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Vanishing of DHKK complexities for singularity categories and generation of syzygy modules

Tokuji Araya, Kei-ichiro Iima, Ryo Takahashi

TL;DR

This work examines the vanishing behavior of DHKK complexities in the singularity category $D_{sg}(R)$ of a noetherian ring $R$, focusing on when the nonvanishing set $\\Delta(R)$ is bounded. By developing techniques to generate high syzygies and maximal Cohen–Macaulay modules from a single module through extension closures and summands, the authors establish bounded nonvanishing ranges in several ring classes, including local rings with isolated singularities, semilocal rings of small dimension, and rings with J-0/J-2 conditions. They prove generation results for $\\Omega^d(\\operatorname{mod}R)$ and, under CM/Gorenstein hypotheses with canonical modules, show that $\\operatorname{CM}(R)$ can be captured by extensions of a finite module, leading to bounded DHKK nonvanishing ranges. The findings provide structural constraints on categorical entropy-like invariants in singularity categories and yield practical criteria for when DHKK complexities must vanish beyond a finite interval, with implications for the study of singularities and syzygy generation in commutative algebra.

Abstract

Let R be a commutative noetherian ring. In this paper, we study, for the singularity category of R, the vanishing of the complexity $δ_t(X,Y)$ in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich. We prove that the set of real numbers t such that $δ_t(X,Y)$ does not vanish is bounded in various cases. We do it by building the high syzygy modules and maximal Cohen-Macaulay modules out of a single module only by taking direct summands and extensions.

Vanishing of DHKK complexities for singularity categories and generation of syzygy modules

TL;DR

This work examines the vanishing behavior of DHKK complexities in the singularity category of a noetherian ring , focusing on when the nonvanishing set is bounded. By developing techniques to generate high syzygies and maximal Cohen–Macaulay modules from a single module through extension closures and summands, the authors establish bounded nonvanishing ranges in several ring classes, including local rings with isolated singularities, semilocal rings of small dimension, and rings with J-0/J-2 conditions. They prove generation results for and, under CM/Gorenstein hypotheses with canonical modules, show that can be captured by extensions of a finite module, leading to bounded DHKK nonvanishing ranges. The findings provide structural constraints on categorical entropy-like invariants in singularity categories and yield practical criteria for when DHKK complexities must vanish beyond a finite interval, with implications for the study of singularities and syzygy generation in commutative algebra.

Abstract

Let R be a commutative noetherian ring. In this paper, we study, for the singularity category of R, the vanishing of the complexity in the sense of Dimitrov, Haiden, Katzarkov and Kontsevich. We prove that the set of real numbers t such that does not vanish is bounded in various cases. We do it by building the high syzygy modules and maximal Cohen-Macaulay modules out of a single module only by taking direct summands and extensions.
Paper Structure (4 sections, 20 theorems, 16 equations)

This paper contains 4 sections, 20 theorems, 16 equations.

Table of Contents

  1. Introduction
  2. DHKK complexities for a general triangulated category
  3. Vanishing of DHKK complexities and generating syzygies
  4. Vanishing of DHKK complexities and generating maximal Cohen--Macaulay modules

Key Result

Theorem 1.2

Suppose that $(R,\mathfrak{m},k)$ is a local ring with an isolated singularity. Let $G$ be a generator of $\operatorname{D_{sg}}(R)$. Let $X$ be an object of $\operatorname{D_{sg}}(R)$. Then the following assertions hold true.

Theorems & Definitions (45)

  • Conjecture 1.1: Takahashi
  • Theorem 1.2: Takahashi
  • Theorem 1.3
  • Corollary 1.4
  • Theorem 1.5
  • Definition 2.2
  • Lemma 2.3
  • Definition 2.4: Dimitrov--Haiden--Katzarkov--Kontsevich
  • Definition 2.5
  • Lemma 2.6
  • ...and 35 more