  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 $R$.
