Diagonal subalgebras of residual intersections

Abstract

Let k be a field, S be a bigraded k-algebra, and S∆ denote the diagonal subalgebra of S corresponding to ∆ = {(cs, es) | s ∈ Z}. It is known that the S∆ is Koszul for c, e ≫ 0. In this article, we find bounds for c, e for S∆ to be Koszul, when 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 property of the diagonal subalgebras of their Rees algebras.