Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers

Mathematics – Commutative Algebra

Scientific paper

Rate now

  [ 0.00 ] – not rated yet Voters 0   Comments 0

Details

24 pages; revised final version to appear in Journal of Algebraic Combinatorics; Lemma 6.5 and Theorem 6.6 corrected and combi

Scientific paper

We use the correspondence between hypergraphs and their associated edge ideals to study the minimal graded free resolution of squarefree monomial ideals. The theme of this paper is to understand how the combinatorial structure of a hypergraph H appears within the resolution of its edge ideal I(H). We discuss when recursive formulas to compute the graded Betti numbers of I(H) in terms of its sub-hypergraphs can be obtained; these results generalize our previous work on the edge ideals of simple graphs. We introduce a class of hypergraphs, which we call properly-connected, that naturally generalizes simple graphs from the point of view that distances between intersecting edges are ``well behaved''. For such a hypergraph H (and thus, for any simple graph), we give a lower bound for the regularity of I(H) via combinatorial information describing H, and an upper bound for the regularity when H = G is a simple graph. We also introduce triangulated hypergraphs, a properly-connected hypergraph which is a generalization of chordal graphs. When H is a triangulated hypergraph, we explicitly compute the regularity of I(H) and show that the graded Betti numbers of I(H) are independent of the ground field. As a consequence, many known results about the graded Betti numbers of forests can now be extended to chordal graphs.

No associations

LandOfFree

Say what you really think

Search LandOfFree.com for scientists and scientific papers. Rate them and share your experience with other people.

Rating

Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers 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 Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and Monomial ideals, edge ideals of hypergraphs, and their graded Betti numbers will most certainly appreciate the feedback.

Rate now

     

Profile ID: LFWR-SCP-O-622036

  Search
All data on this website is collected from public sources. Our data reflects the most accurate information available at the time of publication.