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...

Filter by
Sorted by
Tagged with
1 vote
0 answers
40 views

Show the formal power series ring is a faithfully flat algebra.

Suppose $S=P^{-1}F[x]$, the localization of $F[x]$ at $P$ where $F$ is a field and $P=(x)\backslash\{0\}$. Let $\hat{S}=F[[x]]$, the formal power series ring. Clearly there is a homomorphism $\Phi: S\...
Dick Grayson's user avatar
  • 1,393
1 vote
0 answers
28 views

Flat base change preserves the non-degeneracy (Proposition 9.2 in Commutative Algebra, Matsumura)

Let $f : A \rightarrow B$ and $g : A \rightarrow C$ be homomorphisms of Noetherian rings.Suppose 1) $B \otimes_A C$ is Noetherian, 2) $f$ is flat and 3) $g$ is non-degenerate. Then $1_B \otimes g : B ...
RHspqr's user avatar
  • 147
2 votes
1 answer
54 views

Is the quotient of an infinite product of fields by the direct sum injective?

Let $k$ be any field and consider $R=\prod_{\mathbb{N}}k$ the product of copies of $k$ indexed over the natural numbers. The direct sum $\sum_{\mathbb{N}}k=I$ is an ideal of $R$, hence we can form the ...
N.B.'s user avatar
  • 2,109
2 votes
1 answer
72 views

A Criterion for Faithfully Flatness via Intersection Property

Let $R \subset S$ be an injective map of commutative rings, $M \subset N $ an inclusion of $R$-modules. We identify $M, N$ respectively as $M \otimes_R 1_S \subset N \otimes_R 1_S \subset N \otimes S$...
user267839's user avatar
  • 7,377
2 votes
1 answer
59 views

Pullback of a flat module is flat

Let $f\colon (X,\mathcal{O}_X)\rightarrow (Y,\mathcal{O}_Y)$ be a map of ringed spaces. In the Stacks Project, in the proof of Tag 09U8, they basically claim at some point, that for any flat $\mathcal{...
Asvr_esn's user avatar
  • 167
0 votes
0 answers
38 views

If $M$ is a flat $R-$module and $I$ is an ideal of $R$, why is $I\otimes_R M$ isomorphic to $IM$?

This is often given as an equivalent property to $M$ being a flat $R-$module (for instance in one of the answers here Are $I\otimes_{R}J$ and $IJ$ isomorphic as $R$-modules?). Certainly we can always ...
Aaron Andersen's user avatar
2 votes
0 answers
44 views

"Explicit" showing of non-flatness for a fractional ideal

I am not very familiar with flatness so I was trying to get a feel of what precisely is the crux of the notion, using objects I am used to. Sadly, most of the counter examples I could find where for ...
KeiOh's user avatar
  • 344
3 votes
1 answer
86 views

Interpretation of $B\otimes M$ when $B\subset A$

Let us fix the setting to avoid ambiguity. Let $R$ be a commutative, unital ring. All tensor products will be considered for modules over $R$. Let $A$ be an $R$-module, $B$ a submodule. Now, is $A\...
Academic's user avatar
  • 307
2 votes
2 answers
95 views

[Cor 5.3.24 in Qing Liu]If $X$ is a projective scheme over a DVR $A$, then any coherent sheaf of ideal of $\mathcal{O}_X$ is flat over $A$?

The problem comes from the proof of Corollary 5.3.24 in Qing Liu's book Algebraic Geometry and Arithmetic curve. Let $\DeclareMathOperator{\Spec}{Spec}S=\Spec A$ be the spectrum a discrete valuation ...
Z Wu's user avatar
  • 1,723
0 votes
1 answer
50 views

Is projective morphism over a discrete valuation ring always flat?

Let $A$ be a discrete valuation ring with a uniformizing parameter $t$. Let $f:X\to \DeclareMathOperator{\Spec}{Spec}\Spec A$ be a projective morphism (i.e. $X$ is a closed subscheme of $\mathbb{P}^...
Z Wu's user avatar
  • 1,723
2 votes
0 answers
91 views

Miracle flatness on Wikipedia's "Cohen--Macaulay ring"

In Wikipedia's article on Cohen–Macaulay rings, the following geometric version of miracle flatness is stated, see this link: Let $X$ be a connected affine scheme of finite type over a field $K$ (for ...
Daniel W.'s user avatar
  • 1,756
0 votes
0 answers
80 views

Free module over local ring $R$. [duplicate]

People often say that a module $M$ over a not necessarily neotherian local ring $R$ being projective is flat, and also free. However, some refer to the finitely-generatedness of $M$, i.e. $M$ being ...
Pierre MATSUMI's user avatar
1 vote
1 answer
66 views

$\mathbb Z[\xi_{2n}]$ is a pid

Let $\xi_{2n} \in \mathbb C$ a primitive $2n^{th}$ root of unity for some integer $n\ge 2 $. Is the inclusion $\mathbb Z[\xi_{2n}] \hookrightarrow \mathbb C$ flat? It is possible to answer this ...
Conjecture's user avatar
  • 3,138
1 vote
1 answer
91 views

Show the problem of flat base change is local on the source and target

(The picture is from Exercise 5.1.16 in QingLiu's Algebraic Geometry and Arithmetic Curve) All the proofs I saw on any reference usually start with "the problem is local on $T$ and on $S$ so we ...
Z Wu's user avatar
  • 1,723
1 vote
0 answers
85 views

Prove that the polynomial ring $R[x]$ is a flat $R$-module.

Here is the question I am trying to solve: Prove that the polynomial ring $R[x]$ in the indeterminate $x$ over the commutative ring $R$ is a flat $R$-module. My thoughts: We know by corollary 42 in D&...
Intuition's user avatar
  • 3,269
2 votes
0 answers
39 views

$R$ points of derived subgroup of algebraic group

I'm reading through Milne's book on algebraic groups, and in corollary 6.19 he writes: Assume that $G$ is affine or smooth, then (c) for all $k-$algebras $R$, $(\mathcal{D}G)(R)$ consists of the ...
Et-'s user avatar
  • 53
1 vote
0 answers
42 views

How Can We Prove Flatness from an Induced Exact Sequence?

I want to prove that for every short exact sequence $$ O \to A \xrightarrow{f} B \xrightarrow{g} C \to O $$ of $R$-module homomorphisms, if the induced sequence $$ O \to M\otimes_R A \xrightarrow{\...
Mr Prof's user avatar
  • 435
0 votes
1 answer
63 views

Is Every Isomorphism on Tensor Product of Modules an R-module Isomorphism?

We have an R-module M. We also have S, a multiplicatively closed subset of R. I want to prove that there exists a unique R-module isomorphism f: S$^{-1}$R$⊗$M $\to$S$^{-1}$M, defined by f(${\frac rs}$⊗...
Mr Prof's user avatar
  • 435
-1 votes
1 answer
108 views

How to determine flatness

Let $k$ be a field and define $R=k[x,y]/(xy)$ and an $R$-Algebra $A=k[x]\times k[y]=R/(y)\times R/(x)$. I have to answer two questions Is $k[x]=R/(y)$ flat over $A$ Is $A$ flat over $R$ I don't ...
Mike192000's user avatar
4 votes
1 answer
103 views

$\mathbb{Q}$ with trivial $R$-module structure is flat

I am doing my first course in module theory, and our teacher gave us the following exercise, which I am completely stuck on: Let $R$ be the ring $\mathbb{Z}[x]/(x^2-1)$. An $R$-module is an abelian ...
Luis's user avatar
  • 71
4 votes
1 answer
77 views

If $M$ is a flat $A$-module, then is $M$ a flat $A/I$ module?

Assume that $A$ is a commutative ring with unity and let $M$ be a unital $A$-module. Let $I$ be an ideal of $A$ such that $IM = 0$. Then we can give a natural $A/I$-module structure on $M$ satisfying $...
Ajin Shaji Jose's user avatar
2 votes
1 answer
49 views

Let $f: N \rightarrow P$, $M$ faithfully flat iff $\alpha \otimes id_M = 0$

Stacks project lemma 10.39.14 Let $M$ be a flat $R$-module ($R$ is commutative). Then, $M$ is faithfully flat ($- \otimes_R M$ sends non-exact sequences to non-exact sequences) iff for any module ...
David Lui's user avatar
  • 6,327
0 votes
1 answer
69 views

Flatness of $\mathbb{C}[x_1,\ldots,x_n]$ over $\mathbb{C}[f]$, $f \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$

Let $f \in \mathbb{C}[x_1,\ldots,x_n]$, $n \geq 1$. Call $f$ 'good' if $\mathbb{C}[x_1,\ldots,x_n]$ is flat over $\mathbb{C}[f]$. Is it true that every $f \in \mathbb{C}[x_1,\ldots,x_n]$ is good? $f ...
user237522's user avatar
  • 6,469
3 votes
1 answer
44 views

Example of absolutely flat semi-artinian ring

A commutative ring $R$ is said absolutely flat (or alternatively von Neumann regular) if every $R$-module is flat. This property is equivalent to $r \in (r^2)$ for all elements $r\in R$. The ring $R$ ...
N.B.'s user avatar
  • 2,109
1 vote
1 answer
93 views

Is the following ideal prime or the following quotient ring flat?

Consider the ideal $$I := (xy - 4, z^2 - zy - y) \subseteq \mathbb{Z}[x,y,z].$$ I'd would like to know if it is prime. If it is prime, then I believe I can conclude that $\mathbb{Z}[x,y,z]/I$ is a ...
KSAKY's user avatar
  • 121
4 votes
1 answer
159 views

Étale covering of nodal cubic curve, Hartshorne exercise 10.6.

I was trying to solve exercise III.10.6 of Hartshorne and found this post about it. My problem is that I am struggling to show that the morphism $f : X \longrightarrow Y$ (following notation of the ...
Andarrkor's user avatar
  • 612
0 votes
0 answers
33 views

References for examples of flat and non flat morphisms with solutions?

I am looking for exercises/examples of "simple enough" morphisms between varieties being flat/non-flat, with solutions. I have only ever found a handful of examples, however very rarely with ...
Pandir's user avatar
  • 71
1 vote
0 answers
128 views

Flat locus of finite map between integral schemes is not necessarily open

Let $X,Y$ be integral Noetherian schemes. Let $f:X\to Y$ be a finite map of schemes. I recently had to show that the set of points $V\subset Y$ over which $f$ is flat is open, as is for instance ...
David Wiedemann's user avatar
0 votes
1 answer
103 views

Example of a universally injective ring morphism which is not faithfully flat

Is there some classic example of an universally injective ring morphism which is not a faithfully flat morphism? I was not able to find it in any commutative algebra book and neither around here.
Thiago da Silva's user avatar
1 vote
1 answer
47 views

Faithfully flat change of ring in the tensor product [closed]

Suppose that $S'$ is a faithfully flat $S$-algebra, and let $T$ be a $S'$-algebra. My question: is there some theorem that says that the canonical morphism $$T\underset{S}{\otimes}T\rightarrow T\...
Thiago da Silva's user avatar
1 vote
1 answer
166 views

Atiyah MacDonald Exercise 2.2 [duplicate]

In this exercise, we are asked to prove that if $\mathfrak{a}$ an ideal of $A$ and $M$ an $R$-module, then $A/\mathfrak{a} \otimes_{A} M \cong M/\mathfrak{a}M$. There is a hint given that asks to ...
dormordo's user avatar
0 votes
0 answers
39 views

Some problem about affine space and algebraic variety

Actually, I am reading something about control theory, but it contains lots of mathematical descriptions that I can't understand. The contents are the following: Consider the system $\mathcal{D} / k$ ...
Xiaolong Huang's user avatar
0 votes
1 answer
77 views

Flat modules over $R/I$ vs flat modules over $R$

Let $K$ be the field of Hahn series in an indeterminate $t$, with exponents in $\mathbb{R}$ and coefficients in $\mathbb{F}_2$, and where we denote its valuation by $\nu$. Define $$A:=\{a\in K:\nu(a)\...
user829347's user avatar
  • 3,422
0 votes
1 answer
120 views

Equivalent definitions of flat modules

According to my lecture notes, these three definitions of flat modules are equivalent: a) For every exact sequence of $R$-modules $N’ \rightarrow N \rightarrow N’’$, the sequence $N’ \otimes_R M \...
dahemar's user avatar
  • 1,786
2 votes
0 answers
82 views

Why is $()^{n}:\mathbf{G}_{m}\longrightarrow\mathbf{G}_{m}$ surjective in the $fppf$ topology?

Let $X$ be a scheme. Why is $()^{n}:\mathbb{G}_{m}\longrightarrow\mathbf{G}_{m}$ surjective in the category $X_{fppf}$? (Here $\mathbf{G}_{m}(X)=X\otimes_{\mathbb{Z}}\operatorname{Spec}(\mathbb{Z}[T,X]...
The Thin Whistler's user avatar
0 votes
1 answer
539 views

Prove that $M$ is flat $\iff$ each $M_i$ is flat (2.4 Atiyah & MacDonald pg. 31)

Here is the question I want to solve: Let $M_i (i \in I)$ be any family of $A$-modules, let $M$ be their direct sum. Prove that $M$ is flat $\iff$ each $M_i$ is flat. Here is a solution to it found on ...
weird's user avatar
  • 29
0 votes
0 answers
36 views

Finitely generated flat module over a PID is torsion-free? [duplicate]

I’m trying to show that a finitely generated flat module over a PID is torsion-free. I know the converse is a consequence of the structure theorem for finitely generated modules over a PID and the ...
dahemar's user avatar
  • 1,786
0 votes
0 answers
148 views

Question about Dually flat manifolds

I am reading "Methods of information geometry by shun-Ichi-Amari" and I got stuck in the following chapter 3 section 3.3(Dually flat spaces), let me define it first Let $(S,g,\nabla,\nabla^*)...
Andyale's user avatar
  • 95
1 vote
0 answers
294 views

Proof that torsion-free module over a PID is flat

I have the following proof that a torsion-free module over a PID is flat. Let $A$ be a PID, and let $N$ be a torsion-free $A$-module. If $N$ is finitely generated, then $N$ is free (by the ...
Frank's user avatar
  • 2,451
2 votes
1 answer
60 views

Is the following module flat over $A$?

Let $B \to A$ be a surjection where $B$ and $A$ are Artin local rings which are $k$-algebras and both having residue field $k$. Let $M$ be the kernel of the surjection and $M^2 = 0$. This induces an $...
Angry_Math_Person's user avatar
2 votes
2 answers
246 views

Tor and quotient

I have a question about the following portion of the proof of a result The proof is given by a serious of reductions, but I got stuck at the first step, where it claims that if $x\in R$ is a non zero-...
Lao-tzu's user avatar
  • 2,876
6 votes
0 answers
328 views

Exercise 4.3.13 on Frobenius morphisms in Qing Liu's Algebraic Geometry

Let $X$ be a smooth morphism over a scheme $S$ of positive characteristic $p > 0$. The morphism $F_S: S \to S$ induced by the ring homomorphism $O_S \to O_S : a \mapsto a^p$ the absolute Frobenius ...
user267839's user avatar
  • 7,377
0 votes
0 answers
61 views

Let $A$ be a commutative ring. Let $N$ be a flat $A$-module. Let $S⊆A$ be a multiplicatively closed set. $S^{-1}N$ is flat $S^{-1}A$ module?

Let $A$ be a commutative ring. Let $N$ be a flat $A$-module. Let $S⊆A$ be a multiplicatively closed set. How can I prove $S^{-1}N$ is flat $S^{-1}A$ module ? I see inverse argument ($S^{-1}N$ is flat ...
Pont's user avatar
  • 6,011
0 votes
0 answers
34 views

Let $R$ be a commutative ring and let $0→A→B→C→0$ be an exact sequence of $R$ module. In general, $A \otimes_R N→B \otimes_R N$ [duplicate]

Let $R$ be a commutative ring and let $0→A→B→C→0$ be an exact sequence of $R$ module. In general, for any $R-$module $N$,$A \otimes_R N→B \otimes_R N$・・・① is not always injective, My question is, if $...
Pont's user avatar
  • 6,011
1 vote
0 answers
53 views

A right $R$-module $B$ is flat if and only if $0\to B\otimes_R I\to B\otimes_R R$ is exact for every finitely generated left ideal $I$.

A right $R$-module $B$ is flat if and only if $0\to B\otimes_R I\to B\otimes_R R$ is exact for every finitely generated left ideal $I$. The proof of this statement is given in Rotman Advanced Modern ...
one potato two potato's user avatar
3 votes
2 answers
230 views

Property of Flat Modules

I am trying to understand a proof regarding some properties of flat modules. The place I am stuck at, essentially boils down to this situation: Let $M'\overset{\varphi}{\rightarrow}M\overset{\psi}{\...
Squeezelemma's user avatar
1 vote
0 answers
191 views

Tensor products preserve quasiiso between flat complexes

Let $f:M_\bullet \to N_\bullet$ be a quasi-isomorphism between chain complexes of flat $R$-modules for $R$ a unitary ring. Let $P$ be a right $R$-module. Is it true, and how could I prove that $$f\...
arnett's user avatar
  • 786
1 vote
0 answers
47 views

Understanding the implications of the covariant derivative on geometrical shape

I am self studying differential geometry (and information geometry), and came across the following idea from a lecture by Frederic Schuller: https://youtu.be/2eVWUdcI2ho?t=384 Essentially, one can ...
tisPrimeTime's user avatar
1 vote
1 answer
82 views

Is the image of a map of free modules flat?

Let $A$ be a commutative ring, and $\alpha:F_1\to F_2$ a map of free modules. Question: is the image $\text{im}\alpha$ flat? I've looked at the standard references, e.g. Stacks Project, and can't find ...
Pulcinella's user avatar
  • 1,420
3 votes
1 answer
90 views

Classification compact oriented $2$-manifolds with boundary which admit a flat Riemannan structure [closed]

The only compact orientable $n$-manifolds without boundary which can be given a flat Riemannan structure are tori. I was wondering if we could classify the compact oriented $2$-manifolds with boundary ...
Chetan Vuppulury's user avatar

1
2 3 4 5
10