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Lectures on Noncommutative Geometry

Victor Ginzburg

TL;DR

Ginzburg’s Lectures on Noncommutative Geometry provide a comprehensive, category-centric survey of noncommutative geometry, anchored by Morita invariance and the study of associative algebras through their module categories. The work develops foundational tools (bar complex, Hochschild (co)homology, HKR) and extends classical geometric notions to the noncommutative setting via noncommutative differential forms, Karoubi–de Rham theory, and a calculus of NC operations. It then builds deformation-theoretic machinery (star products, Moyal–Weyl, NC-Poisson structures) and connects these with representation theory through the representation functor and traces, illustrating a program to view noncommutative spaces via their Morita-invariant invariants. Overall, the notes assemble a robust toolkit—Bar complexes, (co)homology, Poisson/Gerstenhaber structures, NC differential forms, and representation theory—that underpins both foundational theory and practical constructions in NC geometry and its links to deformation quantization and NC Chern–Weil theory.

Abstract

These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.

Lectures on Noncommutative Geometry

TL;DR

Ginzburg’s Lectures on Noncommutative Geometry provide a comprehensive, category-centric survey of noncommutative geometry, anchored by Morita invariance and the study of associative algebras through their module categories. The work develops foundational tools (bar complex, Hochschild (co)homology, HKR) and extends classical geometric notions to the noncommutative setting via noncommutative differential forms, Karoubi–de Rham theory, and a calculus of NC operations. It then builds deformation-theoretic machinery (star products, Moyal–Weyl, NC-Poisson structures) and connects these with representation theory through the representation functor and traces, illustrating a program to view noncommutative spaces via their Morita-invariant invariants. Overall, the notes assemble a robust toolkit—Bar complexes, (co)homology, Poisson/Gerstenhaber structures, NC differential forms, and representation theory—that underpins both foundational theory and practical constructions in NC geometry and its links to deformation quantization and NC Chern–Weil theory.

Abstract

These Lectures are based on a course on noncommutative geometry given by the author in 2003 at the University of Chicago. The lectures contain some standard material, such as Poisson and Gerstenhaber algebras, deformations, Hochschild cohomology, Serre functors, etc. We also discuss many less known as well as some new results, in particular, noncommutative Chern-Weil theory, noncommutative symplectic geometry, noncommutative differential forms and double-tangent bundles.

Paper Structure

This paper contains 101 sections, 135 theorems, 495 equations.

Table of Contents

  1. Introduction
  2. Morita Equivalence
  3. Categories and functors.
  4. Algebras and spaces.
  5. Morita theorem.
  6. Derivations and Atiyah algebras
  7. Square-zero construction.
  8. Super-derivations.
  9. The tensor algebra of a bimodule.
  10. Picard group of a category.
  11. Atiyah algebra of a vector bundle.
  12. The Bar Complex
  13. Free product of algebras
  14. Second construction of the bar complex (after Drinfeld).
  15. Third construction of the bar complex.
  16. ...and 86 more sections

Key Result

Lemma 2.1.1

Let ${\mathscr C}_1$ and ${\mathscr C}_2$ be two abelian categories, and let $F\colon{\mathscr C}_1\to{\mathscr C}_2$ be an exact, fully faithful, essentially surjective functor. Then $F$ is an equivalence of categories. ∎

Theorems & Definitions (301)

  • Lemma 2.1.1
  • Lemma 2.1.2: Yoneda lemma
  • Definition 2.1.3
  • Definition 2.1.4
  • Lemma 2.1.5
  • Lemma 2.1.6
  • proof
  • Proposition 2.1.7
  • proof : Proof of Proposition \ref{['P:AbCatEqCond']}
  • Definition 2.1.8
  • ...and 291 more