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Reduced Bass Numbers and Auslander's
d-invariant

(Anne-Marie Simon and J.R. Strooker)

We define the reduced Bass number
n^{i}_{A}(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
Ext^{i}_{A}(k,M) to the local cohomology module
H^{i}_{m}(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
n^{d}_{A}(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
n^{d}_{R}, 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 n^{d}_{R} 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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