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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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On faithfully flat and faithfully projective modules

Let $R$ be a commutative Noetherian ring. Let $P,Q$ be some $R$-modules such that $-\otimes_R P $ and $ Hom_R(Q,-) $ are faithfully exact functors i.e., for any sequence of modules $A \xrightarrow{f}...
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Flat $\mathbb{Z}$-modules and algebraic independence

I know that this question is naive, but I really want to find out if this observation is correct. First, let $\alpha_1, \alpha_2, \dotsc, \alpha_k$ be real numbers. Consider the $\mathbb{Z}$-module ...
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How to prove that a morphism is flat

I think the concept of flatness is very theoretical and I find hard to work with it. Hence, I would like to understand some easy examples before going deeper in its study. For instance, I read that ...
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Condition for a finitely generated flat module be projective

Prove that: Let $R$ be a commuatative ring, let $T$ be total quotient ring of $R$. A finitely generated flat $R$-module $M$ is projective if and only if the scalar extension $T\otimes_R M$ is a ...
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At which point does a sphere's edges look flat?

Good Morning, I'm pretty new to the mathematics section of StackExchange so please do forgive me if I make some mistakes in the question. I'm trying to make a spherical worlds game, however there is ...
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Module not flat

I found a example about the module is not a flat module: "Let $R=\mathbb{C}[t]$ be a ring of polynomials variable $t$ with coefficients in $\mathbb{C}$. Consider $R$-module $N=R[x]/\langle tx-t\rangle$...
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Is every local flat extension of a DVR a domain?

The title pretty much says it all. I would like to know if it is true that given a finite flat ring extension $A \rightarrow B$ with $A$ a DVR, and $B$ a local ring, then $B$ is necessarily a domain....
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Prove that if $M$ is flat over $R$ and $N$ is flat over $S$ then $M\otimes N$ is flat over $S$

We have an $M$ right $R$-module and $N$ both left $R$-module and right $S$-module, how could I show that if $M$ is flat over $R$ and $N$ is flat over $S$ then $M\otimes N$ is flat over $S$?
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Tower of ring extensions and flatness

Consider the following ring extensions: $$A\hookrightarrow\ B\hookrightarrow C$$ where we know that: $A$ is a complete DVR, $B$ and $C$ are Noetherian, Two-dimensional local rings and $B\...
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The flat module, module is not flat

Why $\mathbb{Z}$-module $\mathbb{Q}$ is flat and $\mathbb{Z}$-module $\mathbb{Z}_n$ is not flat? P/s: How can I prove them by definition and without functor. Thankyou.
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are the global sections of a flat sheaf over a discrete valuation ring a free module?

Let $f:X\to \operatorname{Spec}\mathbf{Z}_p$ be a smooth proper $\mathbf{Z}_p$-scheme and $F$ a coherent sheaf on $X$ which is flat over $\mathbf{Z}_p$. Further suppose that $H^1(X_p, F_p)=0$ where $...
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Can we deduce the exactness of the pervious modules sequence if the localized exact modules sequence is exact?

My question comes from a proposition: if M is flat, will $S^{-1}M$ be flat? Where $S = R - \mathfrak{p}$, $\mathfrak{p}$ is a prime ideal. Since localization keeps the exactness, I find that what I ...
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Check if the morphism is flat

Consider the following ring morphism $f:\mathbb{Z}[T_1,T_2] \to \mathbb{Z}[T_1,T_2]$ where $T_1 \to T_1^n$ and $T_2 \to T_2^n$. I want to check if $f$ is flat. For this I want to prove $\mathbb{Z}[...
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Is $A[[x_1,x_2,\dots]]$ flat over $A[x_1,x_2,\dots]$? ($A$ noetherian and commutative)

Let $A$ be a noetherian commutative ring with one and $x_1,x_2,\dots$ indeterminates. Question. Is $A[[x_1,x_2,\dots]]$ flat over $A[x_1,x_2,\dots]\ ?$ Recall that $A[[x_1,x_2,\dots]]$ is the set ...
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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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Surjective homomorphism from a faithfully flat module to a regular local ring.

Let $R$ be a regular local ring and let $M$ be a faithfully flat $R$-module. Does there necessarily exist a surjective $R$-module homomorphism from $M$ to $R$? For context, I am computing $\sum_{f\in\...
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Hartshorne's relative dimension in definition of smoothness

In Hartshorne's "Algebraic Geometry" III, Definition p. 268 smoothness is defined by conditions for a morphism $f:X \to Y$ of schemes of finite type over a field $k$ from which the first two are ...
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Theorem about flat module and ideal over a commutative ring

Let $I$ be a right ideal of $R$ (a commutative ring) then $M$ is flat if and only if $$0\to I\otimes_R M\to R \otimes_R M \cong M$$ is exact. ($\Rightarrow$) If $M$ is flat and $i:I \to R$ is ...
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If $G$ is a smooth scheme over $S$ of characteristic $p$, is the relative Frobenius morphism $F_{G/S}$ faithfully flat?

Let $G$ be a smooth scheme over $S$ of characteristic $p$, do we have that the reltaive Frobenius morphism $F_{G/S}$ is faithfully flat? There is an excersice in Liu's book saying that this is true ...
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$\mathfrak a$-adic Completion of a Ring is Flat

Let $A$ be a Noetherian ring and $\mathfrak a \subset A$ an ideal of $A$. Denote by $\hat{A_{\mathfrak a}}:= \varprojlim_n A/\mathfrak a^n$ the $\mathfrak a$-adic completion of $A$ wrt $\mathfrak a$. ...
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Base Change Preserves Flatness

I have a question about a step in the proof of 5.3.17 in Liu's "Algebraic Geometry" (page 201): Why is $B = A[T] / (P(T))$ flat over $A$? Does anybody see a clever base change to a flat module? My ...
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Flat scheme over a Dedekind ring

I have a problem with Proposition 4.3.9 of Qing Liu's algebraic geometry book. It says if R is a Dedekind ring and X is a reduced scheme and we have a dominant morphism $f:X\to \operatorname{spec}R$, ...
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Morphism of varieties with all fibers isomorphic

Let $k$ be an algebraically closed field. Let $f:X \to Y$ be a morphism of $k$-algebraic schemes, that is separated schemes of finite type over $k$. Furthermore: 1) $Y$ is a smooth variety (that is, ...
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Simple question about flat modules

Everything I can find about this is stated in category theory language that I do not understand. If I have an exact sequence $... \rightarrow A \xrightarrow[\text{}]{\text{f}} B \xrightarrow[\text{}]{...
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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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if the base chage of all fibres of a morphism is faithfully flat, do we have flatness of the morphism?

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 ...
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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 ...
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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 ...
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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 ...
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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 ...
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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 $\supset$ ...
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$\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 ...
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Is constancy of fiber degree analytic-local on the source in a finite flat family?

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 ...
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1answer
126 views

When does a quasifinite surjective flat morphism have constant fiber multiplicity near a point?

Let $V \subset \mathbb{A}_{\mathbb{C}}^2 = \operatorname{Spec}\mathbb{C}[x,t]$ be a closed subscheme containing the point $x = t = 0$, and suppose we have a quasifinite flat surjective morphism $\pi \...
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Faithfully Flat imply $aA = aB \cap A$

Let $f: A \to B$ a faithful flat map between rings. Let $a \in A$. I want to know how to prove that $$aA = aB \cap A$$ holds. Ideas: Consider the exact sequence (1) $$0 \to aB \cap A\to A \to A/...
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68 views

Integral Closure Flat

Let $R$ be an integral domain and $A$ it's integral closure. Futhermore $A$ is a finitely generated $R$-module. I want to show that if $A$ is flat over $R$ then already $R = A$ holds. My attempts: I ...
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about flat $R$-modules with different conditions [closed]

Need to know if the following statements are true or false: Let $S$ be a multiplicative set in $R$. Then $(S^{-1}R)[x,y,z]$ is always a flat $R$-module. Let $I,J \subset R$ be proper ideals of $R$ ...
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about flatness of module

I need help in (iii) implies (iv) in this why tor (M,IN)=0 and tor(M,N/IN)=0 ;and (v) implies (1). In (v) implies (i) I understand the proof until the use of artin-rees lemma but after that I don't ...
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about torsion and flat module

Can someone explain me how (i) implies (ii) ? What is that lopping off term does ? And why at n=0 tensor is not exact?
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Reference of a theorem in EGA about faithfully flat ring changes

Is there an english (or german) reference for the proposition (2.5.8) that can be found in EGA, IV? The proposition is more or less the following, simplified to my application Proposition Let $A$ be ...
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1answer
86 views

Is $\Bbb Z[i]$ flat as $\Bbb Z[2i]$-module?

Is $\Bbb Z[i]$ flat as $\Bbb Z[2i]$-module ? (Here $i^2 = -1$). I know it's flat as a $\Bbb Z$-module, since it is torsion-free and $\Bbb Z$ is Dedekind. Actually it is even free as $\Bbb Z$-module. ...
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1answer
44 views

$K(X)$ is not faithfully flat $K[X]$ module.

How can I show that for a field $K,$ $K(X)$ is not faithfully flat as $K[X]$ module. $K(X)$ is a localization of $K[X],$ it is flat $K[X]$ module. But I cannot prove that tensor by $K(X)$ is not a ...
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The dimension of stalks for flat morphisms

I want to prove the conclusion below: If $f:X\to Y$ is a quasi-finite flat morphism, then $\dim \mathcal{O}_{X,x}=\dim \mathcal{O}_{Y,f(x)}$. For the part $\dim \mathcal{O}_{X,x}\leq\dim \mathcal{O}...
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When $P\otimes_R M \to R \otimes_R M$ is injective for every (finitely generated) prime ideals $P$ of $R$?

Let $R$ be a commutative ring. If $M$ is an $R$-module such that for every finitely generated prime ideal $P$ of $R$, the map $i\otimes Id: P\otimes_R M \to R \otimes_R M\cong M$ is injective, then ...
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Restriction of a flat/smooth morphism with nice properties

This is related to this old question, which doesn't seem to address my exact situation. Suppose I have a smooth morphism $\varphi: X\to Y$ of smooth varieties over an algebraically closed field, and ...
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64 views

Flat affine group scheme $G$ over $\mathbb{Z}$ arises from embedding generic fiber $G_{\mathbb{Q}}$ into $GL_{n,\mathbb{Q}}$.

If we have an connected reductive group (reductive probably doesn't matter, affine group scheme is what matters) $G_{\mathbb{Q}}$ over $\mathbb{Q}$, we may construct a flat affine $\mathbb{Z}$- group, ...
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35 views

Sheaves of abealian groups and base change

Let $k$ be a field of characteristic $p > 0$ and $R_1$ and $R_2$ be two $k$-algebras. Let $X$ be a scheme over $k$. Let $f: R_1 \rightarrow R_2$ be a morphism of $k$-algebras induces the morphism ...
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1answer
116 views

Why does a flat finite type morphism of irreducible noetherian schemes map generic pt to generic pt

Basically the title, I came across this statement reading some notes on dimension of fibers of flat maps. (I don't know if Noetherian actually matters for this question) I think that for maps of ...
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1answer
30 views

Isomorphisms in faithfully flat $R$-modules

Given that an $R$-module $M$ is faithfully flat with $R$ commutative. Let $\alpha:M'\to M''$ be any $R$-homomorphism. I want to prove that If id${M}\otimes \alpha$ is an isomorphism, then so is $\...