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Winter 2021 Regularity and cohomology of Pfaffian thickenings
Michael Perlman
J. Commut. Algebra 13(4): 523-548 (Winter 2021). DOI: 10.1216/jca.2021.13.523


Let S be the coordinate ring of the space of n×n complex skew-symmetric matrices. This ring has an action of the group GLn() induced by the action on the space of matrices. For every invariant ideal IS, we provide an explicit description of the modules ExtS(SI,S) in terms of irreducible representations. This allows us to give formulas for the regularity of basic invariant ideals and (symbolic) powers of ideals of Pfaffians, as well as to characterize when these ideals have a linear free resolution. In addition, given an inclusion of invariant ideals IJ, we compute the (co)kernel of the induced map ExtSj(SI,S)ExtSj(SJ,S) for all j0. As a consequence, we show that if an invariant ideal I is unmixed, then the induced maps ExtSj(SI,S)HIj(S) are injective, answering a question of Eisenbud, Mustaţă and Stillman in the case of Pfaffian thickenings. Finally, using our Ext computations and local duality, we verify an instance of Kodaira vanishing in the sense described in the recent work of Bhatt, Blickle, Lyubeznik, Singh and Zhang.


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Michael Perlman. "Regularity and cohomology of Pfaffian thickenings." J. Commut. Algebra 13 (4) 523 - 548, Winter 2021.


Received: 3 November 2017; Revised: 17 December 2018; Accepted: 15 May 2019; Published: Winter 2021
First available in Project Euclid: 18 January 2022

Digital Object Identifier: 10.1216/jca.2021.13.523

Primary: 00A05

Keywords: Pfaffian varieties , regularity , thickenings

Rights: Copyright © 2021 Rocky Mountain Mathematics Consortium


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Vol.13 • No. 4 • Winter 2021
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