Almost ring

In mathematics, almost modules and almost rings are certain objects interpolating between rings and their fields of fractions. They were introduced by 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 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.

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