Seminari del Dipartimento
Algebra Commutativa Closure operations, continuous valuations on monoids and spectral spaces Dott. Samarpita Ray 23-09-2019 - 14:15 Largo San Leonardo Murialdo,1 - Pal.C - Aula 211
In the last twenty years, several notions of what is called the absolute algebraic geometry or the algebraic geometry over the "field with one element" ($mathbb{F}_1$) has been developed. It is in this context that monoids became topologically and geometrically relevant objects of study. Spectral spaces i.e., topological spaces homeomorphic to the spectrum of rings were introduced by Hochster and are widely studied in the literature. In this talk, we will present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. For this purpose, our main tools are finite type closure operations and continuous valuations on monoids which we introduce in this work. In the process, we make a detailed study of different closure operations on monoids. We prove that the collection of all valuation monoids having the same group completion forms a spectral space. We also prove that the valuation spectrum of any monoid gives a spectral space. Finally, we prove that the collection of continuous valuations on a topological monoid whose topology is determined by any finitely generated ideal also gives a spectral space. |