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Properties of cofinite modules and applications to local cohomology. (English) Zbl 0921.13009
The paper contains some nice results about the so-called \(I\)-cofinite modules. Let \(A\) be a noetherian ring, \(M\) an \(A\)-module, \(I\) an ideal of \(A\). \(M\) is called \(I\)-cofinite if \(\text{Supp}(M)\subseteq V(I)\) and \(\text{Ext}^i_A(A/I,M)\) is finitely generated over \(A\) for all \(i\).
The first result is that \(M\) is \(I\)-cofinite iff \(\text{Supp}(M)\subseteq V(I)\) and \(H^i(x_1,\dots,x_n;M)\) is a finite \(A\)-module for all \(i\), where \(x_1,\dots,x_n\) is a system of generators of \(I\). The second one is that if \(M\) is \(I\)-cofinite for some ideal \(I\), then all Bass numbers \(\mu_i(P,M)\) and all Betti numbers \(\beta_i(P,M)\) are finite for all prime ideals \(P\) and all \(i\).
As a corollary, if \(I\) is an ideal such that \(\dim(A/I)=1\) and \(M\) is an \(A\)-module then the Betti numbers of \(H^i_I(M)\) are finite for all \(i\). If \(A\) is local, \(\dim(A)\leq 2,I\) is an ideal of \(A\), \(M\) is a finitely generated \(A\)-module then all \(H^i_I(M)\) are \(I\)-cofinite. Finally, it is shown that for every local 3-dimensional local ring \(A\) there is an ideal \(I\) such that \(H^2_I(A)\) is not \(I\)-cofinite.
Editorial remark: N. Abazari and K. Bahmanpour [Commun. Algebra 42, No. 3, 1270–1275 (2014; Zbl 1291.13028)] gave a counterexample to Theorem 1.6, which seems to be only true if one assumes \(R\) is complete.

13D45 Local cohomology and commutative rings
18G15 Ext and Tor, generalizations, Künneth formula (category-theoretic aspects)
Zbl 1291.13028
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