Extensions Relative to a Serre Class

Consider a class C of projective /^-modules, where R is a commutative ring with identity, which satisfies the conditions of (2), namely that C is closed under the operations of direct sum and isomorphism and C contains the zero module. Following (2) a module M is said to have C-cotype n (respectively C-type n) if it has a projective resolution ...-*.?„->•...->.Po->M->0 with /",-eC for i>n (respectively P, e C for i^ri). Let S be the class of modules of C-cotype 1 , equivalently of C-type infinity. It is assumed throughout that S is a Serre Class. We define an abelian category Sf of modules with the property that C-cotype is homological dimension in y, while in the case C = 0, S is just the category of /^-modules. It follows that all categorical results on homological dimension also hold for cotype. In Theorem 12 the restriction to a Serre class 5 is expressed in terms of the coherence of the ring R. Some examples of such classes are given. Repeated use is made of the following result of (2).