The Ext-algebra for infinitesimal deformations

TL;DR

This paper studies the algebra structure of Ext-algebras for infinitesimal deformations of finite-dimensional algebras, providing explicit constructions and descriptions under certain conditions.

Contribution

It offers a new explicit method to describe the Ext-algebra of deformed algebras in terms of the original algebra's Ext-algebra.

Findings

Explicit construction of minimal projective resolutions for $A_f$

Description of Ext-algebra structure under specific conditions

Connection between deformed and original Ext-algebras

Abstract

Let $f$ be a Hochschild $2$-cocycle and let $A_f$ be an infinitesimal deformation of an associative finite dimensional algebra $A$ over an algebraically closed field $\Bbbk$. We investigate the algebra structure of the Ext-algebra of $A_f$ and, under some conditions on $f$, we describe it in terms of the Ext-algebra of $A$. We achieve this description by getting an explicit construction of minimal projective resolutions in $\mod A_f$.