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On Gorenstein injectivity of top local cohomology modules. (English) Zbl 1251.13017

Let \((R, \mathfrak m)\) be a \(d\)-dimensional commutative noetherian local ring and \(J\) be an ideal. The paper studies Gorenstein-injectivity of the top local cohomology module \(H^{d}_{J}(R)\). For example over complete hypersurface the author shows that \(H^{d}_{J}(R)\) is Gorenstein-injective. An example of the paper states that the hypersurface assumption is really needed. An auxiliary method is a torsion theory defined by a pair of ideals. Over a complete Cohen-Macaulay ring \((R, \mathfrak m)\), the Gorenstein-injectivity of \(H^{d}_{\mathfrak m}(R)\) is equivalent with the injectivity of \(H^{d}_{\mathfrak m}(R)\).

MSC:

13D45 Local cohomology and commutative rings
13D05 Homological dimension and commutative rings
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