On invariants dual to the Bass numbers

Let $R$ be a commutative Noetherian ring, and let $M$ be an $R$-module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers $\mu _i(p,M)$ were defined for all primes $p$ and all integers $i\ge 0$ by use of the minimal injective resolution of $M$. It is well known that $\mu _i(p,M)=\dim _{k(p)}\operatorname {Ext} _{R_p}^i(k(p),M_p)$. On the other hand, if $M$ is finitely generated, the Betti numbers $\beta _i(p,M)$ are defined by the minimal free resolution of $M_p$ over the local ring $R_p$. In an earlier paper of the second author (1995), using the flat covers of modules, the invariants $\pi _i(p,M)$ were defined by the minimal flat resolution of $M$ over Gorenstein rings. The invariants $\pi _i(p,M)$ were shown to be somehow dual to the Bass numbers. In this paper, we use homologies to compute these invariants and show that

for any cotorsion module $M$. Comparing this with the computation of the Bass numbers, we see that $\operatorname {Ext}$ is replaced by $\operatorname {Tor}$ and the localization $M_p$ is replaced by $\operatorname {Hom}_R(R_p,M)$ (which was called the colocalization of $M$ at the prime ideal $p$ by Melkersson and Schenzel).