# Almost ring

In mathematics, **almost modules** and **almost rings** are certain objects interpolating between rings and their fields of fractions. They were introduced by Gerd Faltings (1988) 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 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 ∈ 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 ∈ 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. *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 is denoted by . The assumptions on *m* guarantee that is an exact functor which has both the right adjoint functor and the left adjoint functor . Moreover, is full and faithful. The category of almost modules is complete and cocomplete.

## 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 of a discrete valuation ring in the algebraic closure of its quotient field, and *m* its maximal ideal. For example, let *V* be , i.e. a *p*-adic completion of . 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*.

## References

- Faltings, Gerd (1988), "p-adic Hodge theory",
*Journal of the American Mathematical Society*,**1**(1): 255–299, doi:10.2307/1990970, MR 0924705 - Gabber, Ofer; Ramero, Lorenzo (2003),
*Almost ring theory*, Lecture Notes in Mathematics,**1800**, Berlin: Springer-Verlag, doi:10.1007/b10047, ISBN 3-540-40594-1, MR 2004652