Hilbert-Kunz functions of 2 x 2 determinantal rings

2012/06/05 by Miller, Lance Edward, Swanson, Irena
#13D40 #13H10 #13H15 #13P10 #Algebraic Geometry (math.AG) #Commutative Algebra (math.AC) #FOS: Mathematics

paper · doi:10.48550/arxiv.1206.1015

Abstract

Let k be an arbitrary field (of arbitrary characteristic) and let X = [x_i,j] be a generic m x n matrix of variables. Denote by I₂(X) the ideal in k[X] = k[x_i,j: i = 1, ..., m; j = 1, ..., n] generated by the 2 x 2 minors of X. We give a recursive formulation for the lengths of the k[X]-module k[X]/(I₂(X) + (x_1,1^q,..., x_m,n^q)) as q varies over all positive integers using Grobner basis. This is a generalized Hilbert-Kunz function, and our formulation proves that it is a polynomial function in q. We give closed forms for the cases when m is at most 2, %as well as the closed forms for some other special length functions. We apply our method to give closed forms for these Hilbert-Kunz functions for cases m ≤ 2.

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