Complexity, curvature and homological dimension of modules under linkage

Abstract

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 ${\mathrm{cx}}_R(M)={\mathrm{inj}\mathrm{cx}}_R(N)$ and ${\mathrm{curv}}_R(M)={\mathrm{inj}\mathrm{curv}}_R(N)$. 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 ${CI-dim}_R(M)={CI-dim}_R(N)$. 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.

arXiv comment15 pages, comments are welcome