Diagonal subalgebras of residual intersections

Abstract

Let k \mathsf {k} be a field, let S S be a bigraded k \mathsf {k} -algebra, and let S Δ S_\Delta denote the diagonal subalgebra of S S corresponding to Δ = { ( c s , e s ) | s ∈ Z } \Delta = \{ (cs,es) \; | \; s \in \mathbb {Z} \} . It is known that the S Δ S_\Delta is Koszul for c , e ≫ 0 c,e \gg 0 . In this article, we find bounds for c , e c,e for S Δ S_\Delta to be Koszul when S S is a geometric residual intersection. Furthermore, we also study the Cohen-Macaulay property of these algebras. Finally, as an application, we look at classes of linearly presented perfect ideals of height two in a polynomial ring, show that all their powers have a linear resolution, and study the Koszul and Cohen-Macaulay properties of the diagonal subalgebras of their Rees algebras.