We define the reduced Bass number niA(M) of a module M over a noetherian local ring (A,m,k) as being the dimension of the k-vector space which is the image of the natural map from ExtiA(k,M) to the local cohomology module Him(M).
When the local ring A is a homomorphic image of a Gorenstein local ring and thus have a canonical module K, the top reduced Bass number ndA(M), d = dimA, of a finitely generated module M is also related to its K-preenvelopes. By K-preenvelope we mean a preenvelope with respect to the class of finite direct sums of copies of K in the terminology of Enochs.
Moreover, when the local ring R is Gorenstein, we show that the top reduced Bass number of the finitely generated module M is also related to its minimal hull of finite injective dimension in the terminology of Auslander and Buchweitz. Thus the invariant ndR, d = dimR, is in some sense dual to the Auslander's d-invariant defined by means of maximal Cohen-Macaulay approximations. We show that this duality between ndR and d is reflected in some of their properties. In particular, the invariant d(M) is also related to the free-cover of M.
Other properties of these invariants will be investigated and a vanishing criterion for d(M) will be given in the case when the Gorenstein local ring R is equicharacteristic, as a consequence of the existence of Big Cohen-Macaulay module.
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