Multiplicative stability for the cohomology of finite Chevalley groups
Let G denote a connected split reductive group scheme over \({\mathbb{Z}}\) (e.g. \(G=G\ell_ n)\). Let p be a prime, \({\mathbb{F}}_ p\) the prime field and \({\mathbb{F}}\) the algebraic closure of \({\mathbb{F}}_ p\). As usual, G(k) denotes the discrete group of k-points of G \((k={\mathbb{F}}_{p^ f}\) or \({\mathbb{F}}\) in the situation below). The author establishes a stability result for the cohomology of the groups G(k) with respect to change of fields from \({\mathbb{F}}_{p^ f}\) to \({\mathbb{F}}\). His result has several interesting consequences. For instance, it implies that \(H^*(G({\mathbb{F}});{\mathbb{Z}}/\ell)\) is an algebra retract of \(H^*(G({\mathbb{F}}_{p^ f});{\mathbb{Z}}/\ell)\) if f is sufficiently large in the sense of divisibility, and \(\ell\) a prime different from p. The author also studies the stability behavior of associated categories of elementary abelian \(\ell\)-subgroups of \(G({\mathbb{F}}_{p^ f})\) and G(\({\mathbb{F}})\), respectively; these categories are - using results of \textit{D. Quillen}'s [cf. Ann. Math., II. Ser. 94, 549-572, 573-602 (1971; Zbl 0247.57013)] - closely related to the mod-\(\ell\) cohomology algebras of \(G({\mathbb{F}}_{p^ f})\) and G(\({\mathbb{F}})\). The author's results stated (Lemma 3, Theorem 4, Corollary 5) need however some mild additional hypothesis on G [cf. Erratum: Comment. Math. Helv. (to appear)] it suffices to assume G semisimple and that \(\ell\) not be a torsion prime for the integral cohomology of the Lie group with points G(\({\mathbb{C}})\). An example of a result is Theorem 4, which states that the cohomological restriction map \(H^*(G({\mathbb{F}});{\mathbb{Z}}/\ell)\to H^*(G({\mathbb{F}}_{p^ f});{\mathbb{Z}}/\ell)\) is an isomorphism of algebras modulo their nilradicals, if f is sufficiently large.
scientific article; zbMATH DE number 4075390