By Ofer Gabber, Lorenzo Ramero
This ebook develops thorough and whole foundations for the tactic of just about etale extensions, that's on the foundation of Faltings' method of p-adic Hodge conception. The relevant suggestion is that of an "almost ring". virtually 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 includes all modules annihilated through a hard and fast excellent m of V, pleasurable yes average conditions.
The reader is thought to be acquainted with common express notions, a few easy commutative algebra and a few complex homological algebra (derived different types, simplicial methods). except those common must haves, the textual content is as self-contained as attainable. One novel characteristic of the e-book - in comparison with Faltings' previous remedy - is the systematic exploitation of the cotangent complicated, specially for the examine of deformations of just about algebras.
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Extra resources for Almost Ring Theory
Now, let δ ∈ AnnV (Coker φ) and ε1 , ε2 , ε3 , ε4 ∈ m. By assumpq t tion there is a complex Ar → As → M with ε1 · Coker q = 0, ε2 · Ker q ⊂ Im t. 11 are fulfilled. So, with I := ε2 · V we see that ε1 · ε2 · ε3 · ε4 · δ · Ker φ lies in a finitely generated submodule of Ker φ. But m1 is contained in an ideal generated by finitely many such products ε1 · ε2 · ε3 · ε4 · δ. ♦ Now, it is clear that (c) implies (a) and (b). 12. We show that (a) implies (c). For a given finitely generated subideal m0 ⊂ m, pick another finitely generated m1 ⊂ m such that m0 ⊂ m31 ; find morphisms f : M → M and g : M → M whose kernels and cokernels are annihilated by m1 , and such that M is finitely presented.
One can lift m0 L to a finitely generated submodule of M containing m30 M . Finally, suppose that M a is almost finitely presented. We already know that there is n ∈ N and φ : Rn → M with m0 · Coker φ = 0. 12, Ker φa has a finitely generated submodule L containing m20 · Ker φa . One can lift m0 L to a finitely generated submodule of Ker φ containing m40 · Ker φ. 14. Suppose that m = λ∈Λ mλ , where (mλ | λ ∈ Λ) is a filtered family of subideals such that m2λ = mλ for every λ ∈ Λ. Let R be a V -algebra, M an Rmodule, and denote by Rλa (resp.
Cit. 6. Let M be an almost finitely generated A-module and B a flat Aalgebra. Then AnnB (B ⊗A M ) = B ⊗A AnnA (M ). Proof. 7(i),(v) we reduce to the case of a finitely generated Amodule M . , axk ) for every a ∈ A∗ . Since B is flat, we have Ker(1B ⊗A φ) B ⊗A Ker φ, whence the claim. 7. Let P be one of the properties : “flat”, “almost projective”, “almost finitely generated”, “almost finitely presented”. If B is a P A-algebra, and M is a P B-module, then M is P as an A-module. Proof. Left to the reader.
Almost Ring Theory by Ofer Gabber, Lorenzo Ramero