Articles written in Proceedings – Mathematical Sciences

    • Reflexive modules with finite Gorenstein dimension with respect to a semidualizing module

      Elham Tavasoli Maryam Salimi Siamak Yassemi

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      Let 𝑅 be a commutative Noetherian ring and let 𝐶 be a semidualizing 𝑅-module. It is shown that a finitely generated 𝑅-module 𝑀 with finite $G_{C}$-dimension is 𝐶-reflexive if and only if $M_{\mathfrak{p}}$ is $C_{\mathfrak{p}}$-reflexive for $\mathfrak{p}\in \text{Spec} (R)$ with depth $(R_{\mathfrak{p}})\leq 1$, and $G_{C_{\mathfrak{p}}} - \dim R_{\mathfrak{p}}(M_{\mathfrak{p}})\leq $ depth $(R_{\mathfrak{p}}) - 2$ for $\mathfrak{p}\in \text{Spec} (R)$ with depth $(R_{\mathfrak{p}})\geq 2$. As the ring $R$ itself is a semidualizing module, this result gives a generalization of a natural setting for extension of results due to Serre and Samuel (see Czech. Math. J. 62(3) (2012) 663-672 and Beiträge Algebra Geom. 50(2) (2009) 353-362). In addition, it is shown that over ring 𝑅 with $\dim R\leq n$, where $n\geq 2$ is an integer, $G_{D}-\dim_{R}(Hom_{R}(M,D))\leq n-2$ for every finitely generated 𝑅-module 𝑀 and a dualizing 𝑅-module 𝐷.

    • Relative grade and relative Gorenstein dimension with respect to a semidualizing module


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      Let $R$ be a commutative Noetherian ring, and let $C$ be a semidualizing $R$-module. For $R$-modules $M$ and $N$, the notions ${\rm grade}_{\mathcal{P}_C}(M, N)$ and ${\rm grade}_{\mathcal{I}_C}(M, N)$are introduced as the relative setting of the notion ${\rm grade}(M, N)$ with respect to $C$. Some results about ${\rm grade}_{\mathcal{P}_C}(M, N)$, ${\rm grade}_{\mathcal{I}_C}(M, N)$ and ${\rm grade}(M, N)$ are mentioned. Forfinitely generated $R$-modules $M$ and $N$, we show that ${\rm grade}_{\mathcal{P}_C}(M, N)= {\rm grade}(M, N)$ (${\rm grade}_{\mathcal{I}_C}(M, N) = {\rm grade}(M, N)$), provided we have some special conditions. Also, thenotions of $C$-perfect and $G_C$-perfect $R$-modules are introduced as the relative setting of the notions of perfect and $G$-perfect $R$-modules with respect to $C$, and it is proven that several results for these new concepts are similar to the classical results. Finally, some results about relative grade of tensor and Hom functors with respect to $C$ are given.

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