Illinois Journal of Mathematics

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

Lance Edward Miller and Irena Swanson

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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 $m\le2$.

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Illinois J. Math., Volume 57, Number 1 (2013), 251-277.

First available in Project Euclid: 23 June 2014

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Zentralblatt MATH identifier

Primary: 13D40: Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
Secondary: 13P10: Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases) 13H10: Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) [See also 14M05] 13H15: Multiplicity theory and related topics [See also 14C17]


Miller, Lance Edward; Swanson, Irena. Hilbert–Kunz functions of $2\times2$ determinantal rings. Illinois J. Math. 57 (2013), no. 1, 251--277. doi:10.1215/ijm/1403534495.

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