By Ofer Gabber, Lorenzo Ramero

This publication develops thorough and whole foundations for the tactic of just about etale extensions, that is on the foundation of Faltings' method of p-adic Hodge idea. The valuable suggestion is that of an "almost ring". nearly jewelry are the commutative unitary monoids in a tensor class acquired as a quotient V-Mod/S of the class V-Mod of modules over a set ring V; the subcategory S contains all modules annihilated by way of a set excellent m of V, pleasing definite ordinary conditions.

The reader is thought to be acquainted with basic express notions, a few simple commutative algebra and a few complicated homological algebra (derived different types, simplicial methods). except those normal must haves, the textual content is as self-contained as attainable. One novel function of the ebook - in comparison with Faltings' prior therapy - is the systematic exploitation of the cotangent advanced, specially for the learn of deformations of just about algebras.

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**Extra resources for Almost Ring Theory**

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Let A → B → C be a sequence of morphisms of almost V algebras. C!! -Mod) u v C!! ⊗B!! LB/A → LC/A → LC/B → C!! ⊗B!! σLB/A where the morphisms u and v are obtained by functoriality of L. Proof. It follows directly from loc. cit. 34. Let (Aλ → Bλ )λ∈I be a system of morphisms of V a -algebra indexed by a small filtered category I. colim Bλ!! -Mod) λ∈I colim LBλ /Aλ λ∈I Lcolim Bλ /colim Aλ . λ∈I λ∈I ∼ Proof. 30(i) gives an isomorphism : colim Aλ!! → (colim Aλ )!! (and λ∈I λ∈I likewise for colim Bλ ).

But by hypothesis, on I and on A the U -module structure is unitary, so X = X1 and this is the left unit property. 14. So much for the general nonsense; we now return to almost algebras. As already announced, from here on, we assume throughout that m is a flat V -module. 15) ∼ ExalA1 (B1 , M1 ) × ExalA2 (B2 , M2 ) → ExalA1 ×A2 (B1 × B2 , M ) whenever A1 , A2 are V a -algebras, Bi is a Ai -algebra, Mi is a (unitary) Bi -module, i = 1, 2 and M := M1 ⊕ M2 , viewed as a B1 × B2 -module. 16. If R is a V -algebra, S (resp.

For any V -algebra R the functor Ra -Mod → R-Mod : M → M! commutes with tensor products. If m is a flat V -module, then this functor takes flat Ra -modules to flat R-modules. Proof. Let M be a flat Ra -module and N → N an injective map of R-modules. Denote by K the kernel of the induced map M! ⊗R N → M! ⊗R N ; we have Ka 0. We obtain an exact sequence 0 → m ⊗V K → m ⊗V M! ⊗R N → m ⊗V M! ⊗R N . But one sees easily that m ⊗V K = 0 and m ⊗V M! M! , which shows that M! is a flat R-module. Similarly, let M, N be two Ra -modules.