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.