Some results on Spectral spaces and Spectral sequences

Abstract

This thesis is made up of two independent parts. We begin the first part by showing that the collection of all prime ideals of a monoid or, in other words, the spectrum of a commutative monoid, endowed with the Zariski topology is homeomorphic to the spectrum of a ring, i.e., it is a spectral space. Spectral spaces were introduced by M. Hochster and are widely studied in the literature. On the other hand, the theory of monoids became relevant in the context of what is known as “absolute algebraic geometry”. In this work, we present several naturally occurring classes of spectral spaces using commutative algebra on pointed monoids. We prove that the collection of all ideals as well as the collection of all proper ideals of a monoid are also spectral spaces. The notion of A-sets over a monoid A is the analogue of the notion of modules over a ring. We introduce closure operations on monoids and thereby obtaining natural classes of spectral spaces using finite type closure operations on A-sets. In the process, different notions of closure operations like integral, saturation, Frobenius and tight closures are introduced for monoids inspired by the corresponding closure operations on rings from classical commutative algebra. We also study their persistence and localization properties in detail. Next, we make a study of valuation on monoids and 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. The other part of this thesis is on categorical generalization of certain results in Hopf algebra theory. This relies on the celebrated “ring with several objects” interpretation of a small preadditive category introduced by B. Mitchell. In this framework, “H-category” plays the role of a Hopf module algebra and similarly co-H-category generalizes a Hopf comodule algebra. These Hopf module categories were first considered by Cibils and Solotar. In our work, we study cohomology of certain relevant classes of modules over H-category and co-H-category using Grothendieck spectral sequences.