Almost ring

In mathematics, almost rings are the analogues of commutative rings in the "almost mathematics" introduced by Faltings (1988) in his study of p-adic Hodge theory. Roughly speaking, the word "almost" means "ignore m-torsion for a certain idempotent ideal m".

Definition

Suppose that V is a ring and m an ideal such that m² = m and m ⊗ m is a flat V-module. An almost V module is an element of the category of V-modules modulo the full subcategory of modules killed by m; these form a tensor abelian category. An almost ring (or more precisely an almost V-algebra) is an almost V-module with a bilinear multiplication map satisfying some conditions similar to the axioms for a ring.

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.

References

• Faltings, Gerd (1988), "p-adic Hodge theory", J. Amer. Math. Soc., 1 (1): 255–299, doi:10.2307/1990970, MR 0924705
• Gabber, Ofer; Ramero, Lorenzo (2003), Almost ring theory, Lecture Notes in Mathematics, vol. 1800, Berlin: Springer-Verlag, doi:10.1007/b10047, ISBN 3-540-40594-1, MR 2004652