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

For positive integers m n p, we compute the GLm ×GLn-equivariant description of the local cohomology modules of the polynomial ring S = Sym(m n) 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 ExtS(SIx ¯ ,S), for x¯ a partition and Ix ¯ the ideal generated by the irreducible subrepresentation of S indexed by x ¯ . In particular we determine the regularity of the ideals Ix ¯ , 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

local cohomology, determinantal ideals, regularity
Mathematical Subject Classification 2010
Primary: 13D45
Secondary: 14M12
Received: 27 September 2013
Revised: 25 February 2014
Accepted: 26 March 2014
Published: 16 September 2014
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
Jerzy Weyman
Department of Mathematics
University of Connecticut
Storrs, CT 06269
United States