Mathematics – Commutative Algebra
Scientific paper
2007-08-22
J. Pure Appl. Algebra 212(7) (2008), 1770-1786
Mathematics
Commutative Algebra
22 pages
Scientific paper
Let $\mathcal{C}$ be a clutter with a perfect matching $e_1,...,e_g$ of K\"onig type and let $\Delta_\mathcal{C}$ be the Stanley-Reisner complex of the edge ideal of $\mathcal{C}$. If all c-minors of $\mathcal{C}$ have a free vertex and $\mathcal{C}$ is unmixed, we show that $\Delta_\mathcal{C}$ is pure shellable. We are able to describe, in combinatorial and algebraic terms, when $\Delta_\mathcal{C}$ is pure. If $\mathcal{C}$ has no cycles of length 3 or 4, then it is shown that $\Delta_\mathcal{C}$ is pure if and only if $\Delta_\mathcal{C}$ is pure shellable (in this case $e_i$ has a free vertex for all $i$), and that $\Delta_\mathcal{C}$ is pure if and only if for any two edges $f_1,f_2$ of $\mathcal{C}$ and for any $e_i$, one has that $f_1\cap e_i\subset f_2\cap e_i$ or $f_2\cap e_i\subset f_1\cap e_i$. It is also shown that this ordering condition implies that $\Delta_\mathcal{C}$ is pure shellable, without any assumption on the cycles of $\mathcal{C}$. Then we prove that complete admissible uniform clutters and their Alexander duals are unmixed. In addition, the edge ideals of complete admissible uniform clutters are facet ideals of shellable simplicial complexes, they are Cohen-Macaulay, and they have linear resolutions. Furthermore if $ \mathcal{C}$ is admissible and complete, then $\mathcal{C}$ is unmixed. We characterize certain conditions that occur in a Cohen-Macaulay criterion for bipartite graphs of Herzog and Hibi, and extend some results of Faridi--on the structure of unmixed simplicial trees--to clutters with the K\"onig property without 3-cycles or 4-cycles.
Morey Susan
Reyes Enrique
Villarreal Rafael H.
No associations
LandOfFree
Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of König type does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of König type, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Cohen-Macaulay, Shellable and unmixed clutters with a perfect matching of König type will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-520772