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Almost Ring Theory by Ofer Gabber, Lorenzo Ramero

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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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.

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