Top local cohomology modules and Gorenstein injectivity with respect to a semidualizing module
Let \((R,\mathfrak m)\) be a Cohen-Macaulay complete local ring. It is known that \(R\) is Gorenstein if and only if the top local cohomology module \(H_{\mathfrak m}^{\dim R}(R)\) is isomorphic to \(E(R/\mathfrak{m})\). Recently, Yoshizawa proved that the following are equivalent: i) \(R\) is a Gorenstein; ii) \(H_{\mathfrak m}^{\dim R}(R)\) is an injective \(R\)-module; iii) \(H_{\mathfrak m}^{\dim R}(R)\) is a Gorenstein injective \(R\)-module. (See Theorem 2.6 in the paper: [\textit{T. Yoshizawa}, ``On Gorenstein injectivity of top local cohomology modules'', Proc. Am. Math. Soc. 140, No. 6, 1897--1907 (2012; Zbl 1251.13017)]). Let \(C\) be a semidualizing \(R\)-module. In the paper under review, it is shown that the following statements are equivalent: i) \(C\) is a dualizing \(R\)-module; ii) \(H_{\mathfrak m}^{\dim R}(R)\) is a \(C\)-injective \(R\)-module; iii) \(H_{\mathfrak m}^{\dim R}(R)\) is a \(G_C\)-injective \(R\)-module. iv) \(H_{\mathfrak m}^{\dim R}(R)\) has finite \(G_C\)-injective dimension. The authors did not assume that \(R\) is Cohen-Macaulay, but one should be aware that their given argument for the implication \(iv)\Longrightarrow i)\) is correct only when \(R\) is Cohen-Macaulay.
scientific article; zbMATH DE number 6037954
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