Diagonal Subalgebras of Residual Intersections

Abstract

Let $\mathsf{k}$ be a field, $S$ be a bigraded $\mathsf{k}$-algebra, and $S_{\Delta }$ denote the diagonal subalgebra of $S$ corresponding to $\Delta =\{(cs,es)|s\in \mathbb{Z}\}$. It is know that the $S_{\Delta }$ is Koszul for $c,e\gg 0$. In this article, we find bounds for $c,e$ for $S_{\Delta }$ 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.

Typos and errors have been fixed in the current version. Scheduled to appear in Proc. Amer. Math. Soc