The Hochschild cohomology groups under gluing idempotents

TL;DR

This paper investigates how the Hochschild cohomology groups of finite dimensional monomial algebras change when gluing idempotents, especially in the case of gluing a source and a sink, leading to stable equivalences.

Contribution

It provides a detailed comparison of Hochschild cohomology groups and fundamental groups under the operation of gluing idempotents, including special cases like source-sink gluing.

Findings

Hochschild cohomology groups are affected by idempotent gluing.

Fundamental groups are compared under gluing operations.

Stable equivalences are characterized in the source-sink case.

Abstract

We compare the first Hochschild cohomology groups of finite dimensional monomial algebras under gluing two idempotents. We also compare the fundamental groups and the Hochschild cohomology groups in other degrees. In particular, we will study the case of gluing a source and a sink, that is, when we obtain a stable equivalence.