The s-multiplicity function of 2x2-determinantal rings

Abstract

This article generalizes joint work of the first author and I. Swanson to the $s$-multiplicity recently introduced by the second author. For $k$ a field and $X=[x_{i,j}]$ a $m\times n$-matrix of variables, we utilize Gröbner bases to give a closed form the length $\lambda (k[X]/(I_2(X)+{\mathfrak{m}}^{\lceil sq\rceil }+{\mathfrak{m}}^{[q]}))$ where $s\in \mathbf{Z}[p^{-1}]$, $q$ is a sufficiently large power of $p$, and $\mathfrak{m}$ is the homogeneous maximal ideal of $k[X]$. This shows this length is always eventually a {\it polynomial} function of $q$ for all $s$.

9 pages, Errors fixed