Koszul property of diagonal subalgebras

TL;DR

This paper investigates the Koszul property of diagonal subalgebras derived from ideals generated by regular sequences in polynomial rings, extending previous results to broader cases and rings.

Contribution

It generalizes existing results on the Koszul property of diagonal subalgebras, including cases where c >= d/2 and replacing polynomial rings with Koszul rings.

Findings

K[(I^e)_{ed+c}] is Koszul for c >= d/2 when k=3

Extended results to Koszul rings instead of polynomial rings

Provided conditions under which diagonal subalgebras maintain the Koszul property

Abstract

Let S=K[x_1,...,x_n] be a polynomial ring over a field K and I a homogeneous ideal in S generated by a regular sequence f_1,f_2,...,f_k of homogeneous forms of degree d. We study a generalization of a result of Conca, Herzog, Trung, and Valla [9] concerning Koszul property of the diagonal subalgebras associated to I. Each such subalgebra has the form K[(I^e)_{ed+c}], where c and e are positive integers. For k=3, we extend [9, Corollary 6.10] by proving that K-algebra K[(I^e)_{ed+c}] is Koszul as soon as c >= d/2. We also extend [9, Corollary 6.10] in another direction by replacing the polynomial ring with a Koszul ring.