# 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 ${A \rightarrow B}$.