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Cohomology of bifunctors. (English) Zbl 1153.20042
Consider a field \(k\) of prime characteristic \(p\) and the category \(\mathcal V\) of finite-dimensional \(k\)-vector spaces and \(k\)-linear maps. A functor \(T\colon\mathcal V\to\mathcal V\) is said to be a strict polynomial functor if for each pair \(V,W\in\mathcal V\), there is an associated polynomial mapping \(\operatorname{Hom}_k(V,W)\to\operatorname{Hom}_k(T(V),T(W))\). For \(W\in\mathcal V\), \(T(W)\) then admits the structure of a rational \(\text{GL}(W)\)-module. These functors were introduced by the second author with A. Suslin [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] and played a key role in their work on finite-generation of cohomology of finite group schemes. More generally, functor cohomology has seen growing use as a tool for computing group cohomology. Significant functor cohomology computations were later made by the authors with A. Scorichenko and A. Suslin [Ann. Math. (2) 150, No. 2, 663-728 (1999; Zbl 0952.20035)].
This work is motivated by an interest in determining the cohomology of the general linear group \(\text{GL}(n,k)\) for a finite field \(k\) with coefficients in symmetric or exterior powers of the adjoint representation \(\mathfrak{gl}_n\). These modules correspond to bifunctors rather than functors, and lead the authors to investigate bifunctor cohomology. The authors introduce the notion of strict polynomial bifunctors which are contravariant in the first variable and covariant in the second.
The first fundamental result is an identification of extensions in this category with rational cohomology of the algebraic group \(\text{GL}\). The authors go on to develop a number of nice cohomological results for strict polynomial bifunctors, including some explicit computations. The authors then relate the cohomology of strict polynomial functors to that of ordinary bifunctors, which is then further related to the cohomology of finite groups \(\text{GL}(n,k)\). Finally, the functor cohomology results are used to obtain a number of new explicit computations of the stable (with respect to \(n\)) cohomology of \(\text{GL}(k)\), generalizing results of L. Evens and the second author [Trans. Am. Math. Soc. 270, 1-46 (1982; Zbl 0492.18008)].

MSC:
20G10 Cohomology theory for linear algebraic groups
20J06 Cohomology of groups
18G40 Spectral sequences, hypercohomology
18G60 Other (co)homology theories (MSC2010)
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