By Ofer Gabber, Lorenzo Ramero
This ebook 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 conception. The important suggestion is that of an "almost ring". nearly jewelry are the commutative unitary monoids in a tensor type got as a quotient V-Mod/S of the class V-Mod of modules over a hard and fast ring V; the subcategory S includes all modules annihilated by way of a set excellent m of V, pleasant definite typical conditions.
The reader is thought to be conversant in common express notions, a few uncomplicated commutative algebra and a few complicated homological algebra (derived different types, simplicial methods). except those basic necessities, the textual content is as self-contained as attainable. One novel characteristic of the publication - in comparison with Faltings' prior remedy - is the systematic exploitation of the cotangent advanced, specifically for the research of deformations of virtually algebras.
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4. 3) are required to be unitary (resp. to be algebras) and I is a unitary B-bimodule (resp. e. 6)); we will call ExunC (resp. ExalC ) the corresponding category. 5. For a morphism φ : C → B of C -monoids, and a C -extension X in ExmonC (B, I), we can pullback X via φ to obtain an exact sequence X ∗ φ with a morphism φ∗ : X ∗ φ → X; one checks easily that there exists a unique structure of C -extension on X ∗ φ such that φ∗ is a morphism of C -extension; then X ∗ φ is an object in ExmonC (C, I).
We leave the details to the reader and we proceed to verify the “if” part. For (i), choose 28 Chapter 2: Homological theory a set I and an epimorphism p : A(I) → M . Let Λ be the directed set of finite subsets of I, ordered by inclusion. For S ∈ Λ, let MS := p(AS ). e. e. there exists S ∈ Λ such that εM ⊂ MS , which proves the contention. g. by taking such a presentation of the A∗ -module M∗ and applying N → N a ). The assumption of (ii) gives that colim HomA (M, Mλ ) → HomA (M, M ) is an almost isomorphism, hence, for Λ every ε ∈ m there is λ ∈ Λ and φε : M → Mλ such that pλ ◦ φε = ε · 1M , where pλ : Mλ → M is the natural morphism to the colimit.
I∗ ) in (i) and (ii) above. When B is exact, also I! (or any A!! -module representing I) will do. 2] it is shown how to associate to any ring homomorphism R → S a natural simplicial complex of S-modules denoted LS/R and called the cotangent complex of S over R. 20. Let A → B be a morphism of almost V -algebras. The almost cotangent complex of B over A is the simplicial B!! -module LB/A := B!! ⊗(V a ×B)!! L(V a ×B)!! /(V a ×A)!! 21. B!! -Mod) of simplicial B!! -modules. Indeed, the hyperext functors computed in this category relate the cotangent complex to a number of important invariants.
Almost Ring Theory by Ofer Gabber, Lorenzo Ramero