An ordinary ring may be expressed as a preadditive category with a single object. Accordingly, as introduced by B. Mitchell, an arbitrary small preadditive category may be understood as a “ring with several objects”. In this respect, for a Hopf algebra H, an H-category will denote an “H-module algebra with several objects” and a co-H-category will denote an “H-comodule algebra with several objects”. Modules over such Hopf categories were first considered by Cibils and Solotar. We study the cohomology in such module categories. In particular, we consider H-equivariant modules over a Hopf module category C as modules over the smash extension C#H. We construct Grothendieck spectral sequences for the cohomologies as well as the H-locally finite cohomologies of these objects. We also introduce relative (D,H)-Hopf modules over a Hopf comodule category D. These generalize relative (A,H)-Hopf modules over an H-comodule algebra A. We construct Grothendieck spectral sequences for their cohomologies by using their rational Hom objects and higher derived functors of coinvariants. We will develop these cohomology theories in a manner similar to the “H-finite cohomology” obtained by Guedenon and the cohomology of relative Hopf modules studied by Caenepeel and Guedenon respectively. This is one of the two thesis problems which we plan to discuss in detail.

If time permits, we will also give a brief presentation of the other thesis project. In the last twenty years, several notions of what is called 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. In our work, we abstract out the topological characteristics of the prime 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, introduced by Hochster, are widely studied in the literature. We use ideals and modules over monoids to present many such spectral spaces. We introduce closure operations on monoids and obtain natural classes of spectral spaces using finite type closure operations. In the process, various closure operations like integral, saturation, Frobenius and tight closures are introduced for monoids. We 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.

- All seminars.
- Seminars for 2019

Last updated: 11 Dec 2023