All Questions
Tagged with flatness reference-request
14 questions
1
vote
1
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216
views
Flatness of "derived local system sheaves"
Let $f: Y\longrightarrow X$ be a smooth proper map of smooth proper schemes over $\mathbb{Q}$, and let $\mathcal{F} = R^1_\text{ét}\overline{f}_*\mathbb{Q}_p$ denote the derived pushforward of $\...
- 2,907
3
votes
1
answer
591
views
Finitely generated modules over Noetherian local ring that become isomorphic after faithfully flat base change
Let $(R,\mathfrak m)$ be a Noetherian local ring. Let $S$ be a Noetherian ring which is a faithfully flat $R$-algebra. If $M,N$ are finitely generated $R$-modules such that $M\otimes_R S \cong N \...
- 31
8
votes
0
answers
1k
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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 ...
- 1,915
6
votes
0
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360
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Flat base change in the complex analytic setting
On page 255 of Hartshorne's Algebraic Geometry, it is shown that "cohomology commutes with flat base extension":
Proposition III.9.3: Let $f : X \to Y$ be a separated morphism of finite type of ...
- 61
3
votes
0
answers
214
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Representability of Flattening stratification functor
Let $f:X \to Y$ be a projective morphism between noetherian scheme. Is there any know condition under which there exists a functorial stratification of $Y$ i.e., there exists a filtration by locally ...
- 1,981
11
votes
2
answers
2k
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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 ...
- 10.5k
1
vote
0
answers
191
views
Deformation of projective bundles
Let $\pi:\mathcal{X} \to \mathbb{P}^1$ be a flat, projective family of noetherian schemes with generic fiber a smooth, projective variety. Let $p:\mathcal{Y} \to \mathcal{X}$ be another flat, ...
- 2,126
0
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0
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191
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Fiberwise injective resolution of coherent sheaf
Let $k$ be an algebraically closed field (of characteristic zero) and $X, Y$ be projective $k$-varieties. Let $F$ be a coherent sheaf on $X \times_k Y$, flat over $Y$. Does there exists a coherent $\...
- 2,323
0
votes
0
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182
views
Is the $B$-tensor power of flat $A$-modules, $A$-flat?
Let $k$ be an algebraically closed field and $A, B$ be two finitely generated $k$-algebras. Suppose $B$ is flat over $A$. Let $M$ be a finitely generated $B$-module which is flat over $A$. Is it true ...
- 1,981
3
votes
1
answer
969
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When is the pullback of an injective sheaf injective?
Let $X$ be a Gorenstein (not necessarily smooth) projective $\mathbb{C}$-scheme and $S$ another $k$-scheme. Let $I$ be an injective sheaf on $X$. Denote by $p:X \times_k S \to X$ the natural ...
- 1,981
4
votes
1
answer
512
views
Being Cohen-Macaulay open in Hilbert scheme?
Let $H$ be the Hilbert scheme of closed subschemes of $\mathbb{P}^n$ with a given Hilbert polynomial. I would like to have a reference (preferably from a published paper or book, not stacks project) ...
- 3,031
11
votes
1
answer
579
views
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 ...
- 10.5k
2
votes
0
answers
154
views
Ultraproducts and subobjects of projectives
Usually the question whether the class of projective algebras in a given variety is closed under taking subalgebras seems to be quite hard. In varieties with well understood dual geometry (e. g. ...
- 18.4k
2
votes
1
answer
561
views
Flatness over non-reduced schemes : no geometric characterisation
I know someone already asked about flatness over non-reduced schemes, but I think my question is different.
I'm reading Bosch, Lütkebohmert and Raynaud's "Néron models", and in the second chapter, ...
- 2,054