Families of cellular resolutions, their syzygies, and stability

Syzygies of Oriented Matroids

We construct minimal cellular resolutions of squarefree monomial ideals arising from hyperplane arrangements, matroids and oriented matroids. These are Stanley-Reisner ideals of complexes of independent sets, and of triangulations of Lawrence matroid polytopes. Our resolution provides a cellular realization of Stanley's formula for their Betti numbers. For unimodular matroids our resolutions are related to hyperplane arrangements on tori, and we recover the resolutions constructed by Bayer, Popescu and Sturmfels. We resolve the combinatorial problems posed in their paper by computing Mobius in…

Divisors on graphs, orientations, syzygies, and system reliability

We study various ideals arising in the theory of system reliability. We use ideas from the theory of divisors, orientations and matroids on graphs to describe the minimal polyhedral cellular free resolutions of these ideals. In each case we give an explicit combinatorial description of the minimal generating set for each higher syzygy module in terms of the acyclic orientations of the graph, the $q$-reduced divisors and the bounded regions of the graphic hyperplane arrangement. The resolutions of all these ideals are closely related, and their Betti numbers are independent of the characteristi…

Characteristic dependence of syzygies of random monomial ideals

When do syzygies depend on the characteristic of the field? Even for well-studied families of examples, very little is known. For a family of random monomial ideals, namely the Stanley--Reisner ideals of random flag complexes, we prove that the Betti numbers asymptotically almost always depend on the characteristic. Using this result, we also develop a heuristic for characteristic dependence of asymptotic syzygies of algebraic varieties.

The category of cellular resolutions

Cellular resolutions is a well studied topic on the level of single resolutions and certain specific families of cellular resolutions. One question coming out of the work on families is to understand the structure of cellular resolutions more generally. We give a starting point to understanding higher level structures by defining the category of cellular resolutions. In this paper we study the properties of this category. The main results are in lifting homotopy colimits from topology and Morse theory on cellular resolution being compatible with the category.

多拟阵理想及其渐近合冲

Polymatroidal ideals and their asymptotic syzygies

本文研究多拟阵理想幂的同调移位理想的渐近行为，证明了第一同调移位代数的生成性质与强持久性质，提出并部分验证了高阶同调移位代数生成次数及相伴素数递增链的猜想，同时证明了分量多拟阵理想在第一同调移位下的封闭性。

Minimal and cellular free resolutions over polynomial OI-algebras

Minimal free resolutions of graded modules over a noetherian polynomial ring have been attractive objects of interest for more than a hundred years. We introduce and study two natural extensions in the setting of graded modules over a polynomial OI-algebra, namely minimal and width-wise minimal free resolutions. A minimal free resolution of an OI-module can be characterized by the fact that the free module in every fixed homological degree, say $i$, has minimal rank among all free resolutions of the module. We show that any finitely generated graded module over a noetherian polynomial OI-algeb…