# Questions tagged [flatness]

An $A$-module $M$ is flat when the right-exact _$\otimes_AM$ functor becomes left exact (and therefore exact). This applies to $A$-algebras as the latter are $A$-module, saying that $B$ flat over $A$, or that $A\to B$ is flat. The notion passes to morphisms of schemes: a morphism of schemes $f: X\to Y$ is flat if all the induces stalks local morphisms are flat. This passes also to sheaves, etc...

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### $A \subset B$ be integral and flat extension then it is faithfully flat.

I want to prove the following: $A \subset B$ be integral and flat extension of rings then it is faithfully flat. Clearly enough to show that for every ideal $I$ of $A$, $I^{ec}=I$. Since the ...
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### Flat Modules are Torsion-free

Can anyone help me prove that a flat module is torsion-free? I’ve tried supposing is isn’t and then get the contradiction that it isn’t flat, but I’m not having any good results. Any suggestion is ...
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### What's the canonical definition of isogeny between semi-abelian schemes over base scheme S?

By the book, Degeneration of abelian varieties-[Faltings G , Chai C ], a semi-abelian scheme is a smooth separated commutative group scheme $\pi : G\rightarrow S$ with geometrically connected fibres, ...
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Let $f:X\rightarrow Y$ be a morphism of schemes over $S$ (with possibly Noetherian conditions all over the place). For every point $s\in S$, the morphism base changed to the fiber $f_s:X_s\rightarrow ... 0answers 31 views ### Is subgroup scheme of smooth group scheme flat over base scheme? Let$\mathcal{A},\mathcal{B}$be smooth separated commutative group scheme over$S$, where$S$is Noetherian. Let$\iota:\mathcal{A}\rightarrow\mathcal{B}$be a morphism of group scheme which is ... 0answers 69 views ### Proposition 1.6.6, Etale Cohomology theory, Lei Fu I have difficulties in understanding a gap of the proof of proposition 1.6.6 in Etale Cohomology Theory written by Lei Fu.$\mathrm{Proposition} 1.6.6$Let$g:S'\rightarrow S$be a quasi-compact ... 0answers 45 views ### Check if the following ring homomorphism is flat? Let$k$be a field. Consider the ring morphism$f: k[x,y,x^{-1}, y^{-1}] \to k[t,t^{-1}]$where$x \to t$and$y \to t$. How do we know if$f$is flat or not? Now let$R=k[x,y,x^{-1}, y^{-1}]$. Then ... 1answer 59 views ### When is the blow up morphism flat? The question is as in the title: given a scheme$X$and a closed subscheme$Z$when is it true that the blow up morphism$Bl_ZX \rightarrow X$is flat? I’m mainly concerned with$X$being a smooth ... 1answer 27 views ### tensoring with flat module factors the kernel I want to show that if$F$is a flat$R$-module, then for any$R$-homomorphism$\varphi: M \rightarrow N$, we have $$\ker (1_F \otimes \varphi) \cong F \otimes \ker \varphi$$ The$\supset$... 2answers 54 views ###$\forall\, 0\to N\to M\to F\to 0 $exact,$\forall R$-module$E$,$F$is flat$\implies 0 \to N\otimes E\to M\otimes E\to F\otimes E \to0$is exact. In$R$-mod category,$\forall\, 0\to N\to M\to F\to 0 $is short exact sequence,$\forall R$-module$E$.$\forall\, 0\to N\to M\to F\to 0 $exact,$\forall R$-module$E$,$F$is flat$\implies 0 \to ...
Let $X \subset \mathbb{A}_{\mathbb{C}}^2$ be a closed subscheme, and let $\pi \colon X \to \mathbb{A}_{\mathbb{C}}^1$ be a finite flat map. For a point $p \in X(\mathbb{C})$, is it true that we can ...