2014/07/22 by Timothy B. P. Clark, Clark, Timothy B. P., Sonja Mapes +1 · 1 citation
Computer Science · Mathematics · #Advanced Combinatorial Mathematics #Combinatorics (math.CO) #Commutative Algebra (math.AC) #Commutative Algebra and Its Applications #FOS: Mathematics #Topological and Geometric Data Analysis #math.AC #math.CO
paper · pdf · doi:10.48550/arxiv.1407.5702
arxiv created 2014/07/22 · openalex publication_date 2014/07/22 · arxiv updated 2014/07/23 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let P be a finite partially ordered set with unique minimal element 0. We study the Betti poset of P, created by deleting elements q∈ P for which the open interval (0, q) is acyclic. Using basic simplicial topology, we demonstrate an isomorphism in homology between open intervals of the form (0,p)⊂ P and corresponding open intervals in the Betti poset. Our motivating application is that the Betti poset of a monomial ideal's lcm-lattice encodes both its ℤd-graded Betti numbers and the structure of its minimal free resolution. In the case of rigid monomial ideals, we use the data of the Betti poset to explicitly construct the minimal free resolution. Subsequently, we introduce the notion of rigid deformation, a generalization of Bayer, Peeva, and Sturmfels' generic deformation.