Complexes Over a Complete Algebra of Quotients

Abstract

Let R be a commutative ring with unit and A be a unitary commutative R-algebra. Let As be a generalized algebra of quotients of A with respect to a multiplicatively closed subset S of A. If (A) and (As) denote the categories of complexes and their homomorphisms over A and As respectively, then one easily sees that there exists a covariant functor T:(A) → (AS) such that T is onto and T(X, d) is universal over As whenever (X, d) is universal over A. Actually the category (AS) is equivalent to a subcategory of (A) where contains all those complexes (X, d) over A such that for each s in S, the module homomorphism ϕs: x → sx of Xn into itself is one-one and onto for each n ⩾ 1.