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Duality and equivalence of module categories in noncommutative geometry I

Jonathan Block

TL;DR

This paper introduces a curved-dga based dg-category framework to express and unify dualities in noncommutative and complex geometry through module categories. By constructing the dg-category ${\mathcal{P}}_{\textbf{A}}$ of cohesive modules and proving its homotopy category is triangulated, it recovers the derived category of coherent sheaves on complex manifolds in the Dolbeault setting and extends to gerbes, Lie algebroids, and noncommutative tori. It develops duality structures (Serre, Poincaré) within this framework and provides concrete examples (elliptic curved dgas, various Lie algebroids, generalized Higgs algebroids, and noncommutative tori) illustrating Mukai–type dualities and their relation to Baum–Connes-type phenomena. The results offer a unified, categorical approach to dualities across geometry and operator algebras, with potential for connecting derived equivalences to index-type maps in noncommutative geometry. Overall, the work lays a versatile foundation for translating geometric dualities into dg-category language and applying it to broad classes of spaces and algebroids.

Abstract

This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator $K$-theory). These dualities are expressed in terms of categories of modules. In this paper, we develop a general framework needed to describe these dualities. In various geometric contexts, e.g. complex geometry, generalized complex geometry, and noncommutative geometry, the geometric structure is encoded in a certain differential graded algebra. We develop the module theory of such differential graded algebras in such a way that we can recover the derived category of coherent sheaves on a complex manifold. In this paper and ones to follow we apply this to stating and proving the duality statements mentioned above. After developing the general framework, we look at a (complex) Lie algebroid $\A\to T_\cx X$. One can then consider our analogue of the derived category of coherent sheaves, integrable with respect to the Lie algebroid. We then establish a (Serre) duality theorem for "elliptic" Lie algebroids and for noncommutative tori.

Duality and equivalence of module categories in noncommutative geometry I

TL;DR

This paper introduces a curved-dga based dg-category framework to express and unify dualities in noncommutative and complex geometry through module categories. By constructing the dg-category of cohesive modules and proving its homotopy category is triangulated, it recovers the derived category of coherent sheaves on complex manifolds in the Dolbeault setting and extends to gerbes, Lie algebroids, and noncommutative tori. It develops duality structures (Serre, Poincaré) within this framework and provides concrete examples (elliptic curved dgas, various Lie algebroids, generalized Higgs algebroids, and noncommutative tori) illustrating Mukai–type dualities and their relation to Baum–Connes-type phenomena. The results offer a unified, categorical approach to dualities across geometry and operator algebras, with potential for connecting derived equivalences to index-type maps in noncommutative geometry. Overall, the work lays a versatile foundation for translating geometric dualities into dg-category language and applying it to broad classes of spaces and algebroids.

Abstract

This is the first in a series of papers that deals with duality statements such as Mukai-duality (T-duality, from algebraic geometry) and the Baum-Connes conjecture (from operator -theory). These dualities are expressed in terms of categories of modules. In this paper, we develop a general framework needed to describe these dualities. In various geometric contexts, e.g. complex geometry, generalized complex geometry, and noncommutative geometry, the geometric structure is encoded in a certain differential graded algebra. We develop the module theory of such differential graded algebras in such a way that we can recover the derived category of coherent sheaves on a complex manifold. In this paper and ones to follow we apply this to stating and proving the duality statements mentioned above. After developing the general framework, we look at a (complex) Lie algebroid . One can then consider our analogue of the derived category of coherent sheaves, integrable with respect to the Lie algebroid. We then establish a (Serre) duality theorem for "elliptic" Lie algebroids and for noncommutative tori.

Paper Structure

This paper contains 28 sections, 28 theorems, 158 equations.

Table of Contents

  1. Baum-Connes and Fourier-Mukai
  2. The dg-category ${\mathcal{P}}_\text{\bfseries\sf{A}}$ of a curved dga
  3. dg-categories
  4. Curved dgas
  5. The dg-category ${\mathcal{P}}_{\text{\bfseries\sf{A}}}$
  6. The homotopy category and triangulated structure
  7. Homotopy equivalences
  8. The dual of a cohesive module
  9. Functoriality
  10. Modules over ${\mathcal{P}}_\text{\bfseries\sf{A}}$
  11. Modules over a dg-category
  12. Construction and properties of modules over ${\mathcal{P}}_\text{\bfseries\sf{A}}$.
  13. Complex manifolds
  14. The derived category of sheaves of $\mathcal{O}_X$-modules with coherent cohomology
  15. Gerbes on complex manifolds
  16. ...and 13 more sections

Key Result

Proposition 2.3.3

For $\text{\bfseries\sf{A}}=(\mathcal{A}^\bullet, d, c)$ a curved dga, the category ${\mathcal{P}}_{\text{\bfseries\sf{A}}}$ is a dg-category.

Theorems & Definitions (58)

  • Definition 2.1.1
  • Definition 2.2.1
  • Definition 2.3.1
  • Definition 2.3.2
  • Proposition 2.3.3
  • Lemma 2.3.4
  • Proposition 2.4.1
  • proof
  • Proposition 2.5.1
  • Proposition 2.5.2
  • ...and 48 more