Units and general linear group cohomology for a ring of algebraic integers
Let \(F\) be a number field, \(p\) an odd prime, \(R= {\mathcal O}_F [1/p]\). Suppose for simplicity that \(F\) contains a primitive \(p\)th root of unity, and let \({\mathcal P}\) denote the algebra of Steenrod \(p\)th power operations. A conjecture of Dwyer and Mitchell (1993) gives a precise description of \(H^*(GLR; \mathbb{Z}/p)\) as a \({\mathcal P}\)-Hopf algebra, in terms of the cohomology of a certain étale \(K\)-theory space \(Y_R\) (there is a natural map \(\eta: BGLR^+ \to Y_R\) which should be a \(p\)-adic equivalence according to the Lichtenbaum-Quillen conjectures). In the direction of this conjecture, the author has given in a previous paper [\textit{S. A. Mitchell}, Math. Z. 223, No. 1, 101-122 (1996; Zbl 0885.11063)] a description of the quotient \(H^* GLR/N\), where \(N\) is the ideal generated by classes whose restrictions to all \(GL_nR\) are nilpotent. In the present paper, he describes the quotient \(H^*GLR/I\), where \(I\) is the ideal generated by classes which restrict to zero on every ``\(R\)-torus'' in \(GL_n R\) \((n\geq 1)\). More precisely, he shows that \(H^* GLR/I \cong H^*B U\otimes (H^*U)^{r_2 +s} \otimes P^*_{a_1} \otimes \cdots \otimes P^*_{a_m} \otimes ({\mathcal Q}^*)^k\), where \(k_2\) is the number of complex places of \(F\), \(s\) the number of primes dividing \(p\) in \({\mathcal O}_F\), \(\mathbb{Z}/p^{a_1} \oplus \cdots \oplus \mathbb{Z}/p^{a_m}\) is the \(p\)-part of \(\text{Pic} R\), \(k\) the rank of the pairing \((K_1 R/p) \otimes (K_1R/p) \to{_pBr} R(1)\), \(P_a\) is the Hopf algebra \(H_* BU/(p^a\)-th powers) and \({\mathcal Q}\) the primitively generated polynomial Hopf algebra with one generator in each even degree.
scientific article; zbMATH DE number 1186176