2022/01/27 by Hailong Dao, David Eisenbud, Dao, Hailong +1 · 1 citation
Computer Science · Mathematics · #13C05 #13D02 #13F55 #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Topological and Geometric Data Analysis
paper · pdf · doi:10.48550/arxiv.2201.11263
openalex publication_date 2022/01/27 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study monomial ideals with linear presentation or partially linear resolution. We give combinatorial characterizations of linear presentation for square-free ideals of degree 3, and for primary ideals whose resolutions are linear except for the last step (the "almost linear" case). We also give sharp bounds on Castelnuovo-Mumford regularity and numbers of generators in some cases. It is a basic observation that linearity properties are inherited by the restriction of an ideal to a subset of variables, and we study when the converse holds. We construct fractal examples of almost linear primary ideals with relatively few generators related to the Sierpiński triangle. Our results also lead to classes of highly connected simplicial complexes Δ that can not be extended to the complete dim Δ-skeleton of the simplex on the same variables by shelling.