Faithfully flat module
WebFaithfully flat descent is a technique from algebraic geometry, allowing one to draw conclusions about objects on the target of a faithfully flat morphism. Such morphisms, … 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? ...
Faithfully flat module
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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 … 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 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. WebMar 20, 2024 · For me an R -module M is faithfully flat if it is flat over R and for any R -module N such that M ⊗ N = { 0 }, N = { 0 }. – Mar 20, 2024 at 16:21 Look at kernels and cokernels. Both properties are preserved by tensoring (due to flatness and right exactness) and are zero after tensoring (by assumption). Mar 20, 2024 at 16:28 Add a comment 1 …
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 concept is defined for modules over a non-necessary commutative ring, it is used mainly for commutative algebras. So, this is the only case that is … See more In algebra, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact … See more A module M over a ring R is flat if the following condition is satisfied: for every injective linear map $${\displaystyle \varphi :K\to L}$$ of … See more In this section, R denotes a commutative ring. If $${\displaystyle {\mathfrak {p}}}$$ is a prime ideal of R, the localization at If an R-module M is … See more A flat resolution of a module M is a resolution of the form $${\displaystyle \cdots \to F_{2}\to F_{1}\to F_{0}\to M\to 0,}$$ See more Flatness is related to various other module properties, such as being free, projective, or torsion-free. In particular, every flat module is torsion-free, every projective module is … See more Flatness may also be expressed using the Tor functors, the left derived functors of the tensor product. A left R-module M is flat if and only if $${\displaystyle \operatorname {Tor} _{n}^{R}(X,M)=0}$$ for all $${\displaystyle n\geq 1}$$ and … See more While projective covers for modules do not always exist, it was speculated that for general rings, every module would have a flat cover, that is, every module M would be the epimorphic image of a flat module F such that every map from a flat module onto M factors … See more 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 …
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.
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. … pictures of buffalo snowstormWebLet , 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 ... pictures of buffalo skullsWebJan 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: tophat soccer clubWebA 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 … top hat soccer atlanta gaWebFrom 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 … pictures of buff colored catsWebJan 7, 2016 · Surjective homomorphism from a faithfully flat module to a regular local ring. 2. Question regarding faithfully flat module. 2. A local algebra is flat if and only if it is faithfully flat. 12. 2 out of 3 property for faithfully flat ring maps. 0. change of rings for faithfully flat modules-Bourbaki's proof. pictures of buffalo snowWebNamely, a faithfully flat morphism in algebraic geometry is roughly analogous to a quotient morphism of topogical spaces (i.e. a surjective morphism in which the target is given the quotient topology induced by the topology on the source). top hat software