Compactness of the Alexandrov topology of maximal Cohen-Macaulay modules

TL;DR

Let $R$ be a Cohen--Macaulay local ring. This paper investigates the annihilators of stable categories of maximal Cohen--Macaulay modules and links their Alexandrov topologies to the cohomology annihilator and to $\\operatorname{Sing}(\\widehat{R})$. It proves radical equalities $\\sqrt{[b]{\\mathrm{Ann}(\\underline{\\mathsf{CM}}(R))}}=\\sqrt{[b]{\\mathrm{Ann}(\\underline{\\mathsf{CM}}_0(R))}}=\\sqrt{[b]{\\mathrm{ca}(R)}}=\\bigcap_{\\mathfrak{p}\\in\\operatorname{Sing}(\\widehat{R})} (\\mathfrak{p}\\cap R)$, and derives equivalent conditions for the compactness of the Alexandrov topologies alongside the presence of isolated singularities. It then analyzes hypersurfaces of countable CM-representation type, computing endomorphism-annihilators to establish compactness of $\\mathcal{A}(\\underline{\\mathsf{CM}}(R))$ in that setting, and provides explicit counterexamples for the $\\underline{\\mathsf{CM}}_0(R)$-case when singularities are not isolated. These results connect the geometry of Sing$(R)$ with stable-category structure and extend compactness phenomena beyond the (quasi-)excellent case.

Abstract

Let $R$ be a Cohen-Macaulay local ring. In this paper, we first describe the radicals of annihilators of stable categories of maximal Cohen-Macaulay $R$-modules. We then prove that the Alexandrov topology of the stable category of maximal Cohen-Macaulay $R$-modules is compact provided that the completion of $R$ has an isolated singularity. Finally, we consider the case of a hypersurface of countable CM-representation type.

Theorems & Definitions (30)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.2
• Lemma 2.3
• Proposition 2.4
• proof
• proof : Proof of Theorem \ref{['main result general']}
• Remark 2.5
• Proposition 2.6