Vincent Franjou

General linear and functor cohomology over finite fields

In recent years, there has been considerable success in computing Extgroups of modular representations associated to the general linear group by relating this problem to one of computing Ext-groups in functor categories [F-L-S], [F-S]. In this paper, we extend our ability to make such Ext-group calculations by establishing several fundamental results. Throughout this paper, we work over fields of positive characteristic p. The reader familiar with the representation theory of algebraic objects will recognize the importance of an understanding of Ext-groups. For example, the existence of nonzero Ext-groups of positive degree is equivalent to the existence of objects which are not “direct sums” of simple objects. Indeed, a knowledge of Ext-groups provides considerable knowledge of compound objects. In the study of modular representation theory of finite Chevalley groups such as GLn(Fq), Ext-groups play an even more central role: it has been shown in [CPS] that a knowledge of certain Ext-groups is sufficient to prove Lusztig’s Conjecture concerning the dimension and characters of irreducible representations. We consider two different categories of functors, the category F(Fq) of all functors from finite dimensional Fq-vector spaces to Fq-vector spaces, where Fq is the finite field of cardinality q, and the category P(Fq) of strict polynomial functors of finite degree as defined in [F-S]. The category P(Fq) presents several advantages over the category F(Fq) from the point of view of computing Extgroups. These are the accessibility of injectives and projectives, the existence of a base change, and an even easier access to Ext-groups of tensor products. This explains the usefulness of our comparison in Theorem 3.10 of Ext-groups in the category P(Fq) with Ext-groups in the category F(Fq). Weaker forms of this theorem have been known to us since 1995 and to S. Betley independently

Cohomology of bifunctors

We initiate the study of the cohomology of (strict polynomial) bifunctors by introducing the foundational formalism, establishing numerous properties in analogy with the cohomology of functors, and providing computational techniques. Since one of the initial motivations for the study of functor cohomology was the determination of the cohomology of GL(k) with coefficients in a tensor product of a symmetric and an exterior power of the adjoint representation, we keep this challenging example in mind as we achieve numerous computations which illustrate our methods.

Strict polynomial functors and coherent functors

We build an explicit link between coherent functors in the sense of Auslander [Coherent functors. Proc. Conf. Categorical Algebra (La Jolla, Calif., 1965). Springer, Berlin, pp. 189–231, 1966] and strict polynomial functors in the sense of Friedlander and Suslin (Invent. Math. 127(2), 209–270, 1997). Applications to functor cohomology are discussed.

A duality for polynomial functors

For a prime p, we consider the functors between finite dimensional Fp-vector spaces which are polynomial of degree less than a given integer d + 1. We study the extension groups with the identity functor, I, in the category of those functors. For any such functor F, there is a perfect pairing Exti(F, I) ⊗ ExtN − i(I, F) → ExtN(I, I) ≅ Fp with N = 2p[logpd] − 2. As a consequence, the minimal projective resolution of I among functors of degree less than d + 1 has finite length N.