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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48 .

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Resolutions of monomial ideals and cohomology over exterior algebras
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by Annetta Aramova, Luchezar L. Avramov and Jürgen Herzog PDF
Trans. Amer. Math. Soc. 352 (2000), 579-594 Request permission

Abstract:

This paper studies the homology of finite modules over the exterior algebra $E$ of a vector space $V$. To such a module $M$ we associate an algebraic set $V_E(M)\subseteq V$, consisting of those $v\in V$ that have a non-minimal annihilator in $M$. A cohomological description of its defining ideal leads, among other things, to complementary expressions for its dimension, linked by a ‘depth formula’. Explicit results are obtained for $M=E/J$, when $J$ is generated by products of elements of a basis $e_1,\dots ,e_n$ of $V$. A (infinite) minimal free resolution of $E/J$ is constructed from a (finite) minimal resolution of $S/I$, where $I$ is the squarefree monomial ideal generated by ‘the same’ products of the variables in the polynomial ring $S=K[x_1,\dots ,x_n]$. It is proved that $V_E(E/J)$ is the union of the coordinate subspaces of $V$, spanned by subsets of $\{ e_1,\dots ,e_n \}$ determined by the Betti numbers of $S/I$ over $S$.
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Additional Information
  • Annetta Aramova
  • Affiliation: Institute of Mathematics, Bulgarian Academy of Sciences Sofia 1113, Bulgaria
  • Email: algebra@bgearn.acad.bg
  • Jürgen Herzog
  • Affiliation: FB 6 Mathematik und Informatik, Universität-GHS-Essen Postfach 103764, Essen 45117, Germany
  • MR Author ID: 189999
  • Email: mat300@uni-essen.de
  • Received by editor(s): September 30, 1997
  • Published electronically: July 1, 1999
  • Additional Notes: Work on this paper started while the first and second author visited the third author; the hospitality of the University of Essen is gratefully acknowledged
    The second author was partially supported by a grant from the National Science Foundation
  • © Copyright 1999 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 352 (2000), 579-594
  • MSC (1991): Primary 13D02, 13D40, 16E10, 52B20
  • DOI: https://doi.org/10.1090/S0002-9947-99-02298-9
  • MathSciNet review: 1603874