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DG singular equivalence and singular locus

Leilei Liu, Jieheng Zeng

TL;DR

This work shows that for a commutative Gorenstein Noetherian ring $R$, the singular locus of $\mathrm{Spec}(R)$ can be recovered from the DG enhancement $S_{dg}(R)$ by constructing an affine scheme $X$ with a finite surjection to $\mathrm{Spec}(R/I)$. The construction hinges on the generalized Tate-Hochschild framework: the zeroth Tate-Hochschild cohomology $\mathrm{HH}_{sg}^{0}(R)$, after reduction, contains the coordinate ring of the singular locus, and $\mathrm{HH}_{sg}^{0}(R)_{red}$ is finitely generated over $R/I$; this yields $X=\mathrm{Spec}(\mathrm{HH}_{sg}^{0}(R)_{red})$ with a finite morphism to the singular locus. A key consequence is that if two rings have equivalent DG singularity categories, their singular loci have the same dimension, via isomorphism of the reduced zeroth Tate-Hochschild cohomology. The results bridge DG-category enhancements with concrete geometric invariants, offering a method to compare singularities through noncommutative invariants and providing a DG-categorical route to understanding classical singular loci and their dimensions.

Abstract

For a commutative Gorenstein Noetherian ring $R$, we construct an affine scheme $X$ solely from DG singularity category $S_{dg}(R)$ of $R$ such that there is a finite surjective morphism $X \rightarrow \mathrm{Spec}(R /I)$, where $\mathrm{Spec}(R /I)$ is the singular locus in $\mathrm{Spec}(R)$. As an application, for two such rings with equivalent DG singularity categories, we prove that the singular loci in their affine schemes have the same dimension.

DG singular equivalence and singular locus

TL;DR

This work shows that for a commutative Gorenstein Noetherian ring , the singular locus of can be recovered from the DG enhancement by constructing an affine scheme with a finite surjection to . The construction hinges on the generalized Tate-Hochschild framework: the zeroth Tate-Hochschild cohomology , after reduction, contains the coordinate ring of the singular locus, and is finitely generated over ; this yields with a finite morphism to the singular locus. A key consequence is that if two rings have equivalent DG singularity categories, their singular loci have the same dimension, via isomorphism of the reduced zeroth Tate-Hochschild cohomology. The results bridge DG-category enhancements with concrete geometric invariants, offering a method to compare singularities through noncommutative invariants and providing a DG-categorical route to understanding classical singular loci and their dimensions.

Abstract

For a commutative Gorenstein Noetherian ring , we construct an affine scheme solely from DG singularity category of such that there is a finite surjective morphism , where is the singular locus in . As an application, for two such rings with equivalent DG singularity categories, we prove that the singular loci in their affine schemes have the same dimension.
Paper Structure (12 sections, 10 theorems, 70 equations)

This paper contains 12 sections, 10 theorems, 70 equations.

Table of Contents

  1. Introduction
  2. Backgrounds
  3. Main result
  4. Idea of the proof
  5. Notation and conventions
  6. preliminaries
  7. Singularity category and singular equivalence
  8. DG category and DG functor
  9. The Tate-Hochschild cohomology
  10. Generalized Tate-Hochschild complex of associative algebra
  11. Hochschild cohomology of DG singularity category
  12. Proof of the main result

Key Result

Theorem 1.1

Let $R$ be a commutative Gorenstein Noetherian ring. Let $S_{dg}(R)$ be the DG category described as above. Then there is an affine scheme $X$ constructed solely from $S_{dg}(R)$, and a finite surjective morphism $X \rightarrow \mathrm{Spec}(R/I)$, where $\mathrm{Spec}(R /I)$ is the singular locus i

Theorems & Definitions (26)

  • Theorem 1.1
  • Definition 2.1
  • Definition 2.2
  • Example 2.3: Knorrer's periodicity theorem
  • Definition 2.4
  • Definition 2.5
  • Definition 2.6
  • Definition 3.1
  • Definition 3.2
  • Definition 3.3
  • ...and 16 more