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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When $\mathbb{C}[s,k]$ is normal, where $s$ is an even polynomial and $k$ is odd

Let $\epsilon: \mathbb{C}[x] \to \mathbb{C}[x]$ be the following involution $\epsilon: x \mapsto -x$. Let $s=s(x),k=k(x) \in \mathbb{C}[x]$, $\deg(s), \deg(k) \geq 1$, with $s$ symmetric and $k$ skew-...
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Locally free modules are flat

The following theorem is well-known in commutative algebra: Let $A$ be a ring. Suppose that, for all prime ideals $\frak p$ of $A$, the module $M_\frak p$ is flat over $A_\frak p$. Then $M$ is flat ...
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Finding the dimension of certain tensor product with flat $A$-algebra

Assume $A$ is a noetherian local ring with $\mathfrak{m}_y$ being the unique maximal ideal and $\dim A=0$. We have the exact sequence $$0\to \mathfrak{m}_y\to A\to k(y)\to 0,$$ where $k(y)$ is the ...
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Surjectivity of character module map implies injective module map?

I'm trying to understand modules. And I want to know, by definition of a module $M$, the underlying group structure on $M$ is abelian. Does that mean that every $R$ module $M$ is also a $\mathbb{Z}$ ...
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Flatness of a subring generated by two subrings

Let $S$ be a commutative $\mathbb{C}$-algebra and let $a,b,c,d \in S$. Assume that $R_1:=\mathbb{C}[a+c,b+d] \subseteq S$ is flat and $R_1:=\mathbb{C}[a-c,b-d] \subseteq S$ is flat. Question 1: Is it ...
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Prime ideals of localization of absolutely flat modules

There is an exercise (chapter 3, Exercise 10, page number 44) in atiyah mc Donald. $A$ is absolutely flat iff $A_m$ is a field for each maximal ideal $m$. I actually proved this statement and it is ...
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Is restriction of scalars preserving injective modules equivalent to flatness?

Given any ring homomorphism $R \to S$, if $S$ is a flat right $R$-module, then any injective left $S$-module is also injective as a left $R$-module. Now, I'm wondering whether the converse is true. ...
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When does restriction of scalars induce flat Modules?

Let $f:Y\rightarrow X$ be a ring homomorphism, and let us consider $X$ to be a $Y$-module via restriction of scalars, i.e. given some $y\in Y$, this element acts on $X$ via $f(y)\cdot x$, for $x\in X$ ...
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Is $\mathbb{C}[V]^G \subset \mathbb{C}[V]$ flat?

Let $V$ be a vector space over $\mathbb{C}$ and let $G$ be a finite group acting on $V$ via $G \longrightarrow \operatorname{GL}(V)$. We know that $\mathbb{C}[V]^G \subset \mathbb{C}[V]$ is an ...
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If $A \subseteq B$, then is $M \otimes A \subseteq M \otimes B$?

Let $A, B$ be $R$-modules. If $A \subseteq B$, then is $M \otimes A \subseteq M \otimes B$? What happens if $A$ is given to be a flat module?
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Example of graded module which is not flat

Could someone help me to find: $R_0$ a local noetherian ring, $R$ a graded algebra over $R_0$ generated by $x_1,...,x_n$ of degree 1, and $M$ graded $R$-module, finitely generated, such that $M$ is ...
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finitely generated submodules are flat

Assume that all the finitely generated submodules of $M$ are flat, then why is $M$ also flat? Thank you so much!
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Prove that elementary tensor is non-zero in proof of non-flatness

This is a follow-up question to this question (and the answer there by René Schipperus) about proving that $k[t]$ is non-flat as $k[t^2,t^3]$-module. I have reduced this to showing that $t\otimes t$ ...
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Geometric intuition - flatness of morphism of stacks

I have seen interesting geometric intuition answers about flatness, for example here and here. I am not sure whether it still applies for morphisms of stack, and if yes how? In partitcular, Can we ...
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Help with understanding the definition of a Hodge groupoid

I am reading this PhD thesis and I can't understand definition 6.21: (Hodge groupoid). $(X/S)_{Hod}$ is a groupoid whose object object is $X\times \Bbb A^1_S$ and whose morphism object $N_{Hod}$ is ...
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Is $\mathbb Q[x,z]$ as a $\mathbb Q[x,y]$-module (with morphism $x\mapsto x$, $y\mapsto xz$) flat?

This was an exercise in my class, please help: Put $A = {\mathbb Q}[x,y]$ and $B = {\mathbb Q}[x,z]$. Consider the morphism $f \colon A \to B$ of ${\mathbb Q}$-algebras given by $x \mapsto x$, $y \...
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is $R/I$ flat over $S$ if $R/\sqrt{I}$ is flat over $S$ and $I$ is primary?

Let $R$ be an $S$-algebra and let $I$ be a primary ideal such that $R/\sqrt{I}$ is flat over $S$ where $\sqrt{I}$ is the radical of $I$. Is $R/I$ flat over $S$? I don't mind assuming $R,S$ Noetherian, ...
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$M$ is flat iff $Tor_1(M,N)=0$?

Given a $R$-module $M$, it's flat iff $Tor_1(N,M)=0$ for all $R$-module $N$, which can be deduced from a free resolution of $N$, tensoring with $M$ and applying the definition of flatness. But there ...
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Example of non-flat $R \subseteq R[w]$

Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras such that: (1) $R$ and $S$ are integral domains. (2) $Q(R)=Q(S)$, namely, their fields of fractions are equal. (3) $S=R[w]$, for some $w \...
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Flatness over tensor products

Let $R \subseteq S$ be two commutative $\mathbb{C}$-algebras. Assume that $S$ is flat as a left $S \otimes_R S$-module, via $f: S \otimes_R S \to S$ given by: $f(s_1 \otimes_R s_2)=s_1s_2$. When this ...
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Certain separable ring extensions

Let $k$ be a field of characteristic zero, $R$ a commutative $k$-algebra which is an integral domain having field of fractions $Q(R)$ and $a$ an algebraic element over $R$ ($a$ belongs to some ...
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Exercise 11.1.G Vakil FOAG

I am trying to solve Ex 11.1.G from Ravi Vakil's FOAG. It says if $X$ is an affine scheme over $k$, a field and $K|_k$ is an algebraic field extension, then $X$ is of pure dimension $n$ iff $X_K:=X\...
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Matlis dual of injective module over complete local ring [closed]

Let $M$ be an injective module over a Noetherian complete local ring $(R,\mathfrak m,k)$. Let $E(k)$ denote the injective hull of $k$. Then, is it true that $\text{Hom}_R(M, E(k))$ is a free $R$-...
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On flat morphisms and going-down

I am trying to show that a flat morphism satisfies the going down property. For this, the reference I am following reduces the problem to a certain claim which uses the following result of section 10....
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Module is flat over localization

Let $S \subset R$ be a multiplicative set, and $M$ is an $S^{-1}R$ module. Then I read on stacks project that $M$ is flat $R$-module iff $M$ is flat $S^{-1}R$ module. I see one direction. If $M$ is ...
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Exact sequence of sheaves on nilpotent thickening

I am rather confused by this argument in a lecture in MSRI's deformation theory case. It is Problem 1 and Remark 2.1 in page 17 notes of the notes. It is rather long Here $R[I]$ is a square zero ...
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50 views

Every finitely generated free Module over a commutative Ring is flat

Let $R$ be a commutative Ring with $1$ and $M$ be a finitely generated free $R$-Module. Show that $M$ is flat. By definition, $M$ is flat $\iff$ For every injective $R$-Module Homomorphism $\varphi: N ...
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How can I show that the inclusion $\mathbb{C}[x^2, xy, y^2] \to \mathbb{C}[x, y]$ is not flat?

I want to show that $\mathbb{C}[s, t, u] \to \mathbb{C}[x, y] : s \mapsto x^2, t \mapsto xy, u \mapsto y^2$ is not flat. If $s \otimes y \neq t \otimes x$ in $(x^2, xy, y^2) \otimes_{\mathbb{C}[s, t, ...
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Holonomy of a flat connection around a contractible loop

I'm currently reading a paper written by John Baez [1]. At a certain point he argues, that the holonomy of a flat connection around a contractible loop is the identity, so its trace in the ...
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If $I$ is contained in the Jacobson radical of a Noetherian ring $A$ then $A\to \widehat{A}$ is faithfully flat.

Let $A$ is a noetherian ring, $I\subseteq A$ is an ideal, and $\widehat{A}$ is the $I$-adic completion of $A$. If $I$ is contained in the Jacobson radical of $A$ then I have to show that $A\to \...
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Contractions and Extensions of Ideals and Faithful Flatness

Let $A \subseteq B \subseteq C$ be rings with $B$ a faithfully flat $A$-module and $C$ a faithfully flat $B$-module which is an integral extension of $B$. Given a maximal ideal $I \subseteq C$, is it ...
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A question involving faithful flatness, support of a module, and Spectrum of a ring

The following theorem is taken from Matsumura's Commutative Ring Theory [M] Theorem 7.3(i) and the paragraph before it. My questions only concern the proof of the Theorem below. A ring homomorphism $...
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A question related to tensor product of $R$-modules

I am currently reading Künneth tensor formula over arbitrary ring, and try to find out the structure of connecting homomorphism appears in the homology long exact sequence. At some point, I have ...
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$A \subset B$ be a faithfully flat extension of domains and $B$ is integrally closed then $A$ is also integrally closed.

Let $A \subset B$ be a faithfully flat extension of integral domains. If $B$ is integrally closed then I have to show that $A$ is also integrally closed. Assuming $L,K$ be the field of fractions of ...
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Is $k[x,xy] \subseteq k[x,y]$ a flat ring extension?

Let $k$ be a field of characteristic zero. Is $k[x,xy] \subseteq k[x,y]$ a flat ring extension? I guess that the answer is no? Though I am not sure how to prove this. Perhaps applying this ...
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Why this flatness condition is needed? (Serge lang's Algebra)(p.618, chapter 16, proposition 3.7)

[My Question] Why "$F$ is flat is needed in this proof? [My attempt] Let $f : \mathcal{a} \otimes F \rightarrow \mathcal{a}F $ be defined by $f(\sum_i x_i \otimes b_i) = \sum_ix_i b_i$ . (By ...
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60 views

Flat extension of local rings with a specified extension of residue field [closed]

Let $(R, \mathfrak m_R, k)$ be a Noetherian local ring and $K$ be a field containing $k$. Then is it true that there is a Noetherian local ring $(S, \mathfrak m_S)$ and a flat ring homomorphism $f: ...
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surjectivity condition for induced map of prime ideals

Let $A,B$ be commutative rings and $f: A \rightarrow B$ a flat ring homomorphism. I would like to show that: if or all $\mathfrak{m} \in$ MaxSpec($A$) one has ($f(\mathfrak{m})$)=:$\mathfrak{m}^e \...
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Flatness of the $p$-th power roots ideals in a perfect ring

Let $A$ be a perfect ring of characteristic $p>0$. If $x\in A$ is a nonzero divisor then $$ (x^{1/p^{\infty}})=\bigcup_{e=1}^{\infty}(x^{1/p^{e}})A $$ is an $A$-flat ideal. Reading Hochster's ...
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When is $K[X_1,X_2,…,X_n] \to K[Y_1,Y_2,…,Y_m]$ a flat morphism

Let $K$ be a field and $\varphi: K[X_1,X_2,...,X_n] \to K[Y_1,Y_2,...,Y_m]$ a polynomial ring morphism. Assume $n, m \ge 2$. By definition $\varphi$ endows $K[Y_1,Y_2,...,Y_m]$ with a $K[X_1,X_2,...,...
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Is the affine cone of a flat projective scheme again flat?

I'm trying to solve Hartshorne Chap.III Ex.9.5.(c): The biggest problem is to show the existence of a closed subscheme $\tilde{X}\subset \mathbb{P}_{T}^{n+1}$ such that $\tilde{X}_{t} = \operatorname{...
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When is relative Hilbert scheme $\text{Hilb}_{X/S}^{p(t)}$ flat over $S$?

Let $S$ be a scheme of finite type over the complex numbers $\mathbb{C}$ and let $X\subset S\times\mathbb{P}^r$ be a projective family over $S$. In the book "Geometry of Algebraic Curves Volume II" ...
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Polynomial map $K[t] \to K[t]$ induces flat module structure

Let $K$ be a field and $\phi: K[t] \to K[t], t \mapsto t^n$, $n \ge 2$. This $\phi$ makes the second $K[t]$ to a $K[t]$-module by $a,b \in K[t]: a \cdot b:=\phi(a)b$. Why this $K[t]$-module structure ...
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141 views

Quotient of polynomial ring flat $R$-algebra

Let $R$ be a Noetherian ring and let $P(T) \in R[T]$ be a monic polynomial. Let $A:= R[T]/(P(T))$. (1) Is $A$ a flat $R$-algebra and why? (About this, I think that's true, below I give an argument ...
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reference request-Absolutely Flat Rings

I want to study about Absolutely Flat rings, but I am unable to get a detailed theory about that. Can anyone here suggest me some reference in which I can get a detailed theory about absolutely flat ...
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105 views

$\operatorname{Hom}_B$ of flat modules is flat over $A$?

Let $(A, \mathfrak{m}) \rightarrow (B,\mathfrak{n})$ be a local homomorphism of noetherian local rings and let $M$ be a finitely generated $B$-module flat over $A$. Suppose moreover that $B$ is also ...
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Flat cohomology as a subspace of local Galois cohomology

Consider an elliptic curve $E$ over a local field $k$ with good reduction and with Neron model $\mathcal{E}/\mathcal{O}_k.$ The Galois group $G_k$ acts on $E[m]$, so we have a cohomology group $H = H^...
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44 views

If $B$ is a flat $A$-algebra then $B_{\mathrm{red}}$ is a flat $A_{\mathrm{red}}$-algebra

Given a commutative ring $A$ we denote by $A_{\mathrm{red}}$ the ring $A/\mathrm{Nil}(A)$ where $\mathrm{Nil}(A)$ is the nilradical of $A$. It is not difficult to see that this construction is ...
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41 views

Faithfully flat module and exact sequences

I have the following problem. Let $M$ be a faithfully flat module. I need to show that if sequence $$0\longrightarrow N'\otimes M \stackrel{\varphi\otimes{id_{M}}}\longrightarrow N\otimes M\...
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Natural map $M\rightarrow M\otimes_R R'$ injective - Missing prerequisite in Bosch's book?

I am working my way through Bosch's book "Algebraic Geometry and Commutative Algebra" and found an exercise in chapter 4.4, exercise 3, which I think is missing some prerequisite. The original text is ...

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