Abstract. Let $R$ be a commutative Noetherian ring, $\Phi $ a system of ideals of $R$, and $M$ a finitely generated $R$-module. Suppose that ${\eufm a}\in\Phi $ and $t$ is a non-negative integer. It is shown that if $\mathop{\rm Ext} _R^i(R/{\eufm a},H_{\Phi }^j(M))$ is finitely generated for all $i$ and all $j< t$, then $\mathop{\rm Ext} _R^i(R/{\eufm a},H_{\Phi }^t(M))$ is finitely generated for $i=0,1$. In particular, if $R$ is a local ring of dimension at most $2$, then $\mathop{\rm Ext} _R^i(R/{\eufm a},H_{\Phi }^j(M))$ is finitely generated for all $i,j$.

AMS Subject Classification (1991): 13D45, 13E99

Keyword(s): General local cohomology modules, Cofinite modules

Received December 13, 2007, and in revised form March 30, 2008. (Registered under 6029/2009.)