Bifunctor cohomology and cohomological finite generation for reductive groups.

*(English)*Zbl 1196.20053Let \(G\) be a reductive linear algebraic group over a field \(k\). 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.

The main result of this paper is the impressive fact that any such \(G\) has the CFG property.

Over a field of characteristic zero, the fact is well-known from invariant theory, so the case of prime characteristic is the focus here. This problem (or special cases thereof) has seen significant study by many people, particularly in full generality by the second author. A nice discussion is given of some of this history along with equivalent formulations of the theorem. Also, some consequences are given for the cohomology module \(H^*(G,M)\) for a Noetherian \(A\)-module \(M\) on which \(G\) acts compatibly.

The CFG property for an arbitrary such \(G\) is first reduced to the case of the general linear group \(\text{GL}_n\) over an algebraically closed field (of prime characteristic). The second author [in CRM Proceedings & Lecture Notes 35, 127-138 (2004; Zbl 1080.20039)] had previously shown the result for some small \(n\) via the construction of certain universal cohomology classes in \(H^*(\text{GL}_n,\Gamma^*(\mathfrak{gl}_n^{(1)}))\) satisfying certain divided power relations, where \(\Gamma^*\) 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. The existence of such classes (without the divided power relations) was shown in general by the first author [in Duke Math. J. 151, No. 2, 219-249 (2010; Zbl 1196.20052)]. 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 proof of the CFG property in part parallels the argument of Friedlander and Suslin.

The authors present two proofs of how the CFG property follows from the existence of the universal cohomology classes constructed by Touzé. One proof makes further investigation of bifunctor cohomology beyond the aforementioned work of Touzé, obtaining further classes and relations, and then follows the argument in the aforementioned work of van der Kallen. The second proof simply uses the universal classes as constructed by Touzé and an inductive argument with Frobenius kernels.

The main result of this paper is the impressive fact that any such \(G\) has the CFG property.

Over a field of characteristic zero, the fact is well-known from invariant theory, so the case of prime characteristic is the focus here. This problem (or special cases thereof) has seen significant study by many people, particularly in full generality by the second author. A nice discussion is given of some of this history along with equivalent formulations of the theorem. Also, some consequences are given for the cohomology module \(H^*(G,M)\) for a Noetherian \(A\)-module \(M\) on which \(G\) acts compatibly.

The CFG property for an arbitrary such \(G\) is first reduced to the case of the general linear group \(\text{GL}_n\) over an algebraically closed field (of prime characteristic). The second author [in CRM Proceedings & Lecture Notes 35, 127-138 (2004; Zbl 1080.20039)] had previously shown the result for some small \(n\) via the construction of certain universal cohomology classes in \(H^*(\text{GL}_n,\Gamma^*(\mathfrak{gl}_n^{(1)}))\) satisfying certain divided power relations, where \(\Gamma^*\) 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. The existence of such classes (without the divided power relations) was shown in general by the first author [in Duke Math. J. 151, No. 2, 219-249 (2010; Zbl 1196.20052)]. 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 proof of the CFG property in part parallels the argument of Friedlander and Suslin.

The authors present two proofs of how the CFG property follows from the existence of the universal cohomology classes constructed by Touzé. One proof makes further investigation of bifunctor cohomology beyond the aforementioned work of Touzé, obtaining further classes and relations, and then follows the argument in the aforementioned work of van der Kallen. The second proof simply uses the universal classes as constructed by Touzé and an inductive argument with Frobenius kernels.

Reviewer: Christopher P. Bendel (Menomonie)

##### MSC:

20G10 | Cohomology theory for linear algebraic groups |

14L24 | Geometric invariant theory |

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

##### Keywords:

cohomological finite generation; rational cohomology; reductive groups; cohomology of general linear groups; strict polynomial bifunctors; bifunctor cohomology; universal cohomology classes; cohomology rings; divided powers
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\textit{A. Touzé} and \textit{W. van der Kallen}, Duke Math. J. 151, No. 2, 251--278 (2010; Zbl 1196.20053)

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