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 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. When R is selfinjective and not necessarily graded, we study connections between periodic modules M, complexity of M and existence of non-nilpotent elements of positive degree in the Ext-algebra of M. Characterizations of periodic algebras are given. (C) 2016 The Author(s). Published by Elsevier B.V.