Minimal Injective and Flat Resolutions of Modules over Gorenstein Rings

Abstract

Over a commutative Noetherian ring R, the Bass invariants μ i ( p, M) were defined for any module M and any prime p ∈ Spec R by H. Bass ( Math. Z. 82, 1963, 8-28). In the first part of this paper, we study these numbers further. We are concerned primarily with the modules having a certain vanishing property of their Bass numbers. For instance, we show that R is Gorenstein if and only if μ i ( p, F) = 0 whenever ht( p) ≠ i for any nat module F. In the second part, we define the invariants π i ( p, M) for any prime p ∈ Spec R and module M by a minimal flat resolution of M. As with the Bass invariants, we can characterize Gorenstein rings and modules by a vanishing property of these numbers. For instance, injective modules are just those modules M having π i ( p, M)= 0 for all prime p with ht( p) ≠ i and all i ≥ 0. Finally, we introduce strongly cotorsion modules and show that these modules M are just those modules having π i ( p, M) = 0 for all prime p with ht( p) > i and all i ≥ 0. From this paper we will see that nat covers defined by E. Enochs ( Israel J. Math. 39, 1981, 189-209) behave in a manner dual to the behavior of injective envelopes.