Questions tagged [flatness]
The flatness tag has no usage guidance.
185
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Are submersions of differentiable manifolds flat morphisms?
Let $\pi \colon M\to N$ be a smooth map between real smooth manifolds. Then $C^\infty(M)$ forms a module over $C^\infty(N)$ (via pullback). Is this module flat when $\pi$ is a submersion?
Recall that ...
38
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1
answer
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Flatness in Algebraic Geometry vs. Fibration in Topology
I am currently trying to get my head around flatness in algebraic geometry. In particular, I'm trying to relate the notion of flatness in algebraic geometry to the notion of fibration in algebraic ...
27
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Can a quotient ring R/J ever be flat over R?
If $R$ is a ring and $J\subset R$ is an ideal, can $R/J$ ever be a flat $R$-module? For algebraic geometers, the question is "can a closed immersion ever be flat?"
The answer is yes: take $J=...
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Does projective imply flat?
Let $\mathcal C$ be an abelian category equipped with a closed symmetric monoidal structure. This implies in particular that the monoidal structure $\otimes$ is right exact in each variable. I care ...
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Homotopy flat DG-modules
A right DG-module $F$ over an associative DG-algebra (or DG-category) $A$ is said to be homotopy flat (h-flat for brevity) if for any acyclic left DG-module $M$ over $A$ the complex of abelian groups $...
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Every finitely generated flat module over a ring with finitely many 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, ...
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Is the support of a flat sheaf flat?
Note: in the following, all scheme/algebra morphisms should be assumed essentially of finite type.
Geometric version: Let $X$ be a scheme flat over $S$ (both noetherian), and let $\mathscr{F}$ be a ...
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Is there a notion of "flat vector bundle over a topological space"?
I am reading this paper and at the top of page 5 the author makes reference to categories consisting of flat complex vector bundles over $X$ where $X$ is an arbitrary topological space. However, the ...
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Does smoothness descend along flat morphisms?
Suppose $f:X\to Y$ is a flat morphism of schemes. If $X$ is smooth at $x$, must $Y$ be smooth at $f(x)$?
If $f$ is locally finitely presented, then it is open (using EGA IV 1.10.4), so after ...
12
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Given a family of curves, when does there exist a fibered surface over Spec Z parametrizing them?
Let $X_p$ be a projective curve over the finite field $\mathbf{F}_p$ (i.e. a projective $\mathbf{F}_p$-scheme pure of dimension 1) for every prime number $p$. Let $X_\mathbf{Q}$ be a projective curve ...
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Relationship between monodromy representations and isomorphism of flat vector bundles
This question is somehow related to this one.
Let $M$ be a smooth (compact, if you wish) connected manifold.
Then, it is well known that there is an equivalence between the isomorphism classes of ...
11
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flatness of power series rings
It is known that $A[[X]]$ is flat if $A$ is noetherian (see for example Bourbaki, Algèbre commutative, Ch. III, §3, Cor. 3 p. 146).
What happens if A is not noetherian? Is there an easy counter-...
11
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intuition about the "section after base-change" for flat descent and exactness of the Amitsur complex
Suppose $A \rightarrow B$ is a faithfully flat map of rings. Then the Amitsur complex is exact:
$0 \rightarrow A \rightarrow B \rightarrow B \otimes_A B \rightarrow \dots$
(the second map is $id \...
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Idea behind Grothendieck's proof that formally smooth implies flat?
From this answer I learned that Grothendieck proved the following result.
Theorem. Every formally smooth morphism between locally noetherian schemes is flat.
The book Smoothness, Regularity, and ...
11
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Can flatness be specified by a natural coherent sheaf?
More precisely:
Given a finite-type morphism $f \colon X \to Y$ of nice schemes (say, both of finite type over a field), is there a "natural" coherent sheaf $\mathcal F_f$ on $X$ such that the ...
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Resolution of singularities for flat families.
Is there a resolution of singularities for flat families?
More precisely, if $X \rightarrow \mathbb{A} ^n$ is a flat map, does there exist a map $Y \rightarrow X$ such that, for every $p \in \mathbb{...
10
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Iteration of a morphism and flatness
Let $A$ be a Noetherian local ring, $f:A \rightarrow A$ be a local ring morphism. Assume some power of $f$ is a flat morphism, must $f$ be flat as well?
Motivation: Kunz's theorem shows the result is ...
10
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Is there a direct proof that affine schemes are fppf quasi-compact?
Let $A$ be a (commutative) ring. A family $(B_i)_{i\in I}$ of $A$-algebras is said to be an fppf cover if it satisfies three properties: (1) each $B_i$ is flat as an $A$-module, (2) each $B_i$ is ...
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How general are Gröbner degenerations?
While working with flat degenerations of flag varieties and Schubert varieties I've noticed that among the numerous known constructions there doesn't seem to be a single one that doesn't turn out to ...
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Flatness from constancy of dimension of fibers
Let $M$ be a finitely presented module over the ring $R$. Suppose that for all primes $P\subset R$ the $k(P)$ vector space $M\otimes _Rk(P)$ has a dimension $d(P)$ independent of $P$.
Can I conclude ...
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Associated graded and flatness
Let $M$ be a filtered module over a filtered algebra $A$, and suppose $gr(M)$ is flat over $gr(A)$, where $gr$ means the associated graded module and algebra, respectively.
What can one say in ...
9
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2
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When can one extend a flat family from a subscheme to the whole scheme?
Is there a nice condition on a closed subscheme $Y$ of $X$ such that for every flat family $Z\to Y$, there is a flat family $W\to X$ whose restriction to $Y$ is $Z$? In particular, I'm interested in ...
9
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1
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Homotopical interpretation of flatness?
I have read a discussion (in a less common language) which discussed a homotopical interpretation of flatness, which went something like:
A map of commutative algebras is flat if pushing it out ...
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Why did Ravenel define a ring spectrum to be flat if its smash-square splits into copies of itself?
In appendix A.2 of the orange book, Ravenel defines a ring spectrum $E$ to be flat if $E\wedge E$ is equivalent to a coproduct of suspensions of $E$. (Call this definition (1).) I've seen this ...
8
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3
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Lifting a complete intersection in $\mathbb{P}^n_{\mathbb{F}_p}$ to $\mathbb{Z}_p$
Suppose that you are given a (not necessarily smooth) projective variety $X \subseteq \mathbb{P}^n_{\mathbb{F}_p}$ of codimension $d$ that is a complete intersection. In other words, it can be defined ...
8
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3
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Quotient of flat module is flat - a property in Mumford's Red book
Hi,
In Mumford's "Red Book of Varieties and Schemes", Chapter III, Paragraph 10 (entitled "Flat and smooth morphisms"), the following property is stated:
Let $M$ be a $B$-module, and $B$ an algebra ...
8
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2
answers
480
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Faithful flatness and non-commutative algebras
$\DeclareMathOperator\Spec{Spec}$When dealing with commutative algebras, a usefull criterion for faithful flatness is the following:
Let $f:A\rightarrow B$ be a morphism of commutative algebras. Then $...
8
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1
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Extending faithfully flat covers of closed subschemes to open neighborhoods
I am curious about the analogue of this question, as stated in the title. Namely,
If $Z \subset X$ is a closed subscheme and $Y \to Z$ is faithfully flat (let's also say of finite presentation), ...
8
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1
answer
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Triviality of direct multiples of flat complex vector bundles
Atiyah Patodi and Singer [Spectral asymmetry and Riemannian geometry III] write that if $E$ is a complex flat bundle (non holomorphic, just smooth and complex) on a compact manifold $X$ (more ...
8
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1
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Does a flat compactification always exist?
Let $\pi:X\to S$ be a separated flat morphism of finite type of Noetherian schemes. Does $\pi$ necessarily factor as an open immersion followed by a proper flat morphism? The analogue of this question ...
8
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0
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Classification of flat Riemannian three manifold
By a theorem of L. Bieberbach we know that that every closed flat Riemannian manifold is a quotient of a torus via action of a finite group $\Gamma$. In this question we are interested in the ...
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A Hartogs-type criterion for flatness
Let $U$ be a smooth affine connected variety over $\mathbb C$ and let $V\subset U$ be an open whose complement is of codimension at least two.
Now, let $Y$ be a smooth quasi-affine connected variety ...
7
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1
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Injective flat module
Let $R$ be a (right noetherian) ring. Is there always a right $R$-module which is both flat and injective? If $R$ is an integral domain, then the answer is indeed yes, as the quotient field is such.
...
7
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1
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Are irreducible components of a flat family flat?
Let $f:X\rightarrow Y$ be a flat morphism of schemes of finite type over a field $k$, and assume $Y$ is irreducible. Let $X_1, \dots, X_n$ be the scheme-theoretic irreducible components of $X$ (i.e., ...
7
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Failure of universal flatification
Raynaud and Gruson proved a beautiful "flatification" theorem (5.2.2): If $S$ is a quasicompact, quasiseparated scheme, and $X$ is a finitely presented $S$-scheme, $M$ is an $\mathcal O_X$-module of ...
7
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2
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Is $(x^2y,xy^2)$ log smooth?
Consider the map
$$f:\mathbb C^2\to\mathbb C^2$$
$$(x,y)\mapsto(x^2y,xy^2)$$
We can view $f$ as induced by the map of monoids $g:\mathbb Z^2_{\geq 0}\to\mathbb Z^2_{\geq 0}$ given by the matrix $(\...
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Flatness, "Continuously varying fibers", and bordism
It is commonly said that a flat map of schemes $f : X \rightarrow Y$ is like a map with "continuously varying fibers". We see a hint at this in the result that $\text{dim} X_y$ is constant when $f$ is ...
7
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Self-flat modules
(This is inspired by this question and asked out of pure curiosity.)
Let $R$ be a commutative ring. Let $M$ and $N$ be $R$-modules. Then, $N$ is called $M$-flat if whenever $M'\rightarrow M$ is a ...
7
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0
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Lifting flat modules over ring quotients
Let $R$ be a commutative ring, $I$ its ideal, and $\bar{R}=R/I$. For which flat $\bar{R}$-modules $\bar{F}$ is there a flat $R$-module $F$ such that $F \otimes_R R/I \simeq \bar{F}$?
By Lazard's ...
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Flatness for family of hypersurfaces
Let $X \to Y$ be a family of hypersurfaces in a constant $\mathbb{P}^n$, i.e. $X \subset Y \times \mathbb{P}^n$ is locally on $Y$ given by one equation of degree $d$ in $\mathbb{P}^n$.
Is $X \to Y$ ...
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Exactness of an additive left Kan extension
Let $\phi:R\to S$ be a flat ring homomorphism and consider the induced adjoint pair
$$\phi_!:R-Mod\rightleftarrows S-Mod:\phi^*,$$
where $\phi_!=(S\otimes_R -)$. The right adjoint $\phi^*$ is easily ...
6
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3
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Torsion-free tensor powers
Let $R$ be an integral domain. If $M$ is an $R$-module such that every tensor power of $M$ over $R$ is $R$-torsion-free, then is $M$ necessarily flat as an $R$-module? If not, then does this ...
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Tori in Compact Riemannian Symmetric Spaces
The closure of a 1-parameter subgroup in a compact Lie group is a torus. To what extent does this result generalize to compact Riemannian symmetric spaces? In other words, is the closure of a geodesic ...
6
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1
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Flatness of schemes
I am learning about flatness for the first time and I cannot wrap my head around why the definition with tensor products of a flat module implies geometrically that 1-parameter families of schemes ...
6
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1
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The inverse limit of locally free module
A is an I-adic complete Noetherian ring. M is a finitely generated A module. For any n>0, $M/I^nM$ is a finitely generated locally free A/I^n-module. Is M necessarily a locally free A-module?
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Flatness of objects in a prestable $\infty$-category
I wonder what is the correct concept of flatness of objects in a prestable $\infty$-category with appropriate conditions?
The typical example is the following. Let $R$ be a connective $\mathbb E_1$-...
5
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2
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When can a finite map be blown up to a flat one?
Let $f:X\to Y$ be a generically finite proper morphism of varieties. There is some locus in $Y$ over which the fiber of $f$ is positive dimensional, so we blow it up, along with the preimage of it in ...
5
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1
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Under what hypotheses are schematic fixed points of a flat deformation themselves flat?
This is something of a follow-up question to this one; I hope people won't think this is a duplicate. At least in my head, it seems like a distinct enough question to merit a fresh start.
All my ...
5
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1
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Multiple of a flat family of subschemes is flat
Let $X$ be a fixed curve (e.g. a Noetherian, projective scheme of dimension 1, of finite type over an algebraically closed field $k$) and let $S$ be an arbitrary parameter scheme over $k$. Let $D \...
5
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Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum
Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A_{q_n}$ as in this paper. Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of ...