Almost Ring Theory

Abstract

ALMOST RING THEORY 15 2.2.18. Note that the essential image of M → M * is closed under limits. Next recall that the forgetful functor A * -Alg → Set (resp. A * -Mod → Set) has a left adjoint A * [−] : Set → A * -Alg (resp. A (−) : Set → A * -Mod) that assigns to a set S the free A * -algebra A * [S] (resp. the free A * -module A (S) * ) generated by S. If S is any set, it is natural to write A[S] (resp. A (S) ) for the A-algebra (A * [S]) a (resp. for the A-module (A (S) * ) a . This yields a left adjoint, called the free A-algebra functor Set → A-Alg (resp. the free A-module functor Set → A-Mod) to the "forgetful" functor A-Alg → Set (resp. A-Mod → Set) B → B * .2.2.19. Now let R be any V -algebra; we want to construct a left adjoint to the localisation functor R-Mod → R a -Mod. For a given R a -module M, letWe have the natural map (unit of adjunction) R → R a * , so that we can view M ! as an R-module. Proposition 2.2.21. Let R be a V -algebra.(i) The functor R a -Mod → R-Mod defined by (2.2.20) is left adjoint to localisation.(ii) The unit of the adjunction M → M a ! is a natural isomorphism from the identity functor 1 R a -Mod to the composition of the two functors.Proof. (i) follows easily from (2.2.4) and (ii) follows easily from (i).Corollary 2.2.22. Suppose that m is a flat V -module. Then we have :(i) the functor M → M ! is exact;(ii) the localisation functor R-Mod → R a -Mod sends injectives to injectives.Proof. By proposition 2.2.21 it follows that M → M ! is right exact. To show that it is also left exact when m is a flat V -module, it suffices to remark that M → M * is left exact. Now, by (i), the functor M → M a is right adjoint to an exact functor, so (ii) is clear.

Le lemme d’Abhyankar perfectoide

“…Nous ferons appelà quelques résultats de cette théorie initiée par G. Faltings et développée dans l'ouvrage fondamental de O. Gabber et L. Ramero [13].…”

“…Comme dans [13], nous supposerons quem := m ⊗ V m est plat sur V. Cette hypothèse est stable par changement de base [13, rem. 2.1.4], et plus faible que la platitude de m (qui entraîne m ∼ =m).…”

“…[8] [24]). Elle utilise le langage de la "presquealgèbre" introduit par Faltingsà ce propos, dont le principe consiste, dans le cadre (V, m) constitué d'un anneau et d'un idéal idempotent,à "négliger" les V-modules annulés par m [13].…”

Almost étale extensions of Fontaine rings and log-crystalline cohomology in the semi-stable reduction case

“…We refer to [9] for the fundamentals of almost ring theory, in particular the categories of almost modules and almost rings. We will always refer to morphisms in these almost categories with the adjective "almost", e.g.…”

“…Then one uses Faltings' theory of almost étale extensions to show that the intermediate cohomology theory H * (X , A crys,n ) almost satisfies Poincaré duality and Künneth formula, hence by standard arguments is almost isomorphic to crystalline cohomology (here the term 'almost' is used in the sense of almost ring theory ( [7], [9])). Since X is smooth, the group on the left is canonically isomorphic to étale cohomology of XK tensored with A cris and compatibility with Poincaré duality gives a one-sided inverse to the almost defined transformation to crystalline cohomology, up to a power of an element t ∈ A cris .…”