On the diagonal subalgebra of an Ext algebra

TL;DR

This paper investigates the properties of the diagonal subalgebra of an Ext algebra for Koszul algebras, revealing its structure, applications to Hochschild cohomology, and its relation to the Koszul dual.

Contribution

It introduces and analyzes the diagonal subalgebra of Ext algebras for Koszul algebras, connecting it to Hochschild cohomology and the graded center of the Koszul dual.

Findings

The diagonal subalgebra $ riangle_M$ has specific algebraic properties.

$ riangle_R$ is isomorphic to the graded center of the Koszul dual of $R$.

Applications to Hochschild cohomology and module periodicity are established.

Abstract

Let $R$ be a Koszul algebra over a field $k$ and $M$ be a linear $R$-module. We study a graded subalgebra $\Delta_M$ of the Ext-algebra $\operatorname{Ext}_R^*(M,M)$ called the diagonal subalgebra and its properties. Applications to the Hochschild cohomology ring of $R$ and to periodicity of linear modules are given. Viewing $R$ as a linear module over its enveloping algebra, we also show that $\Delta_R$ is isomorphic to the graded center of the Koszul dual of $R$.