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Extensions relative to a Serre class

Consider a class C of projective R-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 … → Pn → P0 → M → 0 with Pi ∈ C for i>n (respectively Pi ∈ C for i≦n). 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 of modules with the property that C-cotype is homological dimension in while in the case C = 0, S is just the category of R-modules. It follows that all categorical results on homological dimension also hold for cotype.