[Paper Review] Duality and equivalence of module categories in noncommutative geometry I

Jonathan Block|arXiv (Cornell University)|Sep 13, 2005

Algebraic structures and combinatorial modelsMathematics23 references66 citations

TL;DR

This paper introduces a differential graded category framework to unify dualities in noncommutative geometry, algebraic geometry, and mathematical physics. By constructing a curved differential graded algebra from Dolbeault data, it realizes the derived category of coherent sheaves on complex manifolds and establishes Serre duality for noncommutative complex tori, linking the Baum-Connes conjecture to derived equivalences.

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 $\A o T_\cx X$. 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.

Motivation & Objective

• To develop a general framework for duality statements in noncommutative geometry, algebraic geometry, and mathematical physics.
• To connect the Baum-Connes conjecture in operator K-theory with derived equivalence in algebraic geometry and physics.
• To construct a differential graded category that captures the derived category of coherent sheaves on complex manifolds using global differential geometric data.
• To extend the notion of quasi-isomorphism beyond standard dg-modules to include twisted complexes via a curved dga construction.
• To establish Serre duality for noncommutative complex tori using a dualizing module and trace functional.

Proposed method

• Define a curved differential graded algebra (dga) using the Dolbeault complex of a compact complex manifold, incorporating a group 2-cocycle σ.
• Construct a differential graded category P_A from the dga A, which captures coherent sheaves via global differential geometric structures.
• Introduce a twisted differential ∂̄ on A = A^{0,•}(Λ;σ) using the (1,0)-component of the complex vector space V, ensuring compatibility with duality.
• Equip the dga with a trace τ and define a dualizing module (D, D̄, *, ∫) to realize Serre duality.
• Use the ∫ functional on top-degree forms to define a non-degenerate pairing, verifying the duality axioms.
• Show that the category P_A has a Serre functor given by E ↦ E ⊗ D with induced differential D̄, generalizing classical Serre duality.

Experimental results

Research questions

• RQ1 How can duality statements in noncommutative geometry, such as T-duality and mirror symmetry, be unified under a single categorical framework?
• RQ2 Can the derived category of coherent sheaves on a complex manifold be reconstructed from a global differential geometric structure rather than local holomorphic data?
• RQ3 What is the correct notion of quasi-isomorphism for dg-modules over a Dolbeault dga, and how does it differ from the standard one?
• RQ4 How does the Baum-Connes conjecture relate to derived equivalences in algebraic geometry and physics?
• RQ5 Can Serre duality be extended to noncommutative complex tori, and what structure underlies this duality?

Key findings

• The homotopy category of the differential graded category P_A is equivalent to the derived category of coherent sheaves on a compact complex manifold X.
• The dga A = A^{0,•}(Λ;σ) with trivial σ is isomorphic to the Dolbeault algebra of the dual complex torus X^∨, confirming consistency with classical geometry.
• The dga A is elliptic, with a dualizing module (D, D̄, *, ∫) defined via a trace τ and top-form integration.
• The Serre functor on P_A is given by E ↦ E ⊗ D with induced differential D̄, and it recovers the classical Serre functor when σ = 1.
• The construction avoids the homological defects of standard dg-modules by using a curved dga and a refined category P_A.
• The framework realizes B-branes in string theory as models for coherent sheaves, as used by Bergman, linking to physics.