Title: Some Results on Spectral Spaces and spectral sequences
Speaker: Samarpita Ray (IISc Mathematics)
Date: 13 December 2018
Time: 4 pm
Venue: LH-3, Mathematics Department
In the last twenty years, several notions of what is called the algebraic geometry over
the “field with one element” has been developed. One of the simplest approaches to this
is via the theory of monoid schemes. The concept of a monoid scheme itself goes back to
Kato and was further developed by Deitmar and by Connes, Consani and Marcolli. The idea
is to replace prime spectra of commutative rings, which are the building blocks of
ordinary schemes, by prime spectra of commutative pointed monoids. In our work, we
focus mostly on abstracting out the topological characteristics of the prime spectrum
of a commutative pointed monoid. This helps to obtain several classes of topological
spaces which are homeomorphic to the the prime spectrum of a monoid. Such spaces are
widely studied and are called spectral spaces. They were introduced by M. Hochster.
We 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 like integral,
saturation, Frobenius and tight closures on monoids. We prove that the collection of all
continuous valuations on a topological monoid with topology determined by any finitely
generated ideal is a spectral space.