# 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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### 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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### 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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### 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 ...
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\...