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Let $X$ be a smooth scheme over a (perfect if you want) field $k$. Let $Y$ be a closed subscheme of $X$ that is also smooth over $k$. Is the canonical closed immersion $i:Y \to X$ a smooth morphism?

GMRA
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    In general no, because smooth morphisms are flat. – loch Sep 27 '18 at 17:45
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    I would even say always no, except for embeddings of connected components. – Sasha Sep 27 '18 at 17:50
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    @sasha That is correct. A way overkill result is the fact that a universally injective etale map is automatically an open embedding (evidently if the embedding is smooth its of relative dimension 0). Much more concretely, since everything is reduced we may as well identify the closed subscheme with its underlying space $Z$. Then, since $i$ is smooth, and thus open, $Z$ is clopen. Every reduced closed subscheme with underlying clopen set is automatically an open subscheme. Since clopens are unions of components, the claim follows. – Alex Youcis Sep 27 '18 at 20:09
  • Ah yes! Thank you! This is clear to me now. – GMRA Sep 27 '18 at 21:31

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