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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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$A \subset B$ 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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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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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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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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$K(X)$ 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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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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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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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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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 $\...
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Projective and flat vs. faithfully flat

Let $R$ be a commutative ring with unity and let $M$ be a projective and faithful $R$-module. Then is $M$ faithfully flat ? Is it true at least if $M$ is finitely generated, or say Noetherian ? I ...
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If $A\subset B$ are integral domains with the same field of fractions and $B$ is faithfully flat over $A$, then $A=B$. [duplicate]

I found this exercise in Matsumura, Commutative Ring Theory, (Exercise 7.2). If $A\subset B$ are integral domains with the same field of fractions and $B$ is faithfully flat over $A$, then $A=B$. ...
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If $M$ is a flat $R$-module and $rm=0$ for some $r$ and $m$, show $m=0$.

Let $R$ be a commutative ring and let $M$ be a flat $R$-module. If $r\in R$ is not a zero-divisor and $m\in M$ is such that $rm=0$, prove that $m=0$. I can't seem to figure out how to use flatness ...
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Can it be that $R[[x]]$ is flat over $R$ but not over $R[x]$?

I have been trying to better understand the ascension of flatness. After asking a poor question in general terms (see Does flatness ascend through a free ring map?), I realized that I do not ...
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Does flatness ascend through a free ring map?

Suppose I have a composition of ring maps (rings commutative with identity) $$R \xrightarrow{f} S \xrightarrow{g} T$$ where $f$ and $g$ are monomorphisms, $S$ is $R$-free, and $T$ is (faithfully) $R$...
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63 views

Computing $\operatorname{Tor}^R_1(M,N)$

Let $f$, $g$ be two non-constant polynomials in $\mathbb{Q}[x]$. Let $R = \mathbb{Q}[x]/(fg)$ and consider $M = \mathbb{Q}[x]/(f)$ and $N = \mathbb{Q}[x]/(g)$ as $R$-modules. (a) Find a projective ...
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79 views

Localization and flatness

Let $M$ be a flat $S^{-1}A$-module. I want to show that $M$ is flat as an $A$-module. $\newcommand{\tp}{\otimes}$ My "best" attempt is the following: Let $$0 \to N_1 \to N_2 \to N_3 \to 0$$ be exact (...
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Flatness in a short exact sequence. If the

Let $$0\to L\xrightarrow{\alpha} M \xrightarrow{\beta} N\to 0$$ be a short exact sequence of right $R$-modules and homomorphisms. Show that if $L$ and $N$ are flat then $M$ is flat. Let $f:A\...
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29 views

When flat submodule is direct summand?

Let $M$ be a $\mathbb{Z}$-module. And $N \subset M$ is a submodule. Assume that $N$ is flat as a $\mathbb{Z}$-module. Then I'm wondering if $$ M/N \subset M \Rightarrow N \ is \ direct \ summand \ of ...
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When integral domain is flat?

Let $A$ and $B$ be integral domain. $A$ is integrally closed and $A \rightarrow B$ is integral. -($\star$) This is sufficient to show that the going-down theorem. If $A \rightarrow B$ is flat, then ...
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Submodule of a flat module.

Let $A$ be a commutative ring with unity. Suppose that $A$ is a Noetherian regular ring with dimension equal to 1 and let $M$ be a flat $A$-module. Is it true that any sub-A-module of $M$ is flat?
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155 views

Flat modules over a PID

I know any f.g. flat module over a PID is projective. I am searching about does any flat module over a PID have the same feature? I consider $\mathbb{Q}$ and $\mathbb{Q}/\mathbb{Z}$ which are not free ...
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Difficulty understanding Hartshorne Theorem IV.4.11

I am stuck on a claim in the proof of Theorem IV.4.1 in Hartshorne. Let $X$ be a elliptic curve (a nonsingular projective variety of dimension 1 and genus 1 over an algebraically closed field), $T$ ...
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Hartshorne III Example 9.8.4, how to understand a parameterization of a curve by scheme-language

For each $a\in \mathbb A^1 -\{0\}$ the scheme $X_a$ is given by $$ x=t^2-1,~ ~~y = t^3-t,~~~ z=at $$ and then we have a family over $\mathbb A^1 -\{0\}$. Our goal is to obtain the total family $\bar X$...