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Cofinite modules and local cohomology. (English) Zbl 0893.13005
Summary: We show that if $$M$$ is a finitely generated module over a commutative Noetherian local ring $$R$$ and $$I$$ is a dimension one ideal of $$R$$ (i.e., $$\dim R/I=1)$$, then the local cohomology modules $$H^i_I(M)$$ are $$I$$-cofinite; that is, $$\text{Ext}^j_R (R/I, H^i_I (M))$$ is finitely generated for all $$i,j$$. We also show that if $$R$$ is a complete local ring and $$P$$ is a dimension one prime ideal of $$R$$, then the set of $$P$$-cofinite modules form an abelian subcategory of the category of all $$R$$-modules. Finally, we prove that if $$M$$ is an $$n$$-dimensional finitely generated module over a Noetherian local ring $$R$$ and $$I$$ is any ideal of $$R$$, then $$H^n_I(M)$$ is $$I$$-cofinite.

##### MSC:
 13D45 Local cohomology and commutative rings
##### Keywords:
local cohomology modules
Full Text:
##### References:
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