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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

On the jacobian module
associated to a graph


Author: Aron Simis
Journal: Proc. Amer. Math. Soc. 126 (1998), 989-997
MSC (1991): Primary 13H10; Secondary 13D40, 13D45, 13H15
MathSciNet review: 1425139
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Abstract: We consider the jacobian module of a set $\bold{f}:=\{f_1,\ldots,f_m\} \in R:=k[X_1,\ldots,X_n]$ of squarefree monomials of degree $2$ corresponding to the edges of a connected bipartite graph $G$. We show that for such a graph $G$ the number of its primitive cycles (i.e., cycles whose chords are not edges of $G$) is the second Betti number in a minimal resolution of the corresponding jacobian module. A byproduct is a graph theoretic criterion for the subalgebra $k[G]:=k[\bold{f}]$ to be a complete intersection.


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Additional Information

Aron Simis
Email: aron@ufba.br

DOI: http://dx.doi.org/10.1090/S0002-9939-98-04180-X
PII: S 0002-9939(98)04180-X
Received by editor(s): June 1, 1996
Received by editor(s) in revised form: September 27, 1996
Additional Notes: The author was partially supported by CNPq, Brazil.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 1998 American Mathematical Society