WebMay 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. WebA faithfully at module is obviously also at. The word ‘faithfully at’ is because the de nition implies that A M: Mod(A) !Mod(A) is a fully faithful functor. Lemma 1.2. The following are …
Section 10.80 (058B): Faithfully flat descent for projectivity of ...
WebMay 1, 2024 · Therefore f ^: A p → B q is flat. To prove faithfully flatness, I use the fact that if f ^: A p → B q is flat, then it is faithfully flat f ^ ∗ ( m) ≠ B q for all maximal ideals m ⊂ A p (exercise 16, chapter 3 from Atiyah Macdonald). Since A p is local, its only maximal ideal is p A p, and f ^ ∗ ( p A p) = p A p ⊗ A p B q = A p ... WebMay 29, 2024 · All extensions of fields are faithfully flat. You can use the criterion that every prime ideal of the small field is the inverse image of one in the large field. Share Cite Follow answered May 29, 2024 at 12:48 Angina Seng 156k 28 99 198 Thank you both for your answers, where can I find these cretierion? – user320244 May 29, 2024 at 13:04 thai font name
abstract algebra - Are field extensions faithfully flat?
WebMar 6, 2024 · A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original sequence is exact. Although the … WebLECTURE 18 1. Flatness and completion Let M be an A-module.We say that M is A-flat, respectively A-faithfully flat if, for all sequences of A-modules E →F →G, the sequence is exact implies, respectively is equivalent to, that the sequence E ⊗A M →F ⊗A M →G ⊗A M is exact. For an A-algebra B, we say that B is a flat A-algebra if it is flat as an A … WebSep 10, 2024 · 1 Answer. Yes free modules are faithfully flat, because tensor products commute with direct sums, and a free R -module is isomorphic to some R ( I) (and not R I as you erroneously wrote). Other than Atiyah-Mcdonald's Commutative Algebra already mentioned, you have Nicolas Bourbaki, Commutative Algebra, Ch. 1 Flat Modules, §3 … thai font photoshop