All Questions
Tagged with schemes smoothness
18 questions
0
votes
0
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329
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Smooth morphisms under base change, Qing Liu's proposition 4.3.38
I have a concern about the first assertion in the proof of proposition 4.3.38 of Qing liu's "Algebraic Geometry and Arithmetic Curves". Referring to smooth morphisms, he says "The ...
- 149
6
votes
1
answer
930
views
Is every variety an image of a smooth variety?
Let $X$ be a finite type scheme over a field $k$.
Is it true that there exists a surjective morphism $f : Y \rightarrow X$, where $Y$ is smooth over $k$?
In other words, is every such scheme a ...
- 485
2
votes
0
answers
148
views
etale locally infinitesimal lifting property
For a morphism $X\rightarrow Y$ of qcqs schemes, one has the usual notion of formal smoothness which says that for a pair $(R,I)$ with $I^2=0$, if there is a point $y\in Y(R)$ such that $y_{\vert R/I}$...
- 3,472
2
votes
0
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191
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Morphism between jet spaces smooth
In this article "Introduction to Jet Schemes and Arc Spaces" S. Ishii introduces the spaces of $m$-jets:
Let $X$ be a variety over algebraically closed field $k$. The space $X_m$ of $m$-jets ...
- 6,028
2
votes
0
answers
138
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Formally smooth maps of schemes and factorization systems
I am thinking about how formally smooth maps of schemes relate to factorization systems.
Let $C$ be the category of schemes. Let $E$ be the class of morphisms of schemes consisting of closed ...
user30211
1
vote
0
answers
91
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Factorizations of closed embeddings of smooth schemes
All schemes will be of finite type over a field $k$. Say I have a closed embedding $\iota: X \hookrightarrow Y$ of smooth $S$-schemes for some scheme $S$ (in particular it is a regular embedding). ...
- 595
3
votes
0
answers
294
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Formal smoothness implies local freeness of the sheaf of relative differentials
What is the least restrictive finiteness assumptions guaranteeing that for a formally smooth morphism of schemes $f:X\rightarrow Y$, the sheaf of relative differentials $\Omega _{X/Y}$ would be ...
- 953
2
votes
0
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394
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Blow up along a section of a smooth morphism
Let $C$ be a ground locally notherian and quasiprojective scheme. Let $\pi:S\to B$ be a $C$-morphism of finite type $C$-schemes. We call $\pi$ a $C$-smooth morphism if the morphisms $S\to C$ and $B\to ...
- 43
2
votes
1
answer
257
views
Are the fibers of this morphism geometrically regular?
Let $A\rightarrow B$ be a local morphism of complete noetherian rings making $B$ a formally smooth $A$-algebra. Does the induced morphism $\textrm{Spec}(B)\to\textrm{Spec}(A)$ have geometrically ...
- 255
9
votes
1
answer
943
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on the local structure of schemes
Let $X$ be an integral finite type scheme over $\mathbb{C}$. Let $x\in X$, such that there exists a neighborhood $U$ of $x$, such that the sheaf of differentials $\Omega^{1}_{U}$ decomposes into:
$\...
- 3,472
1
vote
1
answer
342
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Smoothness and smoothness over formal neighborhood
Let $f:X\rightarrow Y$ a locally finitely presented map. Let $x\in X$ and $y=f(x)$.
We assume that the map on the level of fomal neighborhoods $X_{x}\rightarrow Y_{y}$ is formally smooth, can we find ...
- 3,472
2
votes
0
answers
141
views
scheme of sections over complete local ring
Let $f:X\rightarrow S= Spec(k[[\pi]])$ a finite type faithfully flat morphism.
Let $U\subset X$ be an open subset such that $U$ is smooth and surjective on $S$.
We consider the $k$-scheme $X(k[[\pi]]...
- 3,472
1
vote
0
answers
128
views
smooth morphism from a finite type source
Let $f: X\rightarrow Y$ a smooth morphism over a field $k$. We assume that $X$ is locally of finite type, does it imply that $Y$ is also locally of finite type?
- 3,472
2
votes
0
answers
250
views
fpqc, formal smoothness
Based on Possible formal smoothness mistake in EGA, let $X$ and $Y$ $k$-schemes ($k$ a field),
let $f:X\rightarrow Y$ a fpqc morphism such that $f$ is formally smooth and $X$ formally smooth, do we ...
- 3,472
2
votes
0
answers
236
views
descent for formally smooth maps
Let $f:X\rightarrow Y$ a morphism between schemes and $Y'\rightarrow Y$ a fpqc morphism
such that the base change $f'$ of $f$ to $Y'$ is formally smooth, does it imply that $f$ is formally smooth?
- 3,472
18
votes
3
answers
2k
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Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
Is there an example of a variety over the complex numbers with no embedding into a smooth variety?
- 10.5k
16
votes
2
answers
3k
views
Is there an example of a formally smooth morphism which is not smooth?
A morphism of schemes is formally smooth and locally of finite presentation iff it is smooth.
What happens if we drop the finitely presented hypothesis? Of course, locally of finite presentation is ...
- 10.5k
9
votes
1
answer
531
views
Can one check formal smoothness using only one-variable Artin rings?
Let $f:X\rightarrow Y$ be a morphism of schemes over a field $k$. Can one check that $f$ is formally smooth using only Artin rings of the form $k^{\prime}\left[t\right]/t^{n}$, where $k^{\prime}$ is ...
- 10.5k