The dual graph of a ring

Abstract: The dual graph (a.k.a. Hochster-Huneke graph) G(R) of a Noetherian ring R of dimension d is the finite simple graph whose vertices correspond to the minimal primes of R and such that {P,Q} is an edge iff R/(P+Q) has dimension d-1. After showing some basic properties, we will discuss three fundamental results of Grothendieck, Hartshorne, and Hochster-Huneke, concerning the connectedness of G(R). We will also see, given a finite simple graph G, how to construct a Noetherian ring R such that G(R)=R.

In the second part of the talk, we will discuss some recent developments related to the following two questions: 1) How many paths are there between two minimal primes of R? 2) What is the shortest path between two minimal primes of R? By taking the minimum in 1) and the maximum in 2) varying the pair of minimal primes we get two important invariants of the graph G(R): its vertex connectivity and its diameter. Most of the things that I will discuss are contained in works written together with Bruno Benedetti, Barbara Bolognese and Michela Di Marca.

commutative algebra

Audience: researchers in the topic

Fellowship of the Ring

Series comments: Description: National Commutative Algebra Seminar

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