Diagonal subalgebras of residual intersections

Ananthnarayan, H.; Kumar, Neeraj; Mukundan, Vivek

doi:10.1090/proc/14705

Diagonal subalgebras of residual intersections
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by H. Ananthnarayan, Neeraj Kumar and Vivek Mukundan PDF

Proc. Amer. Math. Soc. 148 (2020), 41-52 Request permission

Abstract:

Let $\mathsf {k}$ be a field, let $S$ be a bigraded $\mathsf {k}$-algebra, and let $S_\Delta$ denote the diagonal subalgebra of $S$ corresponding to $\Delta = \{ (cs,es) \; | \; s \in \mathbb {Z} \}$. It is known 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 properties of the diagonal subalgebras of their Rees algebras.

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