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

DOI:
https://doi.org/10.1090/S0002-9947-99-02298-9

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

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

Article copyright:
© Copyright 1999
American Mathematical Society