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A generalization of a theorem of Vogtmann. (English) Zbl 0615.20024
Let A be a Dedekind domain (resp. PID) and let \(G_ n\) be the sequence of groups \(O_{n,n}(A)\) or \(Sp_{2n}(A)\), each embedded in the next in an obvious way. Then for any abelian group of coefficients M, the author proves that the map induced by inclusion from \(H_ i(G_ n;M)\) to \(H_ i(G_{n+1};M)\) is surjective for \(n\geq 2i+5\) (resp. \(n\geq 2i+4)\) and bijective for \(n\geq 2i+6\) (resp. \(n\geq 2i+5)\). The method is fairly standard: construct a highly connected poset on which the G’s act with stabilizers isomorphic to smaller G’s, and then use spectral sequential induction. Many of the constructions and lemmas are given for somewhat more general groups of automorphisms of hyperbolic modules and for A any finite algebra over a commutative ring with noetherian maximal spectrum.
Reviewer: A.Ash

20G10 Cohomology theory for linear algebraic groups
20G30 Linear algebraic groups over global fields and their integers
Full Text: DOI
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