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Hochschild cohomology parametrizes curved Morita deformations

Alessandro Lehmann

TL;DR

This work resolves the curvature problem for first-order deformations by proving that allowing curved Morita deformations yields a bijection between curved deformations and the second Hochschild cohomology $\mathsf{HH}^2(A)$ via the map $\nu$. It builds a precise framework for curved Morita theory using cdg algebras, curved Yoneda embeddings, and Koszul duality to construct and compare deformations, and shows that a curved bimodule induces an equivalence exactly when it induces an equivalence of the $1$-derived categories. The results integrate with a broader deformation-derivation program, connecting Hochschild invariants to categorical square-zero extensions and the $1$-derived category, and provide explicit algebraic models for the groupoid of curved deformations over local Artinian bases. Overall, the paper advances deformation theory in the curved setting and clarifies how curvature data governs Morita-type deformations and their categorical consequences.

Abstract

We show that, if one allows for curved deformations, the canonical map introduced in [KL09] between Morita deformations and second Hochschild cohomology of a dg algebra becomes a bijection. We also show that a bimodule induces an equivalence of curved deformations precisely when it induces an equivalence between the respective 1-derived categories. These results, together with arXiv:2402.08660, solve the curvature problem for first order deformations.

Hochschild cohomology parametrizes curved Morita deformations

TL;DR

This work resolves the curvature problem for first-order deformations by proving that allowing curved Morita deformations yields a bijection between curved deformations and the second Hochschild cohomology via the map . It builds a precise framework for curved Morita theory using cdg algebras, curved Yoneda embeddings, and Koszul duality to construct and compare deformations, and shows that a curved bimodule induces an equivalence exactly when it induces an equivalence of the -derived categories. The results integrate with a broader deformation-derivation program, connecting Hochschild invariants to categorical square-zero extensions and the -derived category, and provide explicit algebraic models for the groupoid of curved deformations over local Artinian bases. Overall, the paper advances deformation theory in the curved setting and clarifies how curvature data governs Morita-type deformations and their categorical consequences.

Abstract

We show that, if one allows for curved deformations, the canonical map introduced in [KL09] between Morita deformations and second Hochschild cohomology of a dg algebra becomes a bijection. We also show that a bimodule induces an equivalence of curved deformations precisely when it induces an equivalence between the respective 1-derived categories. These results, together with arXiv:2402.08660, solve the curvature problem for first order deformations.
Paper Structure (8 sections, 11 theorems, 27 equations)

This paper contains 8 sections, 11 theorems, 27 equations.

Table of Contents

  1. Introduction
  2. Curved Morita deformations
  3. Cdg algebras and modules
  4. Hochschild cohomology and Morita bimodules
  5. Curved Morita deformations
  6. Curved deformations and Hochschild cohomology
  7. Filtered Morita equivalences
  8. The $n$-derived category

Key Result

Lemma 2.2

Equivalence of curved deformations is an equivalence relation, so the set $\operatorname{cDef}_A(k[\varepsilon])$ is well defined.

Theorems & Definitions (24)

  • Definition 2.1
  • Lemma 2.2
  • proof
  • Remark
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Proposition 2.5
  • proof
  • ...and 14 more