On invariants dual to the Bass numbers

Let R R be a commutative Noetherian ring, and let M M be an R R -module. In earlier papers by Bass (1963) and Roberts (1980) the Bass numbers μ i ( p , M ) \mu _i(p,M) were defined for all primes p p and all integers i ≥ 0 i\ge 0 by use of the minimal injective resolution of M M . It is well known that μ i ( p , M ) = dim k ( p ) ⁡ Ext R p i ⁡ ( k ( p ) , M p ) \mu _i(p,M)=\dim _{k(p)}\operatorname {Ext} _{R_p}^i(k(p),M_p) . On the other hand, if M M is finitely generated, the Betti numbers β i ( p , M ) \beta _i(p,M) are defined by the minimal free resolution of M p M_p over the local ring R p R_p . In an earlier paper of the second author (1995), using the flat covers of modules, the invariants π i ( p , M ) \pi _i(p,M) were defined by the minimal flat resolution of M M over Gorenstein rings. The invariants π i ( p , M ) \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 \[ π i ( p , M ) = dim k ( p ) ⁡ Tor i R p ⁡ ( k ( p ) , Hom R ⁡ ( R p , M ) ) \pi _i(p,M)= \dim _{k(p)}\operatorname {Tor}_i^{R_p}(k(p), \operatorname {Hom}_R(R_p,M)) \] for any cotorsion module M M . Comparing this with the computation of the Bass numbers, we see that Ext \operatorname {Ext} is replaced by Tor \operatorname {Tor} and the localization M p M_p is replaced by Hom R ⁡ ( R p , M ) \operatorname {Hom}_R(R_p,M) (which was called the colocalization of M M at the prime ideal p p by Melkersson and Schenzel).