关于Matlis自反模的注记
Notes on Matlis reflexive modules
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中文总结 AI 辅助
本文研究局部Noether环上Matlis自反模，证明了自反模的拟完备性，改进Zöschinger数值判据，并刻画Enochs数为1的条件。
中文摘要 AI 辅助
设$M$表示局部Noether环$(R,\mathfrak{m})$上具有剩余域$\Bbbk$及其内射包$E_R$的模。设$D_R(\cdot) = \operatorname{Hom}_R(\cdot,E_R)$表示Matlis对偶函子。当注入$M \to D_R^2(M)$是同构时，$M$称为Matlis自反模。证明了以下结果：(1) 对于$R$的理想，自反模是拟完备的，推广了Belshoff结果的一部分（见\cite{Br}）。(2) 改进了Zöschinger关于Matlis自反性的数值判据（见\cite{Zh2}）。(3) 如Enochs所示（见\cite{Ee}），对于素理想和内射包$E_R(R/\mathfrak{p})$，$D_R(E_R(R/\mathfrak{p}))$是秩为$\tau_{\mathfrak{p}}$的自由$R_{\mathfrak{p}}$-模的完备化，其中$\tau_{\mathfrak{p}}$是$\mathfrak{p}$的Enochs数。我们刻画了$\tau_{\mathfrak{p}} = 1$的情形，推广了作者的结果（见\cite{Sp16}）。
英文摘要
Let $M$ denote a module over a local Noetherian ring $(R,\mathfrak{m})$ with residue field $\Bbbk$ and its injective hull $E_R$. Let $D_R(\cdot) = \operatorname{Hom}_R(\cdot,E_R)$ denote the Matlis duality functor. Then $M$ is called Matlis reflexive whenever the injection $M \to D_R^2(M)$ is an isomorphism. The following results are proved: (1) For an ideal of $R$ a reflexive module is quasi-complete, generalizing part of Belshoff's results (see \cite{Br}). (2) We improve Zöschinger's numerical criterion for Matlis reflexivity (see \cite{Zh2}). (3) As shown by Enochs (see \cite{Ee}) for a prime ideal and the injective hull $E_R(R/\mathfrak{p})$ it follows that $D_R(E_R(R/\mathfrak{p}))$ is the completion of a free $R_{\mathfrak{p}}$-module of rank $τ_{\mathfrak{p}}$, the Enochs number of $\mathfrak{p}$. We characterize when $τ_{\mathfrak{p}} = 1$, extending author's result (see \cite{Sp16}).
