Representation Stability for Cellular Resolutions

Abstract

This thesis is on cellular resolutions and the invariants of resolutions of monomial ideals. The general area of these topics is combinatorial commutative algebra, and as much of pure mathematics, the studied questions in the thesis are motivated mainly by fascination towards these combinatorial mathematical objects and applying new tools to study them. The questions on resolutions and their invariants have been around for a long time, and over the years they have become a rich topic, with a variety of directions including cellular resolutions. We look at cellular resolutions from a category-theoretic point of view and apply tools from representation stability to study them.In Publication I, we define the category of cellular resolutions and establish the basic properties for it. Among these results are showing that homotopy colimit lifts from topology and that discrete and algebraic Morse maps are morphisms in this category. Having the category of cellular resolutions opens up cellular resolutions for applying tools of representations of categories, and we use these in Publication II to show that that specific families of cellular resolutions have finitely generated syzygies. The main tools used are defining a linear family that satisfies noetherianity properties of representation stability and viewing syzygies as a representation of the category of cellular resolutions. In particular, we show that the powers of maximal monomial ideals of a polynomial ring have finitely generated syzygies. We also touch upon the case of families with cellular resolutions over different rings and show that the finite generation of syzygies applies in this setting in special cases. The last publication covers combinatorial formulas for algebraic invariants of edge ideals of Booth-Lueker graphs.