Universal classes for algebraic groups.

*(English)*Zbl 1196.20052Let \(G\) be a reductive linear algebraic group over a field \(k\) of positive characteristic \(p\). The group \(G\) is said to have the cohomological finite generation (CFG) property if for every finitely generated commutative \(k\)-algebra \(A\) on which \(G\) acts rationally by \(k\)-algebra automorphisms, the cohomology ring \(H^*(G,A)\) is finitely generated as a \(k\)-algebra. W. van der Kallen observed that the CFG property would hold for all such \(G\) if certain universal cohomology classes (in the cohomology of the general linear group \(\text{GL}_n\)) could be constructed. The main result of this paper is the construction of such classes. Proofs of the CFG property are presented by the author and W. van der Kallen [in Duke Math. J. 151, No. 2, 251-278 (2010; Zbl 1196.20053)].

More precisely, the result here is the construction of classes \(c[d]\) in \(H^{2d}(\text{GL}_n,\Gamma^d(\mathfrak{gl}_n^{(1)}))\) where \(\Gamma^d\) denotes the divided power functor and \(\mathfrak{gl}_n^{(1)}\) denotes the adjoint representation (the Lie algebra of \(\text{GL}_n\)) once twisted by Frobenius. Further \(c[1]\) is shown to be non-zero, and there is a cup product relationship between \(c[d]\) and \(c[1]\) for higher \(d\). These classes generalize those constructed by E. M. Friedlander and A. Suslin [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] in their proof of the finite generation of \(H^*(G,k)\) for a finite group scheme \(G\).

The construction of the desired classes is first reduced to large values of \(n\). From there, the problem can be translated to bifunctor cohomology for strict polynomial bifunctors. Note that the work of Friedlander and Suslin made use of strict polynomial functors. Here functors are replaced with bifunctors. Working with bifunctors, explicit coresolutions of \(\Gamma^d(\mathfrak{gl}^{(1)})\) are constructed from which the desired cocycles are obtained. The construction of the resolutions requires investigation of \(p\)-complexes and a tensor product property of complexes obtained from \(p\)-complexes.

More precisely, the result here is the construction of classes \(c[d]\) in \(H^{2d}(\text{GL}_n,\Gamma^d(\mathfrak{gl}_n^{(1)}))\) where \(\Gamma^d\) denotes the divided power functor and \(\mathfrak{gl}_n^{(1)}\) denotes the adjoint representation (the Lie algebra of \(\text{GL}_n\)) once twisted by Frobenius. Further \(c[1]\) is shown to be non-zero, and there is a cup product relationship between \(c[d]\) and \(c[1]\) for higher \(d\). These classes generalize those constructed by E. M. Friedlander and A. Suslin [Invent. Math. 127, No. 2, 209-270 (1997; Zbl 0945.14028)] in their proof of the finite generation of \(H^*(G,k)\) for a finite group scheme \(G\).

The construction of the desired classes is first reduced to large values of \(n\). From there, the problem can be translated to bifunctor cohomology for strict polynomial bifunctors. Note that the work of Friedlander and Suslin made use of strict polynomial functors. Here functors are replaced with bifunctors. Working with bifunctors, explicit coresolutions of \(\Gamma^d(\mathfrak{gl}^{(1)})\) are constructed from which the desired cocycles are obtained. The construction of the resolutions requires investigation of \(p\)-complexes and a tensor product property of complexes obtained from \(p\)-complexes.

Reviewer: Christopher P. Bendel (Menomonie)

##### MSC:

20G10 | Cohomology theory for linear algebraic groups |

18G10 | Resolutions; derived functors (category-theoretic aspects) |

##### Keywords:

cohomology of general linear groups; strict polynomial bifunctors; bifunctor cohomology; coresolutions; \(p\)-complexes; cohomological finite generation; divided powers; universal cohomology classes**OpenURL**

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