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arXiv 1008.4485math.AC
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截口函子的Gorenstein内射性

Gorenstein injectivity of the section functor

Reza Sazeedeh

备注：
It has 9 pages and it will be published in Forum Mathematicum
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摘要：

设$R$是Krull维数为$d$的交换Noether环，且$\frak a$是$R$的任意理想，证明了对于任意Gorenstein内射$R$ -模$G$，$Γ_{\frak a}(G)$是Gorenstein内射的.设$(R,\frak m)$是局部环，$M$是局部环生成的$R$ -模.我们证明了${\rm Gid}{\bf R}Γ_{\frak m}(M)<\infty$当且仅当${\rm Gid}_{\hat{R}}(M\otimes_R\hat{R})<\infty$ .我们还表明，如果${\rm Gfd}_R{\bf R}Γ_{\frak m}(M)<\infty$，那么${\rm Gfd}_RM<\infty$。设$(R,\frak m)$是Cohen-Macaulay局部环，$M$是维数为$n$的Cohen-Macaulay模.证明了若$H_{\frak m}^n(M)$是有限G-内射维数，则Gid $_RH_{\frak m}^n(M)=d-n$ .证明了若$M$是有限G-平坦维数的Matlis自反强挠自由模，则有Gfd $_R\hat{M}<\infty$，其中$\hat{M}$是$\frak m$-adic完备化.

英文摘要：

Let $R$ be a commutative Noetherian ring of Krull dimension $d$ admitting a dualizing complex $D$ and let $\frak a$ be any ideal of $R$, we prove that $Γ_{\frak a}(G)$ is Gorenstein injective for any Gorenstein injective $R$-module $G$. Let $(R,\frak m)$ be a local ring and $M$ be a finitely generated $R$-module. We show that ${\rm Gid}{\bf R}Γ_{\frak m}(M)<\infty$ if and only if ${\rm Gid}_{\hat{R}}(M\otimes_R\hat{R})<\infty$. We also show that if ${\rm Gfd}_R{\bf R}Γ_{\frak m}(M)<\infty$, then ${\rm Gfd}_RM<\infty$. Let $(R,\frak m)$ be a Cohen-Macaulay local ring and $M$ be a Cohen-Macaulay module of dimension $n$. We prove that if $H_{\frak m}^n(M)$ is of finite G-injective dimension, then Gid$_RH_{\frak m}^n(M)=d-n$. Moreover, we prove that if $M$ is a Matlis reflexive strongly torsion free module of finite G-flat dimension, then Gfd$_R\hat{M}<\infty$, where $\hat{M}$ is $\frak m$-adic completion.

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