Title: A survey on spectral spaces
Speaker: Carmelo Finocchiaro (University of Padova)
Abstract: Since the 60s of the last century it has been of interest to find conditions that a topological space must satisfy in order to be homeomorphic to the prime spectrum of some ring. This was the subject of the PhD thesis of M. Hochster, where it was proved that the topological spaces that are homeomorphic to the prime spectrum of a ring - also called spectral spaces - are precisely the spaces satisfying the following properties.
- X is quasi-compact;
- X admits a basis of open and quasi-compact subspaces that is closed under finite intersections;
- Every irreducible closed subspace of X has a unique generic point
While for some classes of spectral spaces, like abstract Riemann surfaces, a class of rings realizing Hochster's theorem was explicitly found, for several other spaces naturally arising in Commutative Ring Theory it is not easy to decide if they are spectral because, in particular, it can be difficult to verify condition (3) of Hochster's characterization. A goal of this survey talk is to present some new perspective about the study of spectral spaces and, in particular a criterion, based on ultrafilters, to decide if a topological space is spectral. Some recent new examples will be discussed.
