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 WebMay 19, 2024 · Viewed 504 times. 12. Proposition 1: Let f: A → B and g: B → C be ring maps such that g is faithfully flat. Then the composition g f is flat (resp. faithfully flat) if and only if f is flat (resp. faithfully flat). Proof: Certainly flatness (resp. faithful flatness) of f implies flatness (resp. faithful flatness) of g f.

Section 10.80 (058B): Faithfully flat descent for projectivity of ...

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 … WebSep 14, 2024 · Faithful flatness for rings. Let R be a ring and let M be a right module over R. We say that M is faithfully flat as a right module if the functor M ⊗ R − from left R … evelyn\u0027s classy clipper

Faithfully Flatness Free Modules - Mathematics Stack Exchange

WebJan 17, 2015 · Faithfully flat module and exact sequences. 1. A reverse directional short exact sequence. Hot Network Questions How far does the direct light of the Companion reach? What is the role of the U.S. Marines under contemporary joint warfare doctrine? ... 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. WebLet A [ x] be the ring of polynomials in one indeterminate over a ring A. Prove that A [ x] is a flat A -algebra. Clearly, we notice that A [ x] = ⨁ m = 0 ∞ A ⋅ ( x m). We showed in the previous exercice that for any family M i ( i ∈ I) of A -modules and M their direct sum, then M is flat iff each M i is flat. evelyn\\u0027s classy clipper