The fii should be identity maps and hence symmetry becomes by We prove a version of faithfully flat descent in rigid analytic geometry, for almost perfect complexes and without finiteness assumptions on the rings involved. You will be redirected to the full text document in the repository in a few seconds, if not click here.click here. Comment #1088 To prove descent of projectivity for countably generated modules, we introduce a “Mittag-Leffler” condition on modules, prove that a countably generated module is projective if and only if it is flat and Mittag-Leffler (Theorem 10.93.3), and then show that the property of being a Mittag-Leffler module descends (Lemma 10.95.1). ′ This is an good example of using descent methods as in EGA (but in an entirely new way). "However, the proof of descent along this more general type of map is incorrect. 58.16 Faithfully flat descent. Since it took me some time to find this reference, I believe it would be worthy to mention that in [G], there isn't a proof of this result. We think of Y as 'above' X, with the Xi projection 'down' onto X. With this language, descent implies a vector bundle on Y (so, a bundle given on each Xi), and our concern is to 'glue' those bundles Vi, to make a single bundle V on X. In mathematics, the idea of descent extends the intuitive idea of 'gluing' in topology. His writeup largely follows the exposition in Raynaud and Gruson’s paper [RG]. In the next few sections we prove, following Raynaud and Gruson [GruRay], that the projectivity of modules descends along faithfully flat ring maps. In this section we discuss faithfully flat descent for quasi-coherent modules. Namely, "However, the proof of descent along this more general type of map is incorrect, as explained by Gruson in [G], although he does not provide a fix for the case of interest.". 2. Given a well-behaved filtration of a module $M$, dévissage allows us to express $M$ as a direct sum of successive quotients of the filtering submodules (see Section 10.84). A module over a unit ring is called faithfully flat if the tensor product functor is exact and faithful . Hyperdescent and étale K-theory, with … {\displaystyle \alpha } We give a self-contained exposition of the proof of faithfully flat descent for projectivity of modules. {\displaystyle f_{ij}=f_{ji}^{-1}} Your email address will not be published. Finally, given an arbitrary module $M$ whose base change by a faithfully flat ring map is projective, we filter $M$ by submodules whose successive quotients are countably generated projective modules, and then by dévissage conclude $M$ is a direct sum of projectives, hence projective itself (Theorem 10.95.5). on October 18, 2014 at 00:50. 542 Faithfully flat descent Let us generalize the descent picture from fields from MATHEMATIC 203 at Oxford University (so that it is fiberwise an isomorphism). One last thing, in the bibliographic entry of [G] should be "sur un anneau". The dimension of $B$ being zero implies $A$ is zero dimensional (fiber dimension formula for flat morphisms) and thus is a field. faithfully flat. However, this excludes many natural examples: for instance, any split ring homomorphism is an effective descent morphism. Later, we shall consider other criteria for (normal) atness that we have not yet ... Then show that ˚is faithfully at. The ideas were developed in the period 1955–1965 (which was roughly the time at which the requirements of algebraic topology were met but those of algebraic geometry were not). Boolean rings and coherent rings A ring A is boolean if x2 = x for x ∈ A. Then A !B is faithfully flat if and only if Spec B !Spec A is surjective. α This extends results of … is fully faithful. . If Bis a faithfully at A algebra, then 0 !A!f B!d B B is exact, where d(b) = 1 b b 1. the fiber product (here an equalizer) of two copies of the projection p. The bundles on the Xij that we must control are V′ and V", the pullbacks to the fiber of V via the two different projection maps to X. I would just add one more sentence to it. f Remark: A faithfully flat extension preserves descent [17]. , A monoid S is boolean if s + s = s for all s. X Title: Faithfully flat descent for projectivity of modules. Comments: 22 pages: To move closer towards the abstract theory we need to interpret the disjoint union of the. Flat morphisms need not be injective, but they are locally injective. That is, we can take essentially same fij, acting on various fibers. Lecture 9 - Faithfully Flat Descent October 15, 2014 1 Descent of morphisms In this lecture we study the concept of ‘faithfully at’ descent, which is the notion that to obtain an object on a scheme X, it is enough to give an object on a faithfully at cover Y of X, together with ‘gluing’ or ‘descent’ data. You can do this by filling in the name of the current tag in the following input field. From the point of view of abstract category theory the work of comonads of Beck was a summation of those ideas; see Beck's monadicity theorem. Faithfully-flat descent: lt;p|>In |mathematics|, the |flat topology| is a |Grothendieck topology| used in |algebraic geome... World Heritage Encyclopedia, the aggregation of the largest online encyclopedias available, and the most definitive collection ever assembled. We are not allowed to display external PDFs yet. When $\dim B = 0$; $A$ has to be a domain otherwise a zero divisor $x \in A$ would be sent to a zero divisor in $B$ ($\phi$ being faithfully flat is necessarily injective) contradicting the fact that $B$ is a field. In the first lecture [FGA], n°190, the general technique of faithfully flat descent is introduced. Content is available under CC BY-SA 3.0 unless otherwise noted. A faithfully flat module is always flat and faithful, but the converse does not hold in general. More precisely, we will prove quasi-coherent modules satisfy effect descent with respect to fpqc coverings. The idea of the proof is to use dévissage à la Kaplansky [Kaplansky] to reduce to the case of countably generated modules. For example, the composition. Johan on November 04, 2014 at 18:15. F Each sheaf F on X gives rise to a descent data: where f Gabber-Ramero's book gives a systematic discussion of this in section 3.4 of their book, see Gabber-Ramero. In [G], Gruson explains what went wrong, although he does not provide a fix for the case of interest.". In [G], Gruson explains what went wrong, although he does not provide a fix for the case of interest. There, descent of projectivity along faithfully flat ring maps is deduced from descent of projectivity along a more general type of ring map ([Example 3.1.4(1) of Part II, GruRay]). Nuno Cardoso The case of the construction of vector bundles from data on a disjoint union of topological spaces is a straightforward place to start. The urgency (to put it that way) of the problem for the geometers accounts for the title of the 1959 Grothendieck seminar TDTE on theorems of descent and techniques of existence (see FGA) connecting the descent question with the representable functor question in algebraic geometry in general, and the moduli problem in particular. A morphism of schemes is called faithfully flat if it is flat and surjective. but since my English is a little bit rusty, I am pretty sure you can come up with a better sentence than these. Let As a reminder, this is tag 058B. One important application to note is change of fiber: if the fij are all you need to make a bundle, then there are many ways to make an associated bundle. Hence we study properties of ascent. Proof. 3. In practice, these forms are created, or shown to exist, either by assumption or in an ad hoc basis. Alex Perry wrote a paper about faithfully flat descent of projectivity for modules, and submitted it to the stacks project. by We do this in Lemma 10.95.1. ′ In other words, does the connection $\nabla$ give rise to a descent datum for $E$ with respect to $F$? Quite often it happens that a certain construction can be carried out only after faithfully flat base change. 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