## 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)

### MSC:

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

### Keywords:

local cohomology modules; finite type of Ext
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### References:

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