Tuyển tập các báo cáo nghiên cứu khoa học ngành toán học tạp chí toán học quốc tế đề tài: Betti numbers of monomial ideals and shifted skew shapes. | Betti numbers of monomial ideals and shifted skew shapes Uwe Nagel Department of Mathematics University of Kentucky 715 Patterson Office Tower Lexington KY 40506-0027 USA uwenagel@ Victor Reinery School of Mathematics University of Minnesota Minneapolis MN 55455 USA reiner@ Submitted Mar 21 2008 Accepted Feb 6 2009 Published Feb 11 2009 Mathematics Subject ClassiEcation 05C65 05C99 13D03 13D25 To Anders Bjorner on his 60th birthday. Abstract We present two new problems on lower bounds for Betti numbers of the minimal free resolution for monomial ideals generated in a Exed degree. The Erst concerns any such ideal and bounds the total Betti numbers while the second concerns ideals that are quadratic and bihomogeneous with respect to two variable sets but gives a more Enely graded lower bound. These problems are solved for certain classes of ideals that generalize in two diEerent directions the edge ideals of threshold graphs and Ferrers graphs. In the process we produce particularly simple cellular linear resolutions for strongly stable and squarefree strongly stable ideals generated in a Exed degree and combinatorial interpretations for the Betti numbers of other classes of ideals all of which are independent of the coefficient Eeld. Partially supported by NSA grant H98230-07-1-0065 and by the Institute for Mathematics its Applications at the University of Minnesota. yPartially supported by NSF grant DMS-0601010. THE ELECTRONIC JOURNAL OF COMBINATORICS 16 2 2009 R3 1 1 Introduction and the main problems The paper concerns minimal free resolutions of an ideal I in a polynomial ring S k x1 . xn which is generated by monomials of a fixed degree. Many of its results were motivated by two new problems Question and Conjecture below which we formulate here. Given a squarefree monomial ideal I generated in degree d it has a uniquely defined minimal generating set of monomials indexed by a collection K of d-subsets of P 1 2 . in the sense that I
