DMS Algebra Seminar

Speaker: Doug Leonard

Title: Induced module orderings

Abstract: This is an audience-participation talk in that I am going to present a case against what Macaulay2 and Singular do in producing free resolutions (and all of you seem to do as well), at least in the context of viewing an ideal I of R as an R-module M, and computing its free resolution. I interreduced the example in Singular’s manual, section 2.3.6 Resolution, to get ideal generators \(x^4 + x^3y + x^2yz\), \(x^2y^2 + xy^2z + y^2z2\),   \( x^2z^2 + 2xz^3\), \(xyz^2 − 4xz^3\) that do not form a Grobner basis (as \(y^2x^4\) and \(xz^5\) are missing). The method I use is really an extension of Faugere’s F5 algorithm, usually billed as a Grobner basis algorithm rather than a syzygy algorithm. The point of the talk is that if there is a useful monomial ordering on R, maybe it induces an R-module ordering on each module of the free resolution that is better than the default (which is usually a TOP, meaning term-overposition, ordering).