On the diagonal subalgebra of an Ext algebra

Let $R$ be a Koszul algebra over a field $k$ and $M$ be a linear $R$-module. We study a graded subalgebra ${\mathrm{\Delta }}_M$ of the Ext-algebra ${\mathrm{Ext}}_R^{\ast }(M,M)$ 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 ${\mathrm{\Delta }}_R$ is isomorphic to the graded center of the Koszul dual of $R$.