suppose there exists a projective S -module P which lifts to an H -module. Furthermore,suppose that there exists a surjective H -map P → k . If M is a finite-dimensional H -module then Ω nS ( M ) stably lifts to H for each n ∈ Z .Proof. Consider the surjective H -module homomorphism P → k . We will prove thatΩ nS ( M ) stably lifts by induction on n ∈ N . For n ≤ n = 0, Ω S ( M ) = M which is an H -module so we can set K = M . Now assume that K n | S ∼ = Ω nS ( M ) ⊕ Q n where K n is an H -module K n and Q n is a projective S -module. Nowdefine K n +1 as the kernel in the short exact sequence obtained by tensoring the complex P → k by K n : 0 → K n +1 → P ⊗ K n → K n → . Then K n +1 is an H -module with K n +1 | S ∼ = Ω n +1 S ( M ) ⊕ Q n +1 for some projective S -module Q n +1 . (cid:3) Let G be a reductive group with subgroups P , B and U as before. The existence ofthe Steinberg representation St r can be used to prove that every projective G r (resp. P r , B r , U r ) module stably lifts to G (resp. P , B , U ). It is only known for p ≥ h − G r -modules lift to G , but there is strong evidence this holds for all p . JON F. CARLSON AND DANIEL K. NAKANO
We can now prove that for reductive groups and their associated Lie type subgroups thatthe syzygies of the trivial module lift stably. The proof also utilizes the existence of theSteinberg representation.
Theorem 5.2.
Let H = G (resp. P , B , U ) and S = G r (resp. P r , B r , U r ). For each n ∈ Z , Ω nS ( k ) stably lifts to H .Proof. It suffices to prove the theorem in the case that H = G and S = G r . The othercases will follow by restriction. Let St r = L (( p r − ρ ) be the Steinberg module, and set P := St r ⊗ L (( p r − ρ ). Then there exists a surjective G -module homomorphism P → k [9, II 10.15 Lemma]. The result now follows by Proposition 5.1 (cid:3) We note that it is not trivial to prove the fact that the left U r -module structure onDist( U r ) lifts to H = U . The conjugation action of U r on Dist( U r ) lifts to U and thereexists a U -module map Dist( U r ) → k under the conjugation action. However, the moduleDist( U r ) is not a projective module under this action (i.e., the conjugation action doesnot lift the left action of Dist( U r ) on itself). Corollary 5.3.
Let H = B (resp. U ), and S = B r (resp. U r ). Then every endotrivial S -module lifts stably to an H -module.Proof. We first consider the case that H = B and S = B r . According to [3, Theorem 6.1,6.2], T ( B r ) ∼ = X ( T r ) × T ( U r ). The one dimensional B r endotrivial modules correspondingelements of X ( T r ) are all B -modules. Therefore, it suffices to prove the statement when H = U and S = U r .Assume that Φ is not A in the case that p = 2 and r = 1. Then any endotrivial B r -module is isomorphic to Ω nB r ( λ ) for some λ ∈ X r ( T ). Since λ lifts to a B -module, byTheorem 5.2, Ω nB r ( λ ) stably lifts to B .In the case when Φ is of type A the endotrivial group T ( B ) is generated by Ω B ( λ )and the simple three dimensional G -module L ( ω ) considered as B -module by restriction.Since L ( ω ) is a B -module all of its syzygies Ω nB ( L ( ω )) stably lift to B by Proposition 5.1. (cid:3) When G is a reductive algebraic group scheme we can state a relationship between a G r -module lifting stably to G and tensor stability as a direct application of [10, Theorem1.1]. This seems to indicate that stably lifting is a stronger form of lifting that tensorstability. Proposition 5.4.
Let G be reductive and let M be a G r -module which lifts stably to G .Suppose that N is a G -module such that N | G r = M ⊕ P where P is a projective G r -module(i.e., a stable lifting of M ). If soc G r M is a G -submodule of N then M is tensor G r -stable. In the next theorem we give a condition on the quotient
H/S which insures that we canlift syzygies.
Theorem 5.5.
Let S be a finite group scheme which is a normal subgroup scheme of H .Assume that (i) If L is a simple H -module, then L | S is a simple S -module, and all simple S -moduleslift to L . NDOTRIVIAL MODULES FOR FINITE GROUP SCHEMES II 9 (ii)
For any simple H -module L there exists a H -module Q ( L ) such that Q ( L ) | S is theprojective cover L | S . (iii) All finite-dimensional modules for
H/S are completely reducible.Let M be a finite-dimensional H -module. Assume that the projective cover P ( M ) of M as an S -module lifts to an H -module and there exists a surjective H -homomorphism P ( M ) → M . Then Ω nS ( M ) lifts to an H -module for all n ∈ Z .Proof. We begin with an observation about the cohomology. For any H -module N , thereexists a Lyndon-Hochschild-Serre (LHS) spectral sequence: E i,j : Ext iH/S ( k, Ext jS ( k, N )) ⇒ Ext i + jH ( k, N ) . Condition (iii) implies that this spectral sequence collapses and hence, the restriction mapExt jH ( k, N ) → Ext jS ( k, N ) H/S is an isomorphism for all j ≥ n ≥ n isnegative can be handled by using a dual argument. For n = 0, we have that Ω S ( M ) ∼ = M and for n = 1, Ω S ( M ) ∼ = ker( P ( M ) → M ). So these modules lift to H .Suppose that Ω nS ( M ) lifts to H . The S -submodule Rad S Ω nS ( M ) is an H -submodule ofΩ nS ( M ) so there exists a surjective H -module map π : Ω nS ( M ) → Ω nS ( M ) / Rad S Ω nS ( M ) . The quotient module Ω nS ( M ) / Rad S Ω nS ( M ) is completely reducible as an S -module. Fur-thermore, it must be completely reducible as an H -module. For if there exists a non-trivialextension of simple H -modules which lives as an H -submodule in Ω nS ( M ) / Rad S Ω nS ( M ),then by condition (i), these simple modules remain simple upon restriction to S and thisextension must split over S (by complete reducibility of the quotient module). Then bythe first observation above, the original extension over H must split.By condition (ii), there exists an H -module Q whose restriction to S is the projectivecover of Ω nS ( M ) / Rad S Ω nS ( M ) with a surjective H -module map γ : Q → Ω nS ( M ) / Rad S Ω nS ( M ).We have a short exact sequence of H -modules:0 → Rad S Ω nS ( M ) → Ω nS ( M ) → Ω nS ( M ) / Rad S Ω nS ( M ) → . Observe that Ext H ( Q, Rad S Ω nS ( M )) = Ext S ( Q, Rad S Ω nS ( M )) H/S = 0. Therefore, in thelong exact sequence in cohomology the mapHom H ( Q, Ω nS ( M )) → Hom H ( Q, Ω nS ( M ) / Rad S Ω nS ( M ))is surjective and we can find and H -module map δ : Q → Ω nS ( M ) such that π ◦ δ = γ andΩ n +1 S ( M ) = ker δ . Consequently, Ω n +1 S ( M ) is an H -module. (cid:3) In the case that H = G r T (resp. P r T , B r T ) and S = G r (resp. P r , B r ) conditions(i)-(iii) of the preceding theorem can be verified (cf. [9, Chapter 9]). Corollary 5.6.
Let H = G r T (resp. P r T , B r T ) and S = G r (resp. P r , B r ). If M is an H -module then for each n ∈ Z , Ω nS ( M ) is an H -module. The conditions (i)-(iii) do not hold for when H = G and S = G r . Nonetheless, we canprove by a direct calculation that all endotrivial G -modules for G = SL lift to G . Theorem 5.7.
Let G = SL . Then every endotrivial G -module lifts to G . Proof.
The category of G -modules has tame representation type and the indecomposablemodules have been determined (cf. [6, Section 3]). The modules of complexity two, whichinclude all endotrivial modules, lift to G .One can also verify this by using the classification of endotrivial modules given in [3].The endotrivial group is Z ⊕ Z and all endotrivial modules are of the form Ω n ( M ) where n ∈ Z and M = k or M = L ( p − M ∼ = M ∗ where M ∗ is the k -dual of M . Hence Ω n ( M ) ∼ = Ω − n ( M ) ∗ , andwithout loss of generality, we may assume that n ≥
0. The minimal projective resolution P • → k can be constructed explicitly. All the terms are tilting modules: P n ∼ = ( T (( n + 1)2( p − n is even ,T (( n +12 )2 p ) if n is odd . Moreover, the syzygies are Weyl modulesΩ n ( k ) ∼ = ( V ( np ) if n is even ,V (( n +12 )2( p − n is odd . Using a similar construction, the minimal projective resolution b P • → L ( p −
2) consists oftilting modules and the syzygies are also Weyl modules. b P n ∼ = ( T (( n + 1) p ) if n is even ,T (( n + 2) p −
2) if n is odd , Ω n ( L ( p − ∼ = ( V (( n + 1) p −
2) if n is even ,V ( np ) if n is odd . (cid:3) An example where Ω S ( k ) does not lift to H In this section we show that syzygies of the trivial module do not, in general, lift to H -modules even in cases where all the projective indecomposable S -modules lift to H .Suppose that G = SL and that p = 2. Let S = B , H = B . The restricted p -Lie algebra u of the unipotent radical of G , has the same representation theory as the infinitesimalunipotent subgroup U ⊆ B . In this context we prove the following. Proposition 6.1.
The second syzygy Ω ( k ) := Ω B ( k ) stably lifts to a B -module, but doesnot lift to a B -module.Proof. The first statement follows from Theorem 5.2. We suppose that Ω ( k ) has a B -structure and prove that this leads to a contradiction.Let V := L ( ω ) be the three dimensional natural representation for G and label thesimple roots ∆ = { α , α } . We will consider the restriction of V to B and B . Set N ∼ = V ⊗ ( − α − α − ω )and N ∼ = V ⊗ ( − α − α − ω ) . NDOTRIVIAL MODULES FOR FINITE GROUP SCHEMES II 11 ( − α − α )( − α − α ) ( − α − α )( − α − α ) ( − α − α )( − α − α ) ( − α − α )( − α ) ( − α ) ❅❅❘ ❅❅❘ ❅❅❘ (cid:0)(cid:0)(cid:0)✠ ❅❅❅❘ (cid:0)(cid:0)✠(cid:0)(cid:0)✠(cid:0)(cid:0)✠ Figure 1.
We can represent Ω ( k ) diagrammatically as in Figure 1. A node ( λ ) is a one-dimensional B i -submodule with highest weight λ . The arrow indicate the action of the simple root-subspace vectors in u := Lie U , and the extensions between the simple one-dimensional B -modules. That is, an arrow that goes down and left represents multiplication by u α ,while an arrow going down and right is multiplication by u α .By analyzing the structure of Ω ( k ) we can conclude thatΩ ( k ) / Rad B Ω ( k ) ∼ = u (1) and Rad B Ω ( k ) ∼ = N ⊕ N as B -modules.The module Ω ( k ) is indecomposable over B . Therefore, if Ω ( k ) has a B -structurethen it is indecomposable over B and represents a non-trivial extension class inExt B (Ω ( k ) / Rad B Ω ( k ) , Rad B Ω ( k )) . This implies that Ext B ( u (1) , N j ) = 0 for j = 1 or 2.Our task is to show by a cohomological calculation that Ext B ( u (1) , N j ) = 0 for j = 1and 2. This provides a contradiction to the assumption that Ω ( k ) has a compatible B -structure. By symmetry we can simply look at the case that j = 1. Apply the LHSspectral sequence E i,j = Ext iB/B ( k, Ext jB ( u (1) , N ) ⇒ Ext i + jB ( u (1) , N ) . (1)Note that Hom B ( k, N ) = 0 so the five term exact sequence (associated to this spectralsequence) yields: E = Ext B ( u (1) , N ) ∼ = Hom B/B ( u (1) , Ext B ( k, N )) . (2)We can utilize the techniques in [12, Lemma 3.1.1, Theorem 3.2.1] to compute the B/B -socle of Ext B ( k, N ). Observe that as a B/B -module:Ext B ( k, N ) ∼ = Ext B ( L ( ω ) ∗ , − α − α − ω ) ∼ = Ext B ( L ( ω ) , − α − α − ω ) . Let − pν be a simple module in the socle where ν ∈ X . Recall that X is the set of weightsand X + is the set of dominant weights. ThenHom B/B ( − pν, Ext B ( L ( λ ) , µ )) ∼ = Hom B/B ( k, Ext B ( L ( λ ) , µ ) ⊗ pν ) ∼ = Hom B/B ( k, Ext B ( L ( λ ) , µ + pν ) . Set λ = ω and µ = 2 α − α − ω . Consider the LHS spectral sequence E i,j = Ext iB/B ( k, Ext jB ( L ( λ ) , µ + pν )) ⇒ Ext i + jB ( L ( λ ) , µ + pν ) . So Hom B ( L ( λ ) , µ + pν ) = 0, because λ − µ / ∈ pX . The associated five term exact sequenceyields an isomorphism given as E , = Hom B/B ( k, Ext B ( L ( λ ) , µ + pν )) ∼ = Ext B ( L ( λ ) , µ + pν ) . There exists another spectral sequence E i,j = Ext iG ( L ( λ ) , R j ind GB ( µ + pν )) ⇒ Ext i + jB ( L ( λ ) , µ + pν ) . We have two cases to consider. Suppose µ + pν ∈ X + . Then by Kempf’s vanishingtheorem, this spectral sequence collapses and we have thatExt B ( L ( λ ) , µ + pν ) ∼ = Ext G ( L ( ω ) , H ( µ + pν )) = Ext G ( V ( ω ) , H ( µ + pν )) = 0 . On the other hand, if µ + pν / ∈ X + , then the five term exact sequence yieldsExt B ( L ( λ ) , µ + pν ) ∼ = Hom G ( L ( ω ) , R ind GB ( µ + pν )) . By results of Andersen [1, Proposition 2.3], we have that µ + pν = s α · ω where α ∈ ∆,and s α is a simple reflection. A direct computation shows that µ − s α · ω = − α + α )and µ − s α · ω = − α . The second condition can not be satisfied because − α / ∈ pX . Therefore, the B/B socleof Ext B ( k, N ) is one-dimensional and is equal to − α + α ).In addition, Ext B ( k, N ) is a subquotient of Hom B ( P ( k ) , N ) (where P ( k ) is theprojective cover of k as B -module), and it has dimension at most two. Furthermore,the T -weights of u (1) are distinct and the B/B -socle of u (1) is − α + α ), so the imageof any any non-zero map in Hom B/B ( u (1) , Ext B ( k, N )) is three-dimensional. We cannow conclude that Hom B/B ( u (1) , Ext B ( k, N )) = 0, and by (2), E = 0. (cid:3) References [1] H.H. Andersen, Extensions of modules for algebraic groups, American J. Math., (1984),489–504.[2] J.F. Carlson, D.J. Hemmer, N. Mazza, The group of endotrivial modules for the symmetric andalternating groups,
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J. Algebra , (1990), 513–526.[12] University of Georgia VIGRE Algebra Group, First cohomology for finite groups of Lie type:simple modules with small dominant weights, arXiv:1010.1203. Department of Mathematics, University of Georgia, Athens, Georgia 30602, USA
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