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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Resolutions of monomial ideals and
cohomology over exterior algebras


Authors: Annetta Aramova, Luchezar L. Avramov and Jürgen Herzog
Journal: Trans. Amer. Math. Soc. 352 (2000), 579-594
MSC (1991): Primary 13D02, 13D40, 16E10, 52B20
Published electronically: July 1, 1999
MathSciNet review: 1603874
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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
Email: mat300@uni-essen.de

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02298-9
PII: S 0002-9947(99)02298-9
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
Article copyright: © Copyright 1999 American Mathematical Society