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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 itsproperties. Applications to the Hochschild cohomology ring of $R$ and toperiodicity of linear modules are given. Viewing $R$ as a linear module overits enveloping algebra, we also show that $\Delta_R$ is isomorphic to thegraded center of the Koszul dual of $R$.

On the diagonal subalgebra of an Ext algebra