Multigraded Betti numbers of simplicial forests

Abstract

We determine (multi)graded Betti numbers of path ideals of lines and star graphs.

Betti numbers of monomial ideals via facet covers

“…In [17, Corollary 3.9] Morey and Villarreal gave a lower bound for regularity of squarefree monomial ideals (see (4) in Section 3.1), improving the previously known bounds by Katzman [11, Lemma 2.2] and Hà and Van Tuyl [9, Theorem 6.5]. Using Corollary 3.4 we can further improve the regularity bound mentioned above.…”

“…Well ordered facet covers generalize the notion of strongly disjoint bouquets introduced by Kimura [12] and our condition generalizes the main result in [12] from graphs to simplicial complexes. Our paper fits within the recent interest in combinatorially bounding or computing homological invariants of squarefree monomial ideals, see for example [3,4,7,9,8,10,11,12,13,15,17,18,21,22].…”

Fibonacci numbers and resolutions of domino ideals

“…As we can find a path ideal containing each domino ideal and because path ideals have this property (see Erey and Faridi [9]), the result follows immediately.…”

Breaking up Simplicial Homology and Subadditivity of Syzygies

“…Considering that it is enough to study the "top degree" Betti numbers (those of degree n, in this case) [EF1,F1], a positive answer to Question 2.1 will establish the subadditivity property for all monomial ideals, since…”