Local cohomology with support in generic determinantal ideals

Raicu, Claudiu; Weyman, Jerzy

doi:10.2140/ant.2014.8.1231

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Local cohomology with support in generic determinantal ideals

Claudiu Raicu and Jerzy Weyman

Vol. 8 (2014), No. 5, 1231–1257

DOI: 10.2140/ant.2014.8.1231

Abstract

For positive integers $m \geq n \geq p$ , we compute the ${GL}_{m} \times {GL}_{n}$ -equivariant description of the local cohomology modules of the polynomial ring $S = Sym (ℂ^{m} \otimes ℂ^{n})$ 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 ${Ext}_{S}^{∙} (S ∕ I_{\underline{x}}, S)$ , for $\underline{x}$ a partition and $I_{\underline{x}}$ the ideal generated by the irreducible subrepresentation of $S$ indexed by $\underline{x}$ . In particular we determine the regularity of the ideals $I_{\underline{x}}$ , 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

Mathematical Subject Classification 2010

Primary: 13D45

Secondary: 14M12

Milestones

Received: 27 September 2013

Revised: 25 February 2014

Accepted: 26 March 2014

Published: 16 September 2014

Authors

Claudiu Raicu
Department of Mathematics University of Notre Dame 255 Hurley Hall Notre Dame, IN 46556 United States Simion Stoilow Institute of Mathematics of the Romanian Academy 21 Calea Grivitei Street 010702 Bucharest Romania

Jerzy Weyman
Department of Mathematics University of Connecticut Storrs, CT 06269 United States

Vol. 8, No. 5, 2014