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Reflexivity and Hochschild Cohomology

Isambard Goodbody

Abstract

Reflexive DG-categories were defined by Kuznetsov and Shinder as generalisations of smooth and proper DG-categories. Over a perfect field, they include all projective schemes and finite dimensional algebras. Smooth and proper DG-categories are the dualizable objects in the symmetric monoidal category of DG-categories localised at Morita equivalences. We show the reflexive DG-categories are the reflexive objects in this monoidal category. Using this perspective we prove that the Hochschild cohomology of a reflexive DG-category is isomorphic to that of its derived category of cohomologically finite modules.

Reflexivity and Hochschild Cohomology

Abstract

Reflexive DG-categories were defined by Kuznetsov and Shinder as generalisations of smooth and proper DG-categories. Over a perfect field, they include all projective schemes and finite dimensional algebras. Smooth and proper DG-categories are the dualizable objects in the symmetric monoidal category of DG-categories localised at Morita equivalences. We show the reflexive DG-categories are the reflexive objects in this monoidal category. Using this perspective we prove that the Hochschild cohomology of a reflexive DG-category is isomorphic to that of its derived category of cohomologically finite modules.
Paper Structure (8 sections, 25 theorems, 68 equations)

This paper contains 8 sections, 25 theorems, 68 equations.

Table of Contents

  1. Introduction
  2. Homotopy Theory of DG-categories
  3. Reflexivity in Closed Symmetric Monoidal Categories
  4. Reflexive DG-categories
  5. Hochschild Cohomology
  6. Corepresentability
  7. Finite Dimensional DGAs and their Koszul duals
  8. An Example

Key Result

Theorem 2.1

$(\mathop{\mathrm{Hqe}}\nolimits, \otimes^{\mathbb{L}}, k)$ is a closed symmetric monoidal category.

Theorems & Definitions (73)

  • Theorem 2.1: Theorem 1.3 Toe06
  • Proposition 2.2
  • proof
  • Remark 2.3
  • Definition 2.4
  • Remark 2.5
  • Theorem 2.6
  • Corollary 2.7
  • proof
  • Remark 2.8
  • ...and 63 more