Graded-Injective Modules and Bass Numbers of Veronese Submodules

Let $R$ be a standard graded, finitely generated algebra over a field, and let $M$ be a graded module over $R$ with all Bass numbers finite. Set $(-)^{(n)}$ to be the $n$-th Veronese functor. We compute the Bass numbers of $M^{(n)}$ over the ring $R^{(n)}$ for all prime ideals of $R^{(n)}$ that are not the homogeneous maximal ideal in terms of the Bass numbers of $M$ over $R$. As an application to local cohomology modules, we determine the Bass numbers of $H_{I\cap R^{(n)}}^i(R^{(n)})$ over the ring $R^{(n)}$ in the case where $H_I^i(R)$ has finite Bass numbers over $R$ and $I$ is a graded ideal.