9 research outputs found
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 of a quiver taking values in monads over a
Grothendieck category . By using adjoint functors between
Eilenberg-Moore categories, we consider two different kinds of modules over
. The first is the category of -modules, which behaves like the category of modules over a ringed space. The
second is the category of cartesian modules, which behave
like quasi-coherent sheaves. We give conditions for and
to be Grothendieck categories. One of our key steps is
finding a modulus like bound for an endofunctor in terms of , where is a generator for
and is a cardinal such that is -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 F<sub>1</sub> -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