Jon F. Carlson

The varieties and the cohomology ring of a module

Let G be a finite group and let K be a field of characteristic p > 0. Recently several studies have focused on the homological invariants of finitely generated KG-modules. In [l] Alperin proposed the study of the complexity of a module. The complexity is related to the degree of the polynomial rate of growth of the terms in a projective resolution of the module (see Definition 2.4). In [2] Alperin and Evens proved that the complexity of a KG-module M is equal to the maximum of the complexities of the restrictions of M to elementary abelian p-subgroups of G. One of the roots of the Alperin-Evens Theorem is Quillen’s Dimension Theorem (see [21] or [22]), which with some difficulty can be interpreted as saying the same thing for the special case of the trivial KG-module. Alperin and Evens [3] and Avrunin [4] have further studied the annihilator in Ext&(K, K) of the cohomology of M, and have produced theorems of a similar nature. These are related to the work in [9] on the structure of the cohomology ring Ext&(M, M). The purpose of this paper is to investigate the relationship between these homological invariants and the structure of the module. We concentrate on the case in which G is an elementary abelian p-group. However, using such results as those mentioned above, we can apply the present work to more general groups. Even as the earlier results indicate that much of the structure and cohomology of a module is revealed in the restrictions to elementary abelian p-subgroups, we show here that, in the case of an elementary abelian group, much of this information can be found by looking at the restrictions to cyclic p-subgroups of the group of units of KG. Some of the results in this paper were announced in [S]. Let G = (xl ,..., xn) be an elementary abelian group of order p”. Suppose that K is algebraically closed. To any finitely generated KG-module M we

Products in negative cohomology

Abstract In this paper, we investigate negative degree elements of group cohomology and their products. We give a definition for arbitrary groups, which generalizes the definitions of Tate for finite groups and Farrell for groups of finite virtual cohomological dimension. We find that for finite groups, if we take coefficients in a field of characteristics p, very often all products between elements of negative degree vanish. In particular, this happens when the depth of the positive cohomology ring is greater than one. The latter is the case, for example, when a Sylow p-subgroup has a non-cyclic center.

The variety of an indecomposable module is connected

Let G be a finite group and let K be an algebraically closed field of characteristic p > 0. The connections between the modular representation theory of finite groups and cohomology theory is currently a topic of great interest. In particular, cohomology theory has been used to attach to each finitely generated KGmodule M a homogeneous affine variety V(M). If we let 17(M) be the corresponding projective variety, then our main result is the following.

Modules and group algebras

1 Augmentations, nilpotent ideals, and semisimplicity.- 2 Tensor products, Homs, and duality.- 3 Restriction and induction.- 4 Projective resolutions and cohomology.- 5 The stable category.- 6 Products in cohomology.- 7 Examples and diagrams.- 8 Relative projectivity.- 9 Relative projectivity and ideals in cohomology.- 10 Varieties and modules.- 11 Infinitely generated modules.- 12 Idempotent modules.- 13 Varieties and induced modules.- References.- List of symbols.

Torsion endo-trivial modules

We prove that the group T(G) of endo-trivial modules for a noncyclic finite p-group G is detected on restriction to the family of subgroups which are either elementary Abelian of rank 2 or (almost) extraspecial. This result is closely related to the problem of finding the torsion subgroup of T(G). We give the complete structure of T(G) when G is dihedral, semi-dihedral, or quaternion.

Projective resolutions and Poincaré duality complexes

Let k be a field lof characteristic p > 0 and let G be a finite group. We investigate the structure of the cohomology ring H*(G, k) in relation to certain spectral sequences determined by systems of homogeneous parameters for the cohomology ring. Each system of homogeneous parameters is associated to a complex of projective fcG-modules which is homotopically equivalent to a Poincaré duality Complex. The initial differentials in the hypercohomology spectral sequence of the complex are multiplications by the parameters, while the higher differentials are matric Massey products. If the cohomology ring is Cohen-Macaulay, then the duality of the complex assures that the Poincaré series for the cohomology satisfies a certain functional equation. The structure of the complex also implies the existence of cohomology classes which are in relatively large degrees but are not in the ideal generated by the parameters. We consider several other questions concerned with the minimal projective resolutions and the convergence of the spectral sequence.

Modules of constant Jordan type

Abstract We introduce the class of modules of constant Jordan type for a finite group scheme G over a field k of characteristic p > 0. This class is closed under taking direct sums, tensor products, duals, Heller shifts and direct summands, and includes endotrivial modules. It contains all modules in an Auslander-Reiten component which has at least one module in the class. Highly non-trivial examples are constructed using cohomological techniques. We offer conjectures suggesting that there are strong conditions on a partition to be the Jordan type associated to a module of constant Jordan type.

Endotrivial modules for finite groups of Lie type.

Abstract 1. Introduction Let G be a finite group and k be a field of characteristic p > 0. An endotrivial kG-module is a finitely generated kG-module M whose k-endomorphism ring is isomorphic to a trivial module in the stable module category. That is, M is an endotrivial module provided where P is a projective kG-module. Now recall that as kG-modules, where M * = Hom k (M, k) is the k-dual of M. Hence, the functor “ ” induces an equivalence on the stable module category and the collection of all endotrivial modules makes up a part of the Picard group of all stable equivalences of kG-modules. In particular, equivalence classes of endotrivial modules modulo projective summands form a group that is an essential part of the group of stable self-equivalences.

ENDOTRIVIAL MODULES FOR THE SYMMETRIC AND ALTERNATING GROUPS

In this paper we determine the group of endotrivial modules for certain symmetric and alternating groups in characteristic

Endotrivial modules for

p