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Let be a commutative noetherian ring, be an ideal of , be non-negative integers and let be an -module such that $\Ext^i_A(A/\frak a,M)$ is finitely generated for all . We define a class $\cS_n(\frak a)$ of modules and we assume that $H_{\frak a}^s(M)\in\cS_{n}(\frak a)$ for all . We show that is -cofinite for all if either or and $\Ext_A^{i}(A/\frak a,H_{\frak a}^{t+s-i}(M))$ is finitely generated for all , and . If is a ring of dimension and $M\in\cS_n(\frak a)$ for any ideal of dimension , then we prove that $M\in\cS_n(\frak a)$ for any ideal of .