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12:17 PM
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Q: Faithful flatness and non-commutative algebras

Fernando Peña VázquezWhen dealing with commutative algebras, a usefull criterion for faithfull flatness is the following: Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $f$ is faithfully flat if and only if $f$ is flat and the associated map $f^*:Spec(B)\rightarrow Spec(A)$ is surjective. This cri...

Would maybe (flatness) be a suitable tag here? (I did not want to retag the question mysefl, since I am not familiar with this topic.) — Martin Sleziak 12 secs ago
In mathematics, in particular in the theory of schemes in algebraic geometry, a flat morphism f from a scheme X to a scheme Y is a morphism such that the induced map on every stalk is a flat map of rings, i.e., f P : O Y , f ( P ) → O X...
In algebra, flat modules include free modules, projective modules, and, over a principal ideal domain, torsion free modules. Formally, a module M over a ring R is flat if 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 sequence if and only if the original sequence is exact. Flatness was introduced by Jean-Pierre Serre (1956) in his paper Géometrie Algébrique et Géométrie Analytique. == Definition == A left module M over a ring R is flat if the following condition is satisfied: for every...
1:08 PM
Sorry, I did not now flatness was a tag on its own. I've added the tag. Thanks! — Fernando Peña Vázquez 4 mins ago
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Q: Faithful flatness and non-commutative algebras

Fernando Peña VázquezWhen dealing with commutative algebras, a usefull criterion for faithfull flatness is the following: Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $f$ is faithfully flat if and only if $f$ is flat and the associated map $f^*:Spec(B)\rightarrow Spec(A)$ is surjective. This cri...


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