The finiteness of coassociated primes of generalized local homology modules

Abstract

We present some finiteness results for co-associated primes of generalized local homology modules. Let M be a finitely generated R-module and N a linearly compact R-module. If N and H I i (N ) satisfy the finiteness condition for co-associated primes for all i < k, then Coass R (H I k (M, N )) is a finite set. On the other hand, if H I i (N ) = 0 for all i < t and Tor R j (M, H I t (N )) = 0 for all j < h, then Tor R h (M, H I t (N )) ∼ = H I h+t (M, N ). Moreover, Coass(H I h+t (M, N )) is also a finite set provided N satisfies the finiteness condition for co-associated primes. Finally, N is a semi-discrete linearly compact R-module such that 0 : N I = 0. Let t = Width I (N ) and h = tor − (M, H I t (N )); it follows that Width I+Ann(M) (N ) = t + h and Coass(H I h+t (M, N )) is a finite set.