Hilbert-Kunz 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 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 $m \le 2$.
- Publication:
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arXiv e-prints
- Pub Date:
- June 2012
- DOI:
- 10.48550/arXiv.1206.1015
- arXiv:
- arXiv:1206.1015
- Bibcode:
- 2012arXiv1206.1015M
- Keywords:
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- Mathematics - Commutative Algebra;
- Mathematics - Algebraic Geometry;
- 13D40;
- 13P10;
- 13H10;
- 13H15
- E-Print:
- 17 pages