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SUMMARY:Matteo Varbaro (University of Genoa\, Italy)
DTSTART:20200611T203000Z
DTEND:20200611T220000Z
DTSTAMP:20260423T035913Z
UID:FOTR/9
DESCRIPTION:Title: <a href="https://researchseminars.org/talk/FOTR/9/">The
  dual graph of a ring</a>\nby Matteo Varbaro (University of Genoa\, Italy)
  as part of Fellowship of the Ring\n\n\nAbstract\nThe dual graph (a.k.a. H
 ochster-Huneke graph) G(R) of a Noetherian ring R of dimension d is the fi
 nite 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.\nAfter showing
  some basic properties\, we will discuss three fundamental results of Grot
 hendieck\, Hartshorne\, and Hochster-Huneke\, concerning the connectedness
  of G(R). We will also see\, given a finite simple graph G\, how to constr
 uct a Noetherian ring R such that G(R)=R.\n\nIn the second part of the tal
 k\, we will discuss some recent developments related to the following two 
 questions:\n1) How many paths are there between two minimal primes of R?\n
 2) What is the shortest path between two minimal primes of R?\nBy taking t
 he 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 Miche
 la Di Marca.\n
LOCATION:https://researchseminars.org/talk/FOTR/9/
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