On the complete relative cohomology of Frobenius extensions

Abstract

Let A be an algebra over a commutative ring K and F a subalgebra. Suppose that the extension A/F is a Frobenius extension. Then in [3, section 3], the complete relative cohomology group Hl^niM, ―) is introduced for an arbitrary left J-module M and reZ. We denote the opposite rings of A and F by A0 and F° respectively. Put P=A KA° and let 5 denote the natural image of F(Z)KF0 in P. Then the extension P/S is also a Frobenius extension. Since A is a left P-module with the natural way, we have HlP,s->(A,―). We will denote this Hu>.s>(A, ―) by Hr(A, F, -) for [6, section 3]. In this paper, we will study this complete relative cohomology H{A, F, ―). In section 1, we will study relative complete resolutions of A and in section 2, we willintroduce the dual of the fundamental exact sequence of [4, Proposition 1 and Theorem 1] for complete relative cohomology groups. In section 3, we will study an internal product like as in [9, section 2] which we will call the cup product. If the basic ring of the Frobenius extension is commutative, the cup product in this paper coincides with the product V in [2, Exercise 2 of Chapter XI]