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Uniformly dominant local rings and Orlov spectra of singularity categories

Ryo Takahashi

Abstract

We define a uniformly dominant local ring as a commutative noetherian local ring with an integer r such that the residue field is built from any nonzero object in the singularity category by direct summands, shifts and at most r mapping cones. We find sufficient conditions for uniform dominance, by which we show Burch rings and local rings with quasi-decomposable maximal ideal are uniformly dominant. For a uniformly dominant excellent equicharacteristic isolated singularity, we get an upper bound of the Orlov spectrum of the singularity category. We prove uniform dominance is preserved under basic operations, and give techniques to construct uniformly dominant local rings. An application of our methods to local rings with decomposable maximal ideal is provided as well.

Uniformly dominant local rings and Orlov spectra of singularity categories

Abstract

We define a uniformly dominant local ring as a commutative noetherian local ring with an integer r such that the residue field is built from any nonzero object in the singularity category by direct summands, shifts and at most r mapping cones. We find sufficient conditions for uniform dominance, by which we show Burch rings and local rings with quasi-decomposable maximal ideal are uniformly dominant. For a uniformly dominant excellent equicharacteristic isolated singularity, we get an upper bound of the Orlov spectrum of the singularity category. We prove uniform dominance is preserved under basic operations, and give techniques to construct uniformly dominant local rings. An application of our methods to local rings with decomposable maximal ideal is provided as well.

Paper Structure

This paper contains 6 sections, 34 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Syzygies over a local ring with decomposable maximal ideal
  3. A fundamental theorem
  4. Syzygies of the residue field
  5. Orlov spectra over uniformly dominant isolated singularities
  6. Producing uniformly dominant local rings

Key Result

Theorem 1.1

Let $(R,\mathfrak{m},k)$ be a complete equicharacteristic local hypersurface of dimension $d$ such that $k$ is algebraically closed and of characteristic $0$. Suppose that $R$ has an isolated singularity. Put $J=\operatorname{jac} R$ and $l=\ell\ell(R/J)$. Then, all the nonzero objects of the singul

Theorems & Definitions (82)

  • Theorem 1.1: Ballard--Favero--Katzarkov BFK
  • Theorem 1.2: Corollaries \ref{['40']} and \ref{['28']}
  • Theorem 1.3: Theorem \ref{['5']} and Corollary \ref{['11']}
  • Theorem 1.4: Nasseh--Takahashi fiber
  • Theorem 1.5: Theorem \ref{['36']}
  • Definition 2.1
  • Proposition 2.2
  • proof
  • Lemma 2.3
  • Lemma 2.4
  • ...and 72 more