Mathematics Stack Exchange
Mathematics Stack Exchange

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...

An $A$-module $M$ is flat when the right-exact _$\otimes_AM$ functor becomes left exact (an 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...

484 questions
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My proof of $I \otimes N \cong IN$ is clearly wrong, but where have I gone wrong?

Ok, I'm reading some thesis of some former students, and come up with this proof, but it doesn't really look good to me. So I guess it should be wrong somewhere. So, here it goes: Let $R$ be a unitary commutative ring, and $I$ be an ideal, and $N$…
user49685
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Motivation behind the definition of flat module

Can someone explain what is the motivation behind the definition of a flat module? I saw the definition but I don't really know why it is important to work with these structures.
Lonely Penguin
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Examples of faithfully flat modules

I'm studying some results about flatness and faithful flatness and I'd like to keep in my mind some examples about faithfully flat modules. In general, free modules are the typical examples. Another (unusual) example of faithfully flat module is…
user11428
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Is the tensor product over $B$ of two flat $A$-modules flat over $A$?

Given a morphism of commutative rings $A\to B$ such that $B$ is a flat $A$-module and given $M$, $N$ two $B$-modules flat as $A$-modules, is the tensor product $M\otimes_B N$ flat over $A$?? The tensor product $M\otimes_A N$ is flat over $A$, the…
MonLau
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Flatness of subring

Let $R$ be a ring with 1, not necessarily commutative, with no zero divisors. Suppose $S$ is a flat extension of $R$. What additional assumptions, if any, would allow us to assert that a subring $R \subseteq T \subseteq S$ must also be flat over…
Mr. Chip
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Every finitely generated flat module over a ring with a finite number of minimal primes is projective

Over a commutative ring $R$, a finite type locally free (weak sense) module for which the rank function is locally constant is projective. If we notice that for each minimal prime $p$ of the ring, the rank function is constant on the adherence of…
brunoh
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Does the fibres being equal dimensional imply flatness?

Let $f: Y \to X$ be a morphism of varieties (proper if necessary). I read from a paper that if all the fibres of $f$ are of the same dimension then $f$ is flat. This seems skeptical for me, and I was wondering more conditions are need to have the…
Li Yutong
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When is $\Bbb{Z}[\zeta_n]$ a PID?

When is $\Bbb{Z}[\zeta_n]$ a PID? I was just wondering if $\Bbb{Z}[\zeta_n]$ is PID or not, where $\zeta_n$ is an $n$th primitive root of unity for arbitrary positive $n$
Departed
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Is it true that a flat module is torsion-free over an arbitrary ring? Does the reverse implication hold for finitely generated modules?

So when you work over a commutative ring, this result is quite well known. I am wondering if the same holds true for an arbitrary ring; that is, if $R$ is some (possibly noncommutative) ring, does the following implication hold: $$\text{Flat module}…
Sam Williams
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Is $R/N(R)$ a faithfully flat $R$-module?

I'm studying recently faithfully flat modules and I'd like to know the following: Is $R/N$ faithfully flat as $R$-module, where $R$ is a commutative ring with unit and $N$ is the ideal of nilpotent elements of $R$?
user93721
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Non-finitely generated, non-projective flat module, over a polynomial ring

Let $R=k[x_1,\ldots,x_n]$. According to the first answer, every finitely generated flat module over an integral domain is necessarily projective. Therefore, the only hope to find a flat non-projective $R$-module $M$ is when $M$ is not finitely…
user237522
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A finitely generated flat $A$-module $M$ is faithfully flat if and only if $\operatorname{Ann}(M)=0$

How one can show that a finitely generated flat $A$-module $M$ is faithfully flat if and only if $\operatorname{Ann}(M)=0$? (Liu, Algebraic Geometry and Arithmetic Curves, Exercise 2.17.) I tried to show that the condition $N$ is an $A$-module…
Jaakko Seppälä
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Integral dependence and (faithfully) flat ring extensions

Let $R\to S$ be a flat ring extension. By theorem 9.5 of the book Commutative Ring Theory written by Matsumura the going-down theorem holds between $R$ and $S$. Is it true (or not) about these theorems: "INCOMPARABILITY THEOREM", "LYING-OVER…
user147308
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Is a submodule of a flat module flat?

This is probably obvious but I am getting stuck thinking about it: Let $A$ be a commutative ring with unity, $M$ a flat $A$-module, and $N\subset M$ a submodule. Is $N$ necessarily flat over $A$? I haven't found a simple counterexample, but I also…
Ben Blum-Smith
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When does intersection commute with tensor product

Given two submodules $U,V \subseteq M$ over a (commutative) ring $R$, and a flat $R$-module $A$, I can interpret $U \otimes_R A$ and $V \otimes_R A$ as submodules of $M \otimes_R A$. Is it necessarily true that $$(U \cap V) \otimes_R A \cong (U…
user404188
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