Document Type
Lecture
Publication Date
3-2-2017
Abstract
Let R be a Noetherian commutative ring of positive dimension. The Hochster-Huneke graph of R (sometimes called the dual graph of Spec R and denoted by G(R)) is defined as follows: the vertices are the minimal prime ideals of R, and the edges are the pairs of prime ideals (P1,P2) with height (P1+P2) = 1. If R satisfies Serres property (S2), then G(R) is connected. In this talk, we provide lower and upper bounds for the maximum diameter of Hochster-Huneke graphs of Stanley-Reisner rings satisfying (S2). These bounds depend on the number of variables and the dimension. Hochster-Huneke graphs of (S2) Stanley-Reisner rings are a natural abstraction of the 1-skeletons of polyhedra. We discuss how our bounds imply new Hirsch-type bounds on 1-skeletons of polyhedra.
Relational Format
presentation
Recommended Citation
Homes, Brent, "On the diameter of dual graphs of Stanley-Reisner rings with Serre (S2) property and Hirsch type bounds on abstractions of polytopes" (2017). Algebra/Number Theory Seminar. 33.
https://egrove.olemiss.edu/math_algebra_number_theory/33
