Algebraic versions of $\mathbb{T}^2$ and of $\mathbb{P}^1\times\mathbb{P}^1$ and Hochschild cohomology

TL;DR

This paper studies Hochschild cohomology of algebraic structures related to geometric surfaces like the torus and quadric, revealing nuanced behaviors in their cohomology and deformation properties.

Contribution

It introduces algebraic models of geometric surfaces and analyzes their Hochschild cohomology, highlighting differences from classical expectations and effects of algebra deformations.

Findings

Cup product behavior differs from monomial algebra intuition.

Deformations can alter cohomology structures without changing dimension.

Derived equivalence does not guarantee similar Hochschild cohomology behaviors.

Abstract

We examine the Hochschild cohomology for triangular algebras that capture some aspects of geometry and topology of the torus and of the quadric surface, and for deformations of these algebras. In particular, this shows that the cup product on the Hochschild cohomology of a triangular algebra does not generally follow the intuition coming from monomial algebras. Our examples also demonstrate that the Hochschild cohomology of a deformation of an algebra may not experience the dimension drop but still have a different cup product structure, and that the Hochschild cohomologies of deformations of two derived equivalent algebras may exhibit noticeably different behaviours.