Crossed products with respect to different kernel functors (Q1063070)

If S is a commutative R-algebra which is faithfully projective as an R- module, then it is well-known that the part of the Brauer group of R split by S is isomorphic to the second Amitsur cohomology group \(H^ 2(S/R,U)\), at least under the condition that the Picard groups of S and \(S\otimes S\) vanish. This last condition may be avoided if we replace the Amitsur complex of the multiplicative units functor U by a more complicated one. We consider two idempotent kernel functors \(\sigma\leq \tau\) on R-mod, and recall the definitions of relative Brauer and Picard groups [cf. \textit{F. Van Oystaeyen} and \textit{A. Verschoren}, ''Relative invariants of rings. The commutative theory'', Pure Appl. Math. 79 (1983; Zbl 0568.13001)]. Then we may introduce a complex with cohomology groups denoted \(H^ n_{\sigma}(S/R,U_{\tau})\), such that \(Br(S/R)\cong H^ 2_{\sigma}(S/R,U_{\tau})\) if \(Pic_{\tau}(S)=Pic_{\tau}(S\perp S)=1\). If \(\tau\) is the kernel functor associated to localization at a certain prime ideal p, then this last condition is fulfilled, so we obtain the result if we take \(\sigma =1\). As an application we may embed Br(R) into lim \(H^ 2_ 1(S/R,U_ p)\). Using Artin's refinement theorem [\textit{M. Artin}, Adv. Math. 7, 282-296 (1971; Zbl 0242.13021)], we may prove that this last group is isomorphic to \(\check H^ 2(R_{et},U)\), as expected.