Local cohomology with support in generic determinantal ideals

Abstract

For positive integers $m\ge n\ge p$, we compute the ${GL}_m\times {GL}_n$-equivariant description of the local cohomology modules of the polynomial ring $S=Sym\left({ℂ}^m\otimes {ℂ}^n\right)$ with support in the ideal of $p\times p$ minors of the generic $m\times n$ matrix. Our techniques allow us to explicitly compute all the modules ${Ext}_S^{\bullet }\left(S∕I_{\underset{¯}{x}},S\right)$, for $\underset{¯}{x}$ a partition and $I_{\underset{¯}{x}}$ the ideal generated by the irreducible subrepresentation of $S$ indexed by $\underset{¯}{x}$. In particular we determine the regularity of the ideals $I_{\underset{¯}{x}}$, and we deduce that the only ones admitting a linear free resolution are the powers of the ideal of maximal minors of the generic matrix, as well as the products between such powers and the maximal ideal of $S$.