Homology of a certain associative algebra

Abstract

Let $R$ be a commutative ring, and let $A$ be an associative $R$-algebra possessing an $R$-free basis $B$. In this paper, we introduce a homology $H_{n}(A,B)$ associated to a pair $(A,B)$ under suitable hypotheses. It depends on not only $A$ itself but also a choice of $B$. In order to define $H_{n}(A,B)$, we make use of a certain submodule of the $(n+1)$-fold tensor product of $A$. We develop a general theory of $H_{n}(A,B)$. Various examples of a pair $(A,B)$ and $H_{n}(A,B)$ are also provided.