Noncommutative tensor triangulated categories and coherent frames

We develop a point-free approach for constructing the Nakano-Vashaw-Yakimov-Balmer spectrum of a noncommutative tensor triangulated category under some mild assumptions. In particular, we provide a conceptual way of classifying radical thick tensor ideals of a noncommutative tensor triangulated category using frame theoretic methods, recovering the universal support data in the process. We further show that there is a homeomorphism between the spectral space of radical thick tensor ideals of a noncommutative tensor triangulated category and the collection of open subsets of its spectrum in the Hochster dual topology

Categories of modules, comodules and contramodules over representations

We study and relate categories of modules, comodules and contramodules over a representation of a small category taking values in (co)algebras, in a manner similar to modules over a ringed space. As a result, we obtain a categorical framework which incorporates all the adjoint functors between these categories in a natural manner. Various classical properties of coalgebras and their morphisms arise naturally within this theory. We also consider cartesian objects in each of these categories, which may be viewed as counterparts of quasi-coherent sheaves over a scheme. We study their categorical properties using cardinality arguments. Our focus is on generators for these categories and on Grothendieck categories, because the latter may be treated as replacements for noncommutative spaces.Comment: Several update

A Gabber type result for representations in Eilenberg-Moore categories

We consider a representation $\mathscr U:\mathbb Q\longrightarrow Mnd(\mathcal C)$ of a quiver $\mathbb Q$ taking values in monads over a Grothendieck category $\mathcal C$. By using adjoint functors between Eilenberg-Moore categories, we consider two different kinds of modules over $\mathscr U$. The first is the category $Mod-\mathscr U$ of $\mathscr U$-modules, which behaves like the category of modules over a ringed space. The second is the category $Cart-\mathscr U$ of cartesian modules, which behave like quasi-coherent sheaves. We give conditions for $Mod-\mathscr U$ and $Cart-\mathscr U$ to be Grothendieck categories. One of our key steps is finding a modulus like bound for an endofunctor $U:\mathcal C\longrightarrow \mathcal C$ in terms of $\kappa(G)$, where $G$ is a generator for $\mathcal C$ and $\kappa(G)$ is a cardinal such that $G$ is $\kappa(G)$-presentable. We conclude with an extension of the classical quasi-coherator construction to modules over a monad quiver with values in Eilenberg-Moore categories

The topological shadow of F₁ -geometry:congruence spaces

In this paper we introduce congruence spaces, which are topological spaces that are canonically attached to monoid schemes and that reflect closed topological properties. This leads to satisfactory topological characterizations of closed morphisms and closed immersions as well as separated and proper morphisms. We study congruence spaces thoroughly and extend standard results from usual scheme theory to monoid schemes: a closed immersion is the same as an affine morphism for which the pullback of sections is surjective; a morphism is separated if and only if the image of the diagonal is a closed subset of the congruence space; a valuative criterion for separated and proper morphisms.</p