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...
212 questions
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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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1answer
39 views
+50
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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41 views
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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21 views
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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1answer
20 views
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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2answers
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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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1answer
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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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26 views
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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1answer
28 views
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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48 views
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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28 views
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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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 ...
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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 ...
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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 ...
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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 ...
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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 $\supset$ ...
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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 ...
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70 views
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
103 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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42 views
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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0answers
61 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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1answer
63 views
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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24 views
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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2answers
28 views
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
83 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. ...
1
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1answer
42 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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38 views
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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42 views
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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0answers
60 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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34 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 ...
2
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1answer
87 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
27 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 $\...
3
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1answer
130 views
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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1answer
53 views
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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1answer
181 views
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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1answer
77 views
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$$
where $f$ and $g$ are monomorphisms, $S$ is $R$-free, and $T$ is (faithfully) $R$...
0
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1answer
63 views
Computing $\operatorname{Tor}^R_1(M,N)$
Let $f$, $g$ be two non-constant polynomials in $\mathbb{Q}[x]$. Let $R = \mathbb{Q}[x]/(fg)$ and consider
$M = \mathbb{Q}[x]/(f)$ and $N = \mathbb{Q}[x]/(g)$ as $R$-modules.
(a) Find a projective ...
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1answer
79 views
Localization and flatness
Let $M$ be a flat $S^{-1}A$-module. I want to show that $M$ is flat as an $A$-module.
$\newcommand{\tp}{\otimes}$
My "best" attempt is the following: Let $$0 \to N_1 \to N_2 \to N_3 \to 0$$ be exact (...
1
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1answer
91 views
Flatness in a short exact sequence. If the
Let $$0\to L\xrightarrow{\alpha} M \xrightarrow{\beta} N\to 0$$ be a short exact sequence of right $R$-modules and homomorphisms. Show that if $L$ and $N$ are flat then $M$ is flat.
Let $f:A\...
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1answer
29 views
When flat submodule is direct summand?
Let $M$ be a $\mathbb{Z}$-module. And $N \subset M$ is a submodule. Assume that $N$ is flat as a $\mathbb{Z}$-module. Then I'm wondering if
$$
M/N \subset M \Rightarrow N \ is \ direct \ summand \ of ...
4
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0answers
113 views
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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0answers
36 views
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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1answer
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 ...
2
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1answer
127 views
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), $T$ ...
2
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0answers
97 views
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 $\bar X$...
