HilbertKunz functions of 2 x 2 determinantal rings
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 I_2(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]/(I_2(X) + (x_{1,1}^q,..., x_{m,n}^q)) as q varies over all positive integers using Grobner basis. This is a generalized HilbertKunz 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 HilbertKunz functions for cases $m \le 2$.
 Publication:

arXiv eprints
 Pub Date:
 June 2012
 arXiv:
 arXiv:1206.1015
 Bibcode:
 2012arXiv1206.1015M
 Keywords:

 Mathematics  Commutative Algebra;
 Mathematics  Algebraic Geometry;
 13D40;
 13P10;
 13H10;
 13H15
 EPrint:
 17 pages