Hochschild cohomology parametrizes curved Morita deformations

TL;DR

This paper establishes a bijection between curved Morita deformations and Hochschild cohomology for dg algebras, and characterizes bimodule-induced equivalences of deformations via derived categories, solving the curvature problem for first order deformations.

Contribution

It demonstrates that allowing curved deformations makes the canonical map between Morita deformations and Hochschild cohomology bijective, and characterizes equivalences of deformations via derived categories.

Findings

Canonical map becomes a bijection with curved deformations.

Bimodule equivalences correspond to equivalences of 1-derived categories.

Solves the curvature problem for first order deformations.

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.