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.
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 ∈ 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. 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 a-Mod, is a localization of V-Mod along this subcategory.
The quotient functor V-Mod → V a-Mod 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.
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. 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 p1/p2 in the quotient is not annihilated by p1/p2 considered as an element of m.
- 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
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