Hilbert–Kunz functions of $2\times2$ determinantal rings

Abstract

Let $k$ be an arbitrary field (of arbitrary characteristic) and let $X=[x_{i,j}]$ be a generic $m\times n$ matrix of variables. Denote by $I_{2}(X)$ the ideal in $k[X]=k[x_{i,j}:i=1,\ldots,m;j=1,\ldots,n]$ generated by the $2\times2$ minors of $X$. Using Gröbner basis, we give a recursive formulation for the lengths of the $k[X]$-module $k[X]/(I_{2}(X)+(x_{1,1}^{q},\ldots,x_{m,n}^{q}))$ as $q$ varies over all positive integers. This is a generalized Hilbert–Kunz function, and our formulation proves that it is a polynomial function in $q$. We apply our method to give closed forms for these Hilbert–Kunz functions for cases $m\le2$.