Definition [4]. Let A be a noetherian ring, [afr ] an ideal of A and M an A-module. M is said to be [afr ]-cofinite if M has support in V([afr ]) and ExtiA(A/[afr ], M) is a finite A-module for each i.

Remark. (a) If 0→M′→M→M″ →0 is exact and two of the modules in the sequence are [afr ]-cofinite, then so is the third one.

This has the following consequence, which will be used several times.

(b) If f[ratio ]M→N is a homomorphism between two [afr ]-cofinite modules and one of the three modules Ker f, Im f and Coker f is [afr ]-cofinite, then all three of them are [afr ]-cofinite.

Example [5, remark 1·3]. If A is local with maximal ideal [mfr ], then an A-module is [mfr ]-cofinite if and only if it is an artinian A-module.