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

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 ...
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How much information about a family is contained in its closed fibres?

One general type of question I am unsure how to approach is "how much information can we deduce about a family given information about its closed fibres?" As a concrete example, say we have ...
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Hartshorne problem III.10.5

Here is a problem I thought I solved but now I think it can't be right. The problem is as follows: Let $X$ be a scheme and $\mathcal{F}$ a coherent sheaf such that every $x\in X$ has an étale ...
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About vanishing of Tor of a morphism over a perfect ring?

Let $R$ be an associative ring with identity. An ideal $I$ of $R$ is said to be T-nilpotent if for every left $R$-module $M$, $IM=M$ implies $M=0$ equivalently for every left $R$-module $M$, $Hom_{R}(...
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A Question on Flatness of Complete Local Rings

Let $f\colon A\to B$ be a morphism of finitely-generated $k$-algebras. Assume that $f$ is flat, and that $\mathfrak{p}=f^{-1}(\mathfrak{q})$ are the corresponding prime ideals. Then we have a flat ...
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Does the multiplication by an ideal commutes with direct limits? [closed]

Let $J$ be an ideal of a ring $R$ and $(M_i, \phi_{ij})_{i,j\in I}$ be a directed system of $R$-modules. Have we the following statement $$J \lim_I M_i \cong \lim_I JM_i$$
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$M$ is flat iff $\operatorname{Tor}(M,R/J(R))=0$?

A commutative ring $R$ is said to be perfect if every flat $R$-module is projective. $J(R)$ denote the radical of Jacobson of $R$ wich is the intersection of all maximal ideals of $R$. I want to know ...
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When the base change functor of modules is full?

Let $\varphi: A \longrightarrow B$ be a morphism of $k$-algebras (with identity and not necessarily commutative) and $k$ a commutative ring (with identity). Let $F:= {}_B B_A \otimes_A {}_A (-) : \...
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Finitness conditions under change of rings?

Let $R \to S$ be a homomorphism of commutative rings, such that $S$ is finitely generated and projective $R$-module. Proof that if $A$ is a finitely presented $S$-module then $A$ is a finitely ...
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Rigorously working with flat limits: lines meeting a curve by specialization

I am trying to get comfortable with flat limits. This question is motivated by Section 3.5.3 of Eisenbud and Harris's '3264 And All That' and Exercises 3.35 and 3.36. This section and the surrounding ...
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Necessary and Sufficient condition for Flatness of Graded Modules

I am currently self studying the textbook Commutative Algebra with a View Toward Algebraic Geometry by David Eisenbud and came across the following exercise: Exercise 6.10 (Flatness of graded modules):...
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Valuative criterion for flatness of sheaves

Let $X \rightarrow S$ a morphism of schemes with $S$ reduced and Noetherian over a field. Let $\mathcal{F}$ be a coherent sheaf on $X$. To show $\mathcal{F}$ is flat over $S$ does it suffice to show ...
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Exercise 1.1c in Hartshorne's Deformation Theory: Is this family of conics flat?

In my attempt to solve Exercise 1.1c in Hartshorne's Deformation Theory book, which says that a family of conics in $\mathbb{P}^2$ parameterised by some finitely generated $k$-algebra $A$ ($k$ is ...
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Under what circumstances are the relative Kahler differentials a flat module?

First let us recall one construction of the relative Kahler differentials. Let $k$ be a ring and $R$ a $k$-algebra. The relative Kahler differentials $\Omega_{R/k}$ are the $R$-module satisfying the ...
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Is local freeness open for curves?

Let $X$ be a nonsingular curve and $S$ a scheme over $k$ algebraically closed, and $\cal{F}$ a coherent sheaf on $X \times S$, flat over $S$ (via the projection). So my question is the following: is ...
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Can $R/I$ be $R$-flat if $R$ is a domain?

Let $R$ be a domain and $I$ a nonzero proper ideal of $R$. Can $R/I$ be flat? For example, I can show if $R$ is Noetherian no such ideal $I$ exists. To see this, if $R/I$ is flat than by considering ...
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About exactness of $Hom$ functor?

We know that a module $P$ is projective if and only if the functor $Hom(P,-):R-Mod \to Ab$ is exact, i.e it preserves epimorphisms: If $\alpha: M \to N$ is an epimorphism of modules then $Hom(P,\alpha)...
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Flatness of a morphism for an infinitesimal deformation

I am trying to learn about infinitesimal deformations and I am particularly looking at Example 1.2.2 (i) from the book Deformations of Algebraic Schemes by Edoardo Sernesi which states the following: ...
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Phantom morphisms and Tor functor?

$\newcommand{\Tor}{\operatorname{Tor}}$In the category of R-modules, a morphism $f:M\to N$ is called a phantom morphism if for every finitely presented module $F$ and every morphism $g:F\to M$, $fg$ ...
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On splitness of epimorphisms?

Let $\alpha: M \to N$ be an epimorphism of left R-modules. Let $Q$ be a left R-module of finite projective dimension d with a morphism $\beta : Q \to N$. We know that if d=0 i.e, $Q$ is projective ...
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About split module epimorphisms

Let $\alpha:M\to N$ be an R-module epimorphism and $X$ be a class of R-modules. $\alpha$ is said to be a split epimorphism with respect to $X$ if the the induced morphism of abelian groups $Hom(F,M)\...
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When every finitely generated submodule of an R-module is cyclic?

We know that if R is a noetherian ring then every submodule of a finitely generated module is finitely generated. In case R is absolutely flat, we have every finitely generated ideal is principal, can ...
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Faithful flatness and split-exactness

Suppose $R\to S$ is a faithfully flat morphism of commutative unital rings, and suppose $A\to B\to C\to 0$ is an exact sequence of finitely-generated $R$-modules. If we know that $0\to A\otimes_R S\to ...
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Change of rings of scalars and projecivity?

Let $f:R\to S$ be a ring homomorphism and M be a left S-module. We can consider M as an R-module via $ r.m := f(r)m $. I know that if M is a flat S-module and S is flat as R-module then M is a flat R-...
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A characterisation of flatness

A left $R$-module $M$ is flat if and only if every morphism $f:K\to M$ , where $K$ finitely presented, factors through a finitely generated projective left $R$-module. I have found a proof of this ...
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Right exactness of tensor product in a long exact sequence

Suppose that $$0\longrightarrow A\longrightarrow B\longrightarrow C\longrightarrow D\longrightarrow0$$ is an exact sequence of $R$-modules. Is it true that $$A\otimes M\longrightarrow B\otimes M\...
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A statement for flat modules analogous to the Baer's Criterion [duplicate]

Recently I came across the following statement about the criterion of flat modules which looks somewhat like Baer's Criterion for injective modules: Statement. Let $R$ be a commutative ring. An $R$-...
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Is the natural projection of flag varieties a flat morphism?

Let $G$ be a reductive group over a field k (of characteristic zero) with parabolic subgroup $P$ and Borel $B \subset P$. Is the natural projection $ \pi: G/B \rightarrow G/P $ of flag varieties then ...
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Let $(R, P)$ be a local ring. If $M$ is a finitely presented module, then $M$ is flat iff $\mathrm{Tor}_{1}^R(M,R/P)=0$.

Let $(R, P)$ be a local ring. If $M$ is a finitely presented module, then $M$ is flat iff $\mathrm{Tor}_{1}^R(M,R/P)=0$. First, choose a minimal set of generators for $M$, and use these we can define ...
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For number field $K$, is maximal order $O_K$ flat over the order $O$?

I am now trying to solve the following question: Suppose $L$ and $K$ are number fields, $K\subset L$, $\Gamma$ is the order of $L$, $O\subset O_K$ is the inverse image of $\Gamma$ via the the ...
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About flatness of modules / algebras

Let $A\to B$ be a ring homomorphism and suppose that $B$ is flat as $A$-module. If $M$ is a flat $B$-module, is it flat as $A$-module? I've been thinking for a while but I couldn't come up with ...
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Confusion about a definition in Atiyah-Macdonald

In chapter 2 of Atiyah-Macdonald are introduced the flat modules, and the algebras. However it is not given any definition of flat algebra, while in exercise 5 and 8, for example, is needed. I don't ...
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Exercise 4.3, Atiyah Macdonald

Let $A$ be an absolutely flat ring, i.e. any $A$-module is flat. Prove that any primary ideal $q\subset A$ is maximal. I have an exact sequence of $A$-modules: $$0\to q\to A\to A/q\to 0.$$ My idea ...
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Support of module and faithfully flat base change

Let $R \subseteq S$ be a faithfully flat extension of Noetherian local rings. Let $M$ be a finitely generated $R$-module such that $\operatorname{Supp}_R(M)=\operatorname{Spec}(R)$. Then, is it true ...
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How to prove that $S^{-1} A$ is a flat A-module

This question was asked in my commutative algebra assignment and I need help in solving it. Prove that if S is a multiplicatively closed subset of A, then $S^{-1} A$ is a flat A-module. Attempt: I ...
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Local flatness criterion: A morphism $X \to Y$ of schemes over $S$ is flat if and only if it is flat on all fibers $X_s \to Y_s$, $s \in S$?

In Nitsure's Part 2. Construction of Hilbert and Quot schemes in Fundamental Algebraic Geometry, there is the following Lemma Lemma 5.21. (3) Let $S$ be a noetherian scheme, and let $f: X \to S$ and $...
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Linear equivalence on pullback of divisors in blowup $\operatorname{Bl}_0(\mathbb{P}^2)$

Let $\pi:\tilde{X}\to \mathbb{P}^2$ be the blowup in the point $0=(0:0:1)$ with coordinates $(x_0:x_1:x_2),(y_0:y_1)$ and let $H_i=V(x_i)\subset \mathbb{P}^2$ be the three hyperplane divisors that are ...
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$M \hookrightarrow M[t^{-1}]$

Take $M$ an $X=\mathbb{C}[[t]]$-module. first question I've to prove that: If $M$ is flat then $M$ is a submodule of $M[t^{-1}]$. Here my attempt. We have: $$M\cong M\otimes_X\mathbb{C}[[t]] \...
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Is an inclusion morphism flat?

I find it really hard to understand the definition of flat. There are so many details, but I didn't find any "intuitive" geometric or algebraic approach to this type of functions. To my ...
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Ideals in the ring of formal power series

It is known that, if $R$ is a commutative Noetherian ring, the ring of formal power series $R[[x_1,...,x_n]]$ is a flat $R$-algebra. This gives us a flat ring homomorphism $\phi:R \rightarrow R[[x_1,.....
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Show that If $A$ is Noetherian, then $A[[x_1,...,x_n]]$ is a faithfully flat $A$-algebra.

Let $A$ be a commutative ring. Show that if $A$ is Noetherian, then $A[[x_1,...,x_n]]$ is a faithfully flat $A$-algebra. I've already shown that $A[[x_1,...,x_n]]$ is a flat $A$-algebra (by showing ...
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$\bigcap_{\mathrm{finite}} (0:_R a_i)M= (0:_R I)M$ for $M$ flat module?

I'm working on a problem that says if $I$ is finitely generated ideal and $M$ is a flat module, then $(0:_M I)=(0:_R I)M$. I have reduced the problem into proving $$\bigcap_{\mathrm{finite}} (0:_R a_i)...
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Exercise 6.6 in Eisenbud's Commutative Algebra on the Family of Projective Plane Curves of Degree d

This was already asked by someone else here: They answered their own question there but I'm not satisfied with their answer for part a. I'll reproduce the question from that link here: Consider $k\...
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criteria for flatness of map $f:A \to A[X]/P(X)=:B$

Let $A$ commutative ring with $1$ and $P(X) \in A[X]$ a non constant polynomial. Can it be exactly characterized when the canonical map $f:A \to A[X]/P(X)=:B$ is (1) flat or (2) induces surjective map ...
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Relative Frobenius of algebraic group and smoothness

I'm quite struggling to understand the proof of Prop.13.52 in Milne's Algebraic groups (rough preliminary version): An (reduced) algebraic group $G$ over $k$ with $\text{char}\,k = p$ is smooth iff ...
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Flat ring extensions and flat modules

Let $R$ be a not necessarily commutative unital ring, and $R \hookrightarrow S$ a unital ring extension such that $S$ is flat as a (right) module over $S$ (i.e. a flat extension). See here for a ...
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non-flatness of $k[\epsilon] \to k$

Let $k$ be a field and $k[\epsilon] = k[x]/(x^2) \to k, x \mapsto 0$ the morphism, which is an example of a non-flat morphism. Although I already know a direct way to show why it's not flat, at ...
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Proof verification: $A$ is absolutely flat if and only if $A_\mathfrak{m}$ is a field

This is a solution to Exercise 3.10 in Atiyah-MacDonald, (i) If $A$ is absolutely flat and $S$ is any multiplicatively closed subset of $A$, then $S^{-1}A$ is absolutely flat. (ii) $A$ is absolutely ...
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Are two nice equidimensional varieties with the same Euler characteristic fibers of a flat morphism?

I'm very amused by the ability of Euler characteristic to withstand any deformation. I would be very impressed if one had a geometric analogue of homotopy (much rougher) that captured all pairs with ...
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Extensions of flat modules and tensor products

Let $R$ be a ring and let $$0 \rightarrow M_1 \rightarrow M_2 \rightarrow M_3 \rightarrow 0$$ be a short exact sequence of flat $R$-modules. Let $N$ be another $R$-module. Question: is the sequence $...
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