The $s$-multiplicity function of $2 \times 2$-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 $X = [ x_{i,j}]$ an $m \times n$-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 polynomial function of $q$ for all $s$.