Leonard L. Scott

Rational and Generic Cohomology

Let G be a semisimple algebraic group defined and split over k,=GF(p). For q=p”, let G(q) be the subgroup of GF(q)-rational points. The main objective of this paper is to relate the cohomology of the finite groups G(q) to the rational cohomology of the algebraic group G. Let I/ be a finite dimensional rational G-module, and, for a non-negative integer e, let V(e) be the G-module obtained by “twisting” the original G-action on V by the Frobenius endomorphism x++xtPel of G. Theorem (6.6) states that, for sufficiently large q and e (depending on I’ and n), there are isomorphisms H”(G, V(e))gH’(G(q), V(e))rH”(G(q), V) where the first map is restriction. In particular, the cohomology groups H”(G(q), V) have a stable or “generic” value H;,,(G, V). This phenomenon had been observed empirically many times (cf. [6, 203). The computation of generic cohomology reduces essentially to the computation of rational cohomology. One (surprising) consequence is that Hi,,(G, V) does not depend on the exact weight lattice for a group G of a given type cf. (6.10), though this considerably affects the structure of G(q). We also obtain that rational cohomology takes a stable value relative to twisting i.e., for sufficiently large E, we have semilinear isomorphisms H”(G, V(E)) % H”(G, V(e)) for all e 2 F. This paper contains many new results on rational cohomology beyond those required for the proof of the main theorem. We mention in particular the vanishing theorems (2.4) and (3.3), and especially the results (3.9) through (3.11) which relate H2(G, V) and Extk( K W) to the structure of Weyl modules. These results explain for example the generic values of H’ determined in [6], cf. (7.6). Also, it is shown in Theorem (3.12) that every finite dimensional rational G-module has a finite resolution by finite dimensional acyclic G-modules. A key ingredient in the proofs is an important theorem of G. Kempf [I93 on the vanishing of cohomology of certain homogeneous line bundles. This result is translated into the language of rational cohomology in (1.2), and is used in

Cohomology of finite groups of lie type, I

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Derived categories and Morita theory

This paper represents a first attempt to construct a Morita theory for derived categories, analogous to the classical theory for module categories of rings [l, lo]. Our motivation comes from the recent importance of derived categories in the representation theory of Lie algebras and algebraic groups [ 3, 111. However, our point of view here is entirely ring-theoretic, aimed at placing into a broader context the recent theory of tilting modules for finite dimensional algebras as developed by Brenner and Butler [S], Bongartz 141, Happel and Ringel [S], and Happel [9]. In Section 1 we briefly recall some standard notation and constructions from the theory of triangulated categories. In Section 2 we introduce the notion of a generalized tilting module for rings and show how it gives rise to equivalences of certain derived categories. The main theorem of this section , Theorem 2.1, represents an extension and generalization of a result a Happel [9], although we obtained it only after thorough study of Happels work as well as that of Bongartz 141. In Section 3 we show that the tilting module conditions arise naturally and necessarily in characterizing certain equivalences of derived categories. Finally in Section 4 we try to place the Morita theory developed so far in the context of a general Morita theory for derived categories, indicating some of the remaining difficulties. We also use part of the tilting module set up to obtain a localization result in the spirit of [2]. We would like to thank Klaus Roggenkamp for directing us to Happels work, and for sharing with us his unpublished manuscript [ 133 on generalized tilting modules for orders (,f= 2 versions in the sense of Theorem (2.1)). All nghtr of reproduchon m any lorm rewvrd

Quillen Stratification for Modules

Let G be a finite group and k a fixed algebraically closed field of characteristic p>O. If p is odd, let H, be the subring of H*(G, k) consisting of elements of even degree; following [20-221 we take H, = H*(G, k) if p=2, though one could just as well use the subring of elements of even degree for all p. H, is a finitely generated commutative k-algebra [13], and we let V, denote its associated affine variety Max H,. If M is any finitely generated kG-module, then the cohomology variety V,(M) of M may be defined as the support in V, of the H,-module H*(G, M) if G is a p-group, and in general as the largest support of H*(G, L@ M), where L is any kG-module [4, 91. A module L with each irreducible kc-module as a direct summand will serve. D. Quillen [20-221 proved a number of beautiful results relating k;; to the varieties V, associated with the various elementary abelian p-subgroups E of G, culminating in his stratification theorem [20, 221. This theorem gives a piecewise description of V, almost explicitly in terms of the subgroups E and their normalizers in G. A well-known corollary is that dim V, =max dim V,, where E

Modular permutation representations

A modular theory for permutation representations and their centralizer rings is presented, analogous in several respects to the classical work of Brauer on group algebras. Some principal ingredients of the theory are characters of indecomposable components of the permutation module over a p-adic ring, modular characters of the centralizer ring, and the action of normalizers of p-subgcoups P on the fixed points of P. A detailed summary appears in [15]. A main consequence of the theory is simplification of the problem of computing the ordinary character table of a given centralizer ring. Also, some previously unsuspected properties of permutation characters emerge. Finally, the theory provides new insight into the relation of Brauers theory of blocks to Greens work on indecomposable modules. The purpose of this article is to present proofs of the results announced in [15]. Statements of these results have been included here, though a number of explanatory remarks and general background references have not been repeated. With the exception of §0, the sections of this paper have been named according to the features of the classical modular theory with which they are most closely related. 0. The centralizer ring. Throughout this paper G is a finite group acting on a finite set fl (perhaps not transitively or faithfully) and p is a fixed prime. If S is any commutative ring with identity, we define the 5-centralizer ring VS(G) = VS(G, fi) to be the collection of all matrices with entries from S that commute with the permutation matrices determined (with respect to some fixed ordering of fi) by elements of G. In case 5 is the ring of rational integers, we write only V(G) fot VS(G) and refer to V(G) as the centralizer ring. The standard basis matrices \A .\T. ate obtained from the full set \0 .\r, of orbits of G on fi x fi by setting the a, ß entry of A . equal to 1 for (a, ß) £ 0 . and 0 otherwise. These matrices always form an 5-basis for VAG). In particular, VS(G) is isomorphic to the tensor product SV(G). Received by the editors August 27, 1971. AMS (MOS) subject classifications (1970). Primary 20C20; Secondary 16A26, 20C05.

A mackey imprimitivity theory for algebraic groups

Let G be an affine algebraic group over an algebraically closed field k and let H be a closed subgroup of G. If V is a rational H-module (a comodule for the coordinate ring of H) there is a now well-known notion of an induced module VI G for G, defined as the space Morph/~(G, V) of all H-equivariant morphisms from G to a finite dimensional subspace of V, with obvious G-action. The question arises, given a rational G-module M, how can one recognize M as an induced module VIG? For a finite group G the answer is the Mackey imprimitivity theorem: the module M is induced if and only if it is a direct sum of subspaces permuted transitively by G (with H the stabilizer of one of these subspaces, called V). One uses this result, for example, in proving the famous Mackey decomposition theorem which describes the restriction of any induced module to a second subgroup L as a direct sum of suitable induced modules; given the imprimitivity theorem, the proof is just a matter of grouping the summands permuted by G into their orbits under L. In the case of algebraic groups the situation is quite different. For some subgroups H, all G-modules are induced. This occurs, for example, if G is connected and k [G/H] ,= k, e.g., if H is parabolic. Also, if G is a connected unipotent group, then a rational G-module M is induced from some proper subgroup if and only if its endomorphism ring contains a two dimensional submodule E, for the conjugation action of G, with E___ k. 1 and such that k. l is precisely the subspace annihilated by the action of the Lie algebra of G on E [27] (cf. also (5.5) below). The latter is an application of a general criterion in case G/H is affine: a rational G-module M is induced if and only if there is an action of the coordinate ring A of G/H compatible with the action of G on both A and M. Note that this generalizes the imprimitivity theorem in the case of finite groups, since then A has a k-basis of [G:H] orthogonal idempotents permuted transitively by G.

Reduced standard modules and cohomology

First cohomology groups of finite groups with nontrivial irreducible coefficients have been useful in several geometric and arithmetic contexts, including Wiless famous paper (1995). Internal to group theory, 1-cohomology plays a role in the general theory of maximal subgroups of finite groups, as developed by Aschbacher and Scott (1985).One can pass to the case where the group acts faithfully and the underlying module is absolutely irreducible. In this case, R. Guralnick (1986) conjectured that there is a universal constant bounding all of the dimensions of these cohomology groups. This paper provides the first general positive results on this conjecture, proving that the generic 1-cohomology H 1 gen (G,L) := lim H 1 (G(q), L) (see Cline, Parshall, Scott, and van der Kallen) (1977) of a finite group G(q) of Lie type, with absolutely irreducible coefficients L (in the defining characteristic of G), is bounded by a constant depending only on the root system. In all cases, we are able to improve this result to a bound on H 1 (G(q),L) itself, still depending only on the root system. The generic H 1 result, and related results for Ext 1 , emerge here as a consequence of a general study, of interest in its own right, of the homological properties of certain rational modules Δ red (λ),∇ red (λ), indexed by dominant weights λ, for a reductive group G. The modules Δ red (λ) and ∇ red (λ) arise naturally from irreducible representations of the quantum enveloping algebra U ζ (of the same type as G) at a pth root of unity, where p > 0 is the characteristic of the defining field for G. Finally, we apply our Ext 1 -bounds, results of Bendel, Nakano, and Pillen (2006), as well as results of Sin (1993), (1992), (1994) on the Ree and Suzuki groups to obtain the (non-generic) bounds on H 1 (G(q),L).

Failure of Nielsen's theorem in higher dimensions

The following is known for closed orientable surfaces. If H: ii/l -+ &! is a map whose n-th power is homotopic to the identity, then H is homotopic to a homeomorphism K with Kqa = identity. The result is known as Nielsen’s Theorem on finite mapping classes. There are doubts (see [S]) as to the correctness of all parts of Nielsen’s arguments in [6]. Different (using complex analysis and the Smith theorems) and valid proofs have been given independently by Fenchel and Macbeath. Because we are dealing with surfaces, each self homotopy equivalence is homotopic to a diffeomorphism, and homotopic diffeomorphisms are diffeotopic. Consequently, Nielsen’s theorem may be equivalently stated as follows: If H: ii + ill is a diffeomorphism whose n-th power is homotopic to the identity then H is diffeotopic to a diffeomorphism K with Km = identity. The theorem is extremely useful in studying periodic maps on 3-manifolds, and there are obvious applications in surface theory. The question has been raised as to what extent Nielsen’s theorem holds for aspherical manifolds of dimensions greater than 2. An aspherical manifold is a closed manifold whose universal covering is contractible. Aspherical manifolds are therefore closed manifolds which are also K(n, 1)‘s. An important and interesting class of aspherical manifolds (generalizing the tori) are the nil-manifolds. A nil-manifold is simply the quotient of a connected contractible nilpotent Lie group by a uniform discrete subgroup. We shall show that in dimensions greater than 2 there exist closed nil-manifolds for which Nielsen’s theorem fails in a very strong sense. Specifically

Some new examples in 1-cohomology

This paper gives some new examples in the 1-cohomology theory of finite groups of Lie type, obtained from both computer calculations and the use of several theoretical results. In particular, the paper gives the first known examples of 1-cohomology groups of dimension greater than 2 for absolutely irreducible faithful modules of a finite group. The computer calculations were made originally while checking special cases of Lusztig’s conjecture on characteristic p representations of algebraic groups, and we take this opportunity to announce in print some results in that direction. (They reinforce Lusztig’s conjecture, even in a strong form suggested by Kato.)  2003 Published by Elsevier Science (USA).

A new approach to the Koszul property in representation theory using graded subalgebras

Given a quasi-hereditary algebra