Almost ring

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, 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 m² = 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^{{{space|hair}}a}-Mod, is a localization of V-Mod along this subcategory.

The quotient functor V-Mod → V^{{{space|hair}}a}-Mod is denoted by $N\; \backslash mapsto\; N^a$. The assumptions on m guarantee that $(-)^a$ is an exact functor which has both the right adjoint functor $M\; \backslash mapsto\; M\_*$ and the left adjoint functor $M\; \backslash mapsto\; M\_!$. Moreover, $(-)\_*$ is full and faithful. The category of almost modules is complete and cocomplete.

The tensor product of V-modules descends to a monoidal structure on V^{{{space|hair}}a}-Mod. An almost module R ∈ V^{{{space|hair}}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.

In the original paper by Faltings, V was the integral closure of a discrete valuation ring in the algebraic closure of its quotient field, and m its maximal ideal. For example, let V be $\backslash mathbb\{Z\}\_p[p^\{1/p^\{\backslash infty\}\}]$, i.e. a p-adic completion of $\backslash operatorname\{colim\}\backslash limits\_n\; \backslash mathbb\{Z\}\_p[p^\{1/p^n\}]$. 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/p2 in the quotient is not annihilated by p^1/p2 considered as an element of m.

• {{citation|mr=0924705|last=Faltings|first= Gerd|authorlink=Gerd Faltings|title=p-adic Hodge theory

|journal=Journal of the American Mathematical Society|volume= 1 |year=1988|issue= 1|pages= 255–299|doi=10.2307/1990970|jstor=1990970}}

• {{citation|mr=2004652|last1=Gabber|first1=Ofer|authorlink=Ofer Gabber|last2= Ramero|first2= Lorenzo|authorlink2=Lorenzo Ramero|title=Almost ring theory|series=Lecture Notes in Mathematics|volume= 1800|publisher= Springer-Verlag|place= Berlin|year= 2003|isbn= 3-540-40594-1|doi=10.1007/b10047 |s2cid=14400790}}

Category:Commutative algebra