Cofiniteness of local cohomology modules for ideals of dimension one. (English) Zbl 0899.13018

The starting point of this paper is the following Grothendieck conjecture: “If \(I\) is an ideal of a noetherian local ring \((A,m,k)\) and \(M\) is a finite \(A\)-module, then \(\operatorname{Hom}_A (A/I,H^i_I (M))\) is of finite type for all \(i\).” This conjecture, which is false in general, gave rise to many researches on the subject; the main contribution of this paper consists in the following theorem:
Let \((A,m,k)\) be a local ring, \(I\) an ideal with \(\dim A/I=1\), \(M\) a finite \(A\)-module. Then, for any finite \(A\)-module \(N\) such that \(\text{Supp}_A(N) \subseteq V(I)\), the module \(\text{Ext}_A^i (N,H_I^j (M))\) is of finite type, for any \(i,j\).
Reviewer: C.Massaza (Torino)


13D45 Local cohomology and commutative rings
18G15 Ext and Tor, generalizations, K√ľnneth formula (category-theoretic aspects)
13D02 Syzygies, resolutions, complexes and commutative rings
13H99 Local rings and semilocal rings
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