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Orlov's Theorem for dg-algebras

Michael K. Brown, Prashanth Sridhar

TL;DR

The paper extends Orlov's landmark relation between singularity categories and derived categories to the setting of differential graded algebras (dg-algebras) by introducing Gorenstein dg-algebras with a Gorenstein parameter a. It develops a noncommutative dg-projective geometry framework, defining D_gr^sg(A) and D_qgr(A) and proving a main theorem that yields explicit semiorthogonal decompositions and fully faithful embeddings depending on a (with a=0 giving an equivalence). The results provide new cases of the Lattice Conjecture in noncommutative Hodge theory for dg-categories and connect to derived categories of sheaves of dg-algebras on Proj(A^0) via a dg-version of Artin–Zhang's theory, including explicit Koszul and weight-graded examples. Overall, the work broadens the reach of Orlov-type equivalences to the dg-context, enabling duality, semiorthogonal decompositions, and topological K-theory consequences in a broader noncommutative geometric setting.

Abstract

A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. We generalize this theorem to the setting of differential graded algebras. As an application, we obtain new cases of the Lattice Conjecture in noncommutative Hodge theory.

Orlov's Theorem for dg-algebras

TL;DR

The paper extends Orlov's landmark relation between singularity categories and derived categories to the setting of differential graded algebras (dg-algebras) by introducing Gorenstein dg-algebras with a Gorenstein parameter a. It develops a noncommutative dg-projective geometry framework, defining D_gr^sg(A) and D_qgr(A) and proving a main theorem that yields explicit semiorthogonal decompositions and fully faithful embeddings depending on a (with a=0 giving an equivalence). The results provide new cases of the Lattice Conjecture in noncommutative Hodge theory for dg-categories and connect to derived categories of sheaves of dg-algebras on Proj(A^0) via a dg-version of Artin–Zhang's theory, including explicit Koszul and weight-graded examples. Overall, the work broadens the reach of Orlov-type equivalences to the dg-context, enabling duality, semiorthogonal decompositions, and topological K-theory consequences in a broader noncommutative geometric setting.

Abstract

A landmark theorem of Orlov relates the singularity category of a graded Gorenstein algebra to the derived category of the associated noncommutative projective scheme. We generalize this theorem to the setting of differential graded algebras. As an application, we obtain new cases of the Lattice Conjecture in noncommutative Hodge theory.
Paper Structure (9 sections, 21 theorems, 26 equations)

This paper contains 9 sections, 21 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Resolutions and derived categories over dg-algebras
  3. Derived categories
  4. Gorenstein dg-algebras
  5. Noncommutative algebraic geometry over a dg-algebra
  6. Derived global sections
  7. Sheaves of dg-algebras
  8. Proof of Theorem \ref{['main']}
  9. Application to the Lattice Conjecture

Key Result

Theorem 1.1

If $X$ is Calabi-Yau, then the categories $\operatorname{D}^{\operatorname{b}}(X)$ and $\operatorname{D}_{\operatorname{gr}}^{\operatorname{sg}}(R)$ are equivalent.

Theorems & Definitions (73)

  • Theorem 1.1: Orlov2009
  • Theorem 1.2: Orlov2009 Theorem 2.5
  • Theorem 1.3
  • Corollary 1.4
  • Definition 2.1
  • Definition 2.2
  • Definition 2.4
  • Remark 2.5
  • Remark 2.6
  • Remark 2.7
  • ...and 63 more