On invariants dual to the Bass numbers
Edgar E. Enochs, Jinzhong Xu · Proceedings of the American Mathematical Society · 1997
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