Diagonal Subalgebras of Residual Intersections

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