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Proceedings of the American Mathematical Society

Published by the American Mathematical Society since 1950, Proceedings of the American Mathematical Society is devoted to shorter research articles in all areas of pure and applied mathematics.

ISSN 1088-6826 (online) ISSN 0002-9939 (print)

The 2020 MCQ for Proceedings of the American Mathematical Society is 0.85.

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The syzygies of some thickenings of determinantal varieties
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by Claudiu Raicu and Jerzy Weyman PDF
Proc. Amer. Math. Soc. 145 (2017), 49-59 Request permission


The vector space of $m\times n$ complex matrices ($m\geq n$) admits a natural action of the group $\operatorname {GL}=\operatorname {GL}_m\times \operatorname {GL}_n$ via row and column operations. For positive integers $a,b$, we consider the ideal $I_{a\times b}$ defined as the smallest $\operatorname {GL}$-equivariant ideal containing the $b$-th powers of the $a\times a$ minors of the generic $m\times n$ matrix. We compute the syzygies of the ideals $I_{a\times b}$ for all $a,b$, together with their $\operatorname {GL}$-equivariant structure, generalizing earlier results of Lascoux for the ideals of minors ($b=1$), and of Akin–Buchsbaum–Weyman for the powers of the ideals of maximal minors ($a=n$). Our methods rely on a nice connection between commutative algebra and the representation theory of the superalgebra $\mathfrak {gl}(m|n)$, as well as on our previous calculation of $\operatorname {Ext}$ modules done in the context of describing local cohomology with determinantal support. Our results constitute an important ingredient in the proof by Nagpal–Sam–Snowden of the first non-trivial Noetherianity results for twisted commutative algebras which are not generated in degree one.
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Additional Information
  • Claudiu Raicu
  • Affiliation: Department of Mathematics, 255 Hurley Hall, University of Notre Dame, Notre Dame, Indiana 46556 – and – Institute of Mathematics “Simion Stoilow” of the Romanian Academy
  • MR Author ID: 909516
  • Email:
  • Jerzy Weyman
  • Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269
  • MR Author ID: 182230
  • ORCID: 0000-0003-1923-0060
  • Email:
  • Received by editor(s): January 27, 2016
  • Received by editor(s) in revised form: March 15, 2016
  • Published electronically: July 12, 2016
  • Communicated by: Irena Peeva
  • © Copyright 2016 American Mathematical Society
  • Journal: Proc. Amer. Math. Soc. 145 (2017), 49-59
  • MSC (2010): Primary 13D02, 14M12, 17B10
  • DOI:
  • MathSciNet review: 3565359