The Finiteness of Coassociated Primes of Generalized Local Homology Modules

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 $\operatorname{Coass}_R(H^I_k(M, N))$ is a finite set. On the other hand, if $H^I_i(N)=0$ for all $i<t$ and ${\operatorname{Tor}}^R_j(M,H^I_t(N))=0$ for all $j<h$, then ${\operatorname{Tor}}^R_h(M,H^I_t(N))\cong H^I_{h+t}(M, N)$. Moreover, $\operatorname{Coass}(H^I_{h+t}(M, N))$ 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 $0:_NI\not=0$. Let $t=\operatorname{Width}_I(N)$ and $h={\operatorname{tor}}_-(M,H^I_t(N))$; it follows that $\operatorname{Width}_{I+\operatorname{Ann}(M)}(N)=t+h$ and $\operatorname{Coass}(H^I_{h+t}(M, N))$ is a finite set.