I.9: Almost mathematics (I)

I describe basics of almost mathematics which is useful in many ways in applications. Often certain maps are “`almost isomorphisms”‘ rather that isomorphisms. Let {K} be a perfectoid field, let {\mathfrak{m} = K^{\circ \circ} \subset K^{\circ}} be the subset of topologically nilpotent elements. It is also the unique maximal ideal of {K^{\circ}} and can be identified with the set {\{x\in K | |x| <1\}}. The idea behind almost mathematics is to do things up to {\mathfrak{m}}-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 {M} be a {K^{\circ}}-module. An element {x \in M} is almost zero if {\mathfrak{m} x = 0}. The module {M} is almost zero if all of its elements are almost zero or equivalently {\mathfrak{m} M = 0}.

One can prove that the full subcategory of almost zero objects in the category K^{\circ}-\mod of {K^{\circ}}-modules is thick, i.e. closed under extensions (and triangulated).

Define the category of almost {K^{\circ}}-modules as K^{\circ a}-\mod = K^{\circ}-\mod / ({\mathfrak{m}}-torsion). We have a localization functor {M \mapsto M^a} from K^{\circ }-\mod to K^{\circ a}-\mod, 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 {M,N} be two {K^{\circ}}-modules. Then

\displaystyle \mathrm{Hom} _{K^{\circ a}}(M^a,N^a) = \mathrm{Hom} _{K^{\circ }}(\mathfrak{m} \otimes M,N)

In particular {\mathrm{Hom} _{K^{\circ a}}(X,Y)} has a natural structure of {K^{\circ}}-module for any two {K^{\circ a}}-modules {X,Y}. The module {\mathrm{Hom} _{K^{\circ a}}(X,Y)} has no almost zero elements.

For two {K^{\circ a}}-modules {M,N} we define {al\mathrm{Hom}(X,Y) = \mathrm{Hom}(X,Y) ^a}. We then have

Proposition 40 The category K^{\circ a}-\mod is an abelian tensor category, where we define kernels, cokernels and tensor products in the unique way compatible with their definition in K^{\circ}-\mod, that is

\displaystyle M^{a} \otimes N^a = (M \otimes N)^a

for any two {K^{\circ}}-modules {M,N}. For any L,M,N \in K^{\circ a}-\mod there is a functorial isomorphism

\displaystyle \mathrm{Hom}(L, al\mathrm{Hom}(M,N))= \mathrm{Hom}(L\otimes M,N)

This means that K^{\circ a} -\mod has all properties of the category of modules over a ring and thus one can define the notion of {K^{\circ a}}-algebra. Any {K^{\circ}}-algebra {R} defines a {K^{\circ a}}-algebra {R^{a}} as the tensor products are compatible (multiplication). Localisation extents to a functor from {R}-modules to {R^a}-modules.

Proposition 41 There is a right adjoint

\displaystyle K^{\circ a}-\mod \mapsto K^{\circ}-\mod : M\mapsto M_*

to the localization functor {M\mapsto M^a} given by the functor of almost elements

\displaystyle M_* = \mathrm{Hom} _{K^{\circ a}}(K^{\circ a},M)

The adjunction morphism {(M_*)^a \mapsto M} is an isomorphism. If {M} is a {K^{\circ}}-module, then {(M^a)_* = \mathrm{Hom} (\mathfrak{m},M)}.

If {A} is a {K^{\circ a}}-algebra, then {A_*} has a natural structure as {K^{\circ}}-algebra and {A^a _* = A}. In particular, any {K^{\circ a}}-algebra comes via localization from a {K^{\circ}}-algebra. Furthermore the functor {M\mapsto M_*} induces a functor from {A}-modules to {A_*}-modules, and one can see that also all {A}-modules come via localization from {A_*}-modules. The category of {A}-modules is again an abelian tensor category, and all properties about the category of {K^{\circ a}}-modules stay true for the category of {A}-modules. Observe that we could equivalently define {A}-algebras as algebras over the category of {A}-modules, or as {K^{\circ a}}-algebras {B} with an algebra morphism {A \rightarrow B}.

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