tailieunhanh - Báo cáo toán học: "Algebraic properties of edge ideals via combinatorial topology"

Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Algebraic properties of edge ideals via combinatorial topology. | Algebraic properties of edge ideals via combinatorial topology Anton Dochtermann TU Berlin MA 6-2 Strafie des 17. Juni 136 10623 Berlin Germany dochterm@ Alexander Engstrom KTH Matematik 100 44 Stockholm Sweden alexe@ Dedicated to Anders Bjorner on the occasion of his 60th birthday. Submitted Oct 22 2008 Accepted Feb 3 2009 Published Feb 11 2009 Mathematics Subject Classifications 13F55 05C99 13D02 Abstract We apply some basic notions from combinatorial topology to establish various algebraic properties of edge ideals of graphs and more general Stanley-Reisner rings. In this way we provide new short proofs of some theorems from the literature regarding linearity Betti numbers and sequentially Cohen-Macaulay properties of edge ideals associated to chordal complements of chordal and Ferrers graphs as well as trees and forests. Our approach unifies and in many cases strengthens these results and also provides combinatorial enumerative interpretations of certain algebraic properties. We apply our setup to obtain new results regarding algebraic properties of edge ideals in the context of local changes to a graph adding whiskers and ears as well as bounded vertex degree. These methods also lead to recursive relations among certain generating functions of Betti numbers which we use to establish new formulas for the projective dimension of edge ideals. We use only well-known tools from combinatorial topology along the lines of independence complexes of graphs not necessarily pure vertex decomposability shellability etc. 1 Introduction Suppose G is a finite simple graph with vertex set n 1 . ng and edge set E G and let S k x1 . xn denote the polynomial ring on n variables over some field k. We define the edge ideal IG c S to be the ideal generated by all monomials XiXj whenever ij 2 E G . The natural problem is to then obtain information regarding the algebraic THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R2 1 invariants of the S-module RG S