Explicit HilbertKunz 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 $m n$ 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 HilbertKunz 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.
 Publication:

arXiv eprints
 Pub Date:
 April 2013
 DOI:
 10.48550/arXiv.1304.7274
 arXiv:
 arXiv:1304.7274
 Bibcode:
 2013arXiv1304.7274R
 Keywords:

 Mathematics  Commutative Algebra;
 13D40
 EPrint:
 Pacific J. Math. 275 (2015) 433442