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 H₈ ₑ^ (₍) ⁱ (R^ (n) ) over the ring R^ (n) in the case where HIⁱ (R) has finite Bass numbers over R and I is a graded ideal.