I describe basics of almost mathematics which is useful in many ways in applications. Often certain maps are “`almost isomorphisms”‘ rather that isomorphisms. Let be a perfectoid field, let be the subset of topologically nilpotent elements. It is also the unique maximal ideal of and can be identified with the set . The idea behind almost mathematics is to do things up to -torsion. I will follow closely Scholze’s [Perfectoid spaces] in exposition, which in turn cites mostly Gabber-Ramero’s book – for precise references consult Section 4 in [Perfectoid spaces].
Definition 38 Let be a -module. An element is almost zero if . The module is almost zero if all of its elements are almost zero or equivalently .
One can prove that the full subcategory of almost zero objects in the category of -modules is thick, i.e. closed under extensions (and triangulated).
Define the category of almost -modules as / (-torsion). We have a localization functor from to , with kernel being exactly the category of almost zero modules. In this way we have defined both objects and morphisms. A result of Gabber-Ramero yields:
Proposition 39 Let be two -modules. Then
In particular has a natural structure of -module for any two -modules . The module has no almost zero elements.
For two -modules we define . We then have
Proposition 40 The category is an abelian tensor category, where we define kernels, cokernels and tensor products in the unique way compatible with their definition in , that is
for any two -modules . For any there is a functorial isomorphism
This means that has all properties of the category of modules over a ring and thus one can define the notion of -algebra. Any -algebra defines a -algebra as the tensor products are compatible (multiplication). Localisation extents to a functor from -modules to -modules.
Proposition 41 There is a right adjoint
to the localization functor given by the functor of almost elements
The adjunction morphism is an isomorphism. If is a -module, then .
If is a -algebra, then has a natural structure as -algebra and . In particular, any -algebra comes via localization from a -algebra. Furthermore the functor induces a functor from -modules to -modules, and one can see that also all -modules come via localization from -modules. The category of -modules is again an abelian tensor category, and all properties about the category of -modules stay true for the category of -modules. Observe that we could equivalently define -algebras as algebras over the category of -modules, or as -algebras with an algebra morphism .