mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...

, almost modules and almost rings are certain objects interpolating between rings and their fields of fractions. They were introduced by in his study of ''p''-adic Hodge theory.

Almost modules

Let ''V'' be a local integral domain with the maximal ideal ''m'', and ''K'' a fraction field of ''V''. The category of ''K''- modules, ''K''-Mod, may be obtained as a quotient of ''V''-Mod by the Serre subcategory of torsion modules, i.e. those ''N'' such that any element ''n'' in ''N'' is annihilated by some nonzero element in the maximal ideal. If the category of torsion modules is replaced by a smaller

subcategory In mathematics, specifically category theory, a subcategory of a category ''C'' is a category ''S'' whose objects are objects in ''C'' and whose morphisms are morphisms in ''C'' with the same identities and composition of morphisms. Intuitivel ...

, we obtain an intermediate step between ''V''-modules and ''K''-modules. Faltings proposed to use the subcategory of almost zero modules, i.e. ''N'' ∈ ''V''-Mod such that any element ''n'' in ''N'' is annihilated by ''all'' elements of the maximal ideal. For this idea to work, ''m'' and ''V'' must satisfy certain technical conditions. Let ''V'' be a ring (not necessarily local) and ''m'' ⊆ ''V'' an idempotent ideal, i.e. an ideal such that ''m''² = ''m''. Assume also that ''m'' ⊗ ''m'' is a flat ''V''-module. A module ''N'' over ''V'' is almost zero with respect to such ''m'' if for all ''ε'' ∈ ''m'' and ''n'' ∈ ''N'' we have ''εn'' = 0. Almost zero modules form a Serre subcategory of the category of ''V''-modules. The category of ''almost V-modules'', ''V''^''a''-Mod, is a localization of ''V''-Mod along this subcategory. The quotient functor ''V''-Mod → ''V''^''a''-Mod is denoted by

N \mapsto N^a

. The assumptions on ''m'' guarantee that

(-)^a

is an exact functor which has both the right adjoint functor

M \mapsto M_*

and the left adjoint functor

M \mapsto M_!

. Moreover,

(-)_*

is full and faithful. The category of almost modules is

complete Complete may refer to: Logic * Completeness (logic) * Completeness of a theory, the property of a theory that every formula in the theory's language or its negation is provable Mathematics * The completeness of the real numbers, which implies t ...

and

cocomplete In mathematics, a complete category is a category in which all small limits exist. That is, a category ''C'' is complete if every diagram ''F'' : ''J'' → ''C'' (where ''J'' is small) has a limit in ''C''. Dually, a cocomplete category is one in w ...

Almost rings

The tensor product of ''V''-modules descends to a monoidal structure on ''V''^''a''-Mod. An almost module ''R'' ∈ ''V''^''a''-Mod with a map ''R'' ⊗ ''R'' → ''R'' satisfying natural conditions, similar to a definition of a ring, is called an almost ''V''-algebra or an almost ring if the context is unambiguous. Many standard properties of algebras and morphisms between them carry to the "almost" world.

Example

In the original paper by Faltings, ''V'' was the

integral closure In commutative algebra, an element ''b'' of a commutative ring ''B'' is said to be integral over ''A'', a subring of ''B'', if there are ''n'' ≥ 1 and ''a'j'' in ''A'' such that :b^n + a_ b^ + \cdots + a_1 b + a_0 = 0. That is to say, ''b'' is ...

of a discrete valuation ring in the algebraic closure of its quotient field, and ''m'' its maximal ideal. For example, let ''V'' be

\mathbb_p^/math>, i.e. a ''p''-adic completion of \operatorname\limits_n \mathbb_p^/math>. Take ''m'' to be the maximal ideal of this ring. Then the quotient ''V/m'' is an almost zero module, while ''V/p'' is a torsion, but not almost zero module since the class of ''p''

^1/''p''² in the quotient is not annihilated by ''p''^1/''p''² considered as an element of ''m''.

References

* * Commutative algebra {{abstract-algebra-stub