#### Vol. 8, No. 5, 2014

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Local cohomology with support in generic determinantal ideals

### Claudiu Raicu and Jerzy Weyman

Vol. 8 (2014), No. 5, 1231–1257
##### Abstract

For positive integers $m\ge n\ge p$, we compute the ${GL}_{m}×{GL}_{n}$-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 $p×p$ minors of the generic $m×n$ 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 $S$.

 To the memory of Andrei Zelevinsky
##### Keywords
local cohomology, determinantal ideals, regularity
Primary: 13D45
Secondary: 14M12