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In the category of finitely generated modules over a local Cohen-Macaulay ring with dualizing module, one can, following Auslander-Buchweitz, consider maximal Cohen-Macaulay approximations and hulls of finite injective dimension. The former give rise to y- (and δ for a Gorenstein ring) invariants, the latter to vd -invariants which are in some sense dual. This last type of invariant has, in the guise of what we call a reduced Bass number, been encountered in the study of the Canonical Element Conjecture (CEC) by Hochster. In this paper we establish the basic properties of these invariants, some known, some not. We go on to explore the connection with CEC. The latter is, through linkage, seen to imply a fact about delta invariants. This in turn shows that CEC guarantees the existence of certain maximal Cohen-Macaulay modules over a Gorenstein ring.

Reduced Bass numbers, Auslanders δ -invariant and certain homological conjectures