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I am trying to understand smooth morphisms of schemes as in https://stacks.math.columbia.edu/tag/01V4

All the definition seems to only take in considerations the source of the morphism, as the stacksprojects says itself, that ``being smooth is local in nature on the source".

I have two questions:

  1. So, if $X$ is a classic smooth variety, then for any scheme $Y$, what are the conditions for a morphism $f: X \to Y$ to be smooth?

  2. Is there an example of smooth morphism $f:X\to Y$ where $X$ is smooth and $Y$ is not smooth?

Thanks in advance.

User43029
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    The intuitive idea is that a smooth morphism is one whose fibres are smooth. There can be morphisms between smooth varieties that are not smooth. – Zhen Lin Apr 29 '23 at 13:50
  • But how about the opposite, could there be a smooth morphism between non-smooth schemes (either source or target)? – User43029 May 03 '23 at 12:17
  • If you have a smooth morphism and the codomain is smooth then the domain is also smooth. You can have a smooth morphism with smooth domain and non-smooth codomain: after all, you can replace the codomain with any scheme containing the original as a disjoint summand. So perhaps the interesting question to ask is what happens when you have a surjective smooth morphism with smooth domain. – Zhen Lin May 03 '23 at 12:36

1 Answers1

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By Stacks, Lemma 063U, if $f: X \rightarrow Y$ is a smooth surjective morphism, with $X$ a smooth variety over a field $k$ (seen as a scheme) and $f$ is a $k$-morphism, then $Y$ is smooth over $k$ at any point in the image of $f$.

This suggests the following counterexample for 2: let $X =\mathbb{A}^1\backslash \{0\}$ (with coordinate $t$) and $Y=\operatorname{Spec}\,k[x,y]/(x^2-y^3)$, and $X \rightarrow Y$ given by $(x,y) \longmapsto (t^3,t^2)$.

$Y$ is smooth at every point except the maximal ideal $(x,y)$, that is, $Y$ is smooth at exactly the points in the image of the morphism $X \rightarrow Y$.

Now, consider a map $f: X \rightarrow Y$ of smooth schemes over a field $k$. Let $x \in X$ be a point. Then the following are equivalent:

  1. $f$ is smooth at $x$
  2. the natural map $\Omega^1_{Y/k,f(x)} \_{O_{Y,f(x)}} O_{X,f(x)} \rightarrow \Omega^1_{X/k,x}$ (where $\Omega^1$ denotes the sheaf of relative differentials) is an isomorphism onto a direct factor.
  3. $\Omega^1_{Y/k} \otimes_{O_{Y,f(x)}} \kappa_X(x)\rightarrow \Omega^1_{X/k,x} \otimes_{O_{X,f(x)}} \kappa_X(x)$ is injective, where $\kappa$ denotes the residue field at a point.
  4. The tangent map to $f$ at $x$ is surjective.

(See Neron models by Bosch, Lutkebohmert and Raynaud, Chapter 2.3, Prop 8 – I added point 4, it is pretty elementary to derive and seems the most intuitive).

Aphelli
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