They were introduced by Gerd Faltings (1988) in his study of p-adic Hodge theory.
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.
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
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.
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.