Explicit Hilbert-Kunz functions of 2 x 2 determinantal rings

Abstract

Let $k[X]=k[x_{i,j}:i=1,...,m;j=1,...,n]$ be the polynomial ring in $mn$ variables $x_{i,j}$ over a field $k$ of arbitrary characteristic. Denote by $I_2(X)$ the ideal generated by the $2\times 2$ minors of the generic $m\times n$ matrix $[x_{i,j}]$. We give a closed formulation for the dimensions of the $k$-vector space $k[X]/(I_2(X)+(x_{1,1}^q,...,x_{m,n}^q))$ as $q$ varies over all positive integers, i.e., we give a closed form for the generalized Hilbert-Kunz function of the determinantal ring $k[X]/I_2[X]$. We also give a closed formulation of dimensions of related quotients of $k[X]/I_2[X]$. In the process we establish a formula for the numbers of some compositions (ordered partitions of integers), and we give a proof of a new binomial identity.

References in corpus (1)

• Hilbert-Kunz functions of 2 x 2 determinantal rings

Cited by in corpus (2)

• Hilbert-Kunz functions of surface rings of type ADE
• The s-multiplicity function of 2x2-determinantal rings