Koszul property of diagonal subalgebras

Abstract

Let $S=K[x_1,\ldots,x_n]$ be a polynomial ring over a field $K$ and $I$ a homogeneous ideal in $S$ generated by a regular sequence $f_1,f_2, \ldots,f_k$ of homogeneous forms of degree~$d$. We study a generalization of a result of Conca et al.~\cite{CHTV} concerning Koszul property of the diagonal subalgebras associated to $I$. Each such subalgebra has the form $K[(I^e)_{ed+c}]$, where $c,e \in \mathbf{N}$. For $k=3$, we extend \cite[Corollary 6.10]{CHTV} by proving that $K[(I^e)_{ed+c}]$ is Koszul as soon as $c \geq {d}/{2}$ and $e >0$. We also extend \cite[Corollary 6.10]{CHTV} in another direction by replacing the polynomial ring with a Koszul ring.