Faithfully flat module
WebFaithfully flat descent. In this section we discuss faithfully flat descent for quasi-coherent modules. More precisely, we will prove quasi-coherent modules satisfy effective …
Faithfully flat module
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WebLet A!B be a faithfully at morphism of rings, and let M be an A-module. Then M0:= M ABis a B-module. Moreover, we can de ne two B ABmodules, given by M0 ABand B AM0. Note that while the underlying sets of these two modules are clearly the same, the actions of B ABare very di erent. Nontheless, in this case we have an isomorphism ˚ M: B = WebOct 16, 2014 · Finally, given an arbitrary module whose base change by a faithfully flat ring map is projective, we filter by submodules whose successive quotients are countably generated projective modules, and then by dévissage conclude is a direct sum of projectives, hence projective itself (Theorem 10.95.6 ).
WebJan 30, 2015 · Tour Start here for a quick overview of the site Help Center Detailed answers to any questions you might have Meta Discuss the workings and policies of this site WebHowever, if A is a principal ideal domain, or more generally a Dedekind domain, then submodules of flat A -modules are flat, because for such a ring A, flat = torsion-free, and it is clear that submodules of torsion-free modules are torsion-free (over any domain). Share Cite Follow edited Nov 14, 2012 at 1:13 answered Nov 13, 2012 at 23:34
WebMar 6, 2024 · Faithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such … WebMar 24, 2024 · In general, a finite module over an infinite ring cannot be faithful, since in this case the infinitely many elements of the ring have to give rise to only a finite number of …
WebOn the other hand, the injective map f A → A gives after tensoring the map f A ⊗ A B → B sending a ⊗ b to φ ( a) ⋅ b. But this is the zero map since f A ⊂ ker ( φ). In particular, this map is not injective, contradicting the flatness of A → B. Let A be a nonzero and absolutely flat ring, let m ⊆ A be a maximal ideal, and let B ...
Webat as an A-module if and only if for every prime ideal Qof B, N Q is at over A P (where Pis the inverse image of Qin A). Remark 2 The family of at morphisms is closed under composition and base change. Example 3 A ring Ais faithfully at as a module over itself. Any free A-module is faithfully at over A. Any localization of Ais at over A. Any direct pronationskeil + supinationskeilWebmodule is faithfully at over A. Any localization of Ais at over A. Any direct factor of A(as a ring) is at over A. A projective A-module is at over A. If A is an integral domain and Iis a … pronails knokke heistWebFormal properties. The tensor product of two faithfully flat modules is faithfully flat. If M is a faithfully flat module over the faithfully flat A -algebra B, then M is faithfully flat … pronaatio supinaatioWebJan 2, 2013 · Although this is the usual property that defines the faithfully flatness, (one of the) Liu's definition(s) which is helpful here is the following: pronails kit semi permanenteWebMay 21, 2016 · Let I be a finitely generated ideal of A: A / I is flat. I 2 = I. I = ( e) where e 2 = e. I can show that 2 3 and that 1 2, and I remember proving the other way before but cannot recall it now. That is, I would like to show that A / I is flat assuming that it is principal and generated by an idempotent. commutative-algebra. pronail jojomay emaarWebLet , be rings and be a -module. Let be a ring morphism. For a prime ideal let , and the corresponding local morphism makes an -module. I want to show: If for any prime ideal , is a flat -module, then is a flat -module. I want to use the fact " is flat over is flat over for all the primes ". (1) In the above problem, I have rather than , and it ... pronails vitamin maskWebFrom an algebraic point of view the G -space X only has good properties if A is left (or right) faithfully flat as a module over B. In the last few years many interesting examples of … pronamel sensodyne kaina