Cohomology of bifunctors

Abstract

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 H *(GL (k), S*(gℓ) ⊗ Λ*(gℓ)), we keep this challenging example in mind as we achieve numerous computations which illustrate our methods.

A construction of the universal classes for algebraic groups with the twisting spectral sequence

“…(2) If B = F gl, the representation F gl(k n , k n ) is the representation F (gl n ) obtained by evaluating F on the adjoint representation gl n of GL n,k . Franjou and Friedlander proved [FF,Thm 1.5] the following generalization of Friedlander and Suslin's isomorphism. For all n ≥ 1 there is a natural map…”

“…In this section, we recall the main facts that we will need about this category. The reader might consult [FS,FF,Kr,T4] for more details on strict polynomial functors and bifunctors. Throughout this section, k is a field of positive characteristic p.…”

Cohomologie du groupe linéaire à coefficients dans les polynômes de matrices

“…Dans [4], Franjou et Friedlander entament l'étude de la cohomologie H * P (GL, B) des bifoncteurs B strictement polynomiaux. Cet articleétend les premiers calculs effectués dans [4].…”

Universal classes for algebraic groups

“…In section 1, we remark that theorem 0.1 reduces to a stable cohomology statement, that is, it suffices to prove it for large values of n. Bifunctor cohomology [4] gives access to the stable rational cohomology of GL n, , and we translate theorem 0.1 in terms of (strict polynomial) bifunctors. More specifically, we show that theorem 0.1 reduces to theorem 1.4, that is, to the computation of some classes in the cohomology of the strict polynomial bifunctors Γ d (gl (1) ).…”

“…Classical functors such as the tensor products ⊗ d , the divided powers Γ d , the symmetric powers S d or the Frobenius twist I (1) are strict polynomial functors. Now the representations we are interested in are not given by strict polynomial functors but by strict polynomial bi functors, contravariant in the first variable and covariant in the second one, as introduced in [4]. Examples of such bifunctors are the bifunctor gl(− 1 , − 2 ) := Hom (− 1 , − 2 ) or postcompositions of gl by strict polynomial functors, such as gl (1) …”