模与理想联系下的复杂性、曲率与同调维数
Complexity, curvature and homological dimension of modules under linkage
AI总结：
本文研究在Gorenstein局部环上，通过Gorenstein完美理想联系的Cohen-Macaulay模的投射与内射复杂性、曲率以及完全交维数之间的关系，并推广了Peskine和Szpiro的经典结果。
摘要：
本文分析了模与理想联系下（投射与内射）复杂性、曲率以及完全交维数的行为。设$R$为Gorenstein局部环。考虑一个Gorenstein完美理想$\mathfrak{a}$（例如，$\mathfrak{a}$由$R$-正则序列生成）。设$M$和$N$是由$\mathfrak{a}$联系的两个Cohen-Macaulay $R$-模。我们证明$\mathrm{cx}_R(M)= \mathrm{inj\,cx}_R(N)$和$\mathrm{curv}_R(M)= \mathrm{inj\,curv}_R(N)$。特别地，当$R$为完全交时，有$\mathrm{cx}_R(M)= \mathrm{cx}_R(N)$和$\mathrm{curv}_R(M)= \mathrm{curv}_R(N)$。此外，我们证明$\mathrm{pd}_R(M)= \mathrm{pd}_R(N)$和$\operatorname{CI-dim}_R(M)= \operatorname{CI-dim}_R(N)$。如果这些维数中任意一个是有限的，则它等于$\mathrm{ht}(\mathfrak{a})$。对于理想的联系也得到了类似的结果。所有这些结果在许多方向上高度推广了Peskine和Szpiro的经典结果。我们构造了几个例子来补充我们的结果。这些例子也展示了诸如“整闭”、“$\mathfrak{m}$-满”和“Burch”等性质在理想联系下的行为。
英文摘要：
In this article, we analyze how (projective and injective) complexity, curvature, and complete intersection dimension behave under linkage of modules and ideals. Let $R$ be a Gorenstein local ring. Consider a Gorenstein perfect ideal $\mathfrak{a}$ (e.g., $\mathfrak{a}$ is generated by an $R$-regular sequence). Let $M$ and $N$ be two Cohen-Macaulay $R$-modules linked by $\mathfrak{a}$. We prove that $\mathrm{cx}_R(M)= \mathrm{inj\,cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{inj\,curv}_R(N)$. In particular, when $R$ is complete intersection, $\mathrm{cx}_R(M)= \mathrm{cx}_R(N)$ and $\mathrm{curv}_R(M)= \mathrm{curv}_R(N)$. Furthermore, we show that $\mathrm{pd}_R(M)= \mathrm{pd}_R(N)$ and $\operatorname{CI-dim}_R(M)= \operatorname{CI-dim}_R(N)$. If any of these dimensions is finite, it is equal to $\mathrm{ht}(\mathfrak{a})$. Similar results are obtained for linkage of ideals. All these results highly extend a classical result of Peskine and Szpiro in many directions. We construct several examples that complement our results. These also show how properties like `integrally closed', `$\mathfrak{m}$-full' and `Burch' behave under linkage of ideals.
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