Closed embeddings of manifold/varieties

In Italian, varietà means both manifold and variety. Here is one of the many theorems that exists for both, and which my coauthor and I have used recently in a work in progress.

Theorem We have that f:X → Y  is a closed embedding if and only if it satisfies all the following conditions:
  1. f is injective on points
  2. f is injective on tangent vectors
  3. f is closed (as a map of topological spaces)

Algebraic version (AV): f is a morphism between schemes of finite type over an algebraically closed field k, we only consider k points. 

Differential version (DV): f is a smooth (holomorphic) map between differentiable (complex) manifolds (all manifolds are without boundary). We write k for R or C, respectively.

A closed embedding means an isomorphism to a closed subscheme AV or submanifold DV of Y. 

Note that the conditions indicated are clearly necessary (only if). 

Proof AV Let Z be the schematic closure of the image. Since f is closed, f:X → Z is surjective; by 1. it is bijective, and since it's closed it's a homeomorphism. Thus there is an induced homomorphism of sheaves of algebras \(O_Z\to f_*O_X\) which by definition of schematic closure is injective.
It remains to prove it's surjective, equivalent to prove surjectivity for \(O_Y\to f_*O_X\). This means for all p in X, \(O_{Y,q} \to O_{X,p}\) is surjective where q:=f(p); equivalently, that \(m_q \to m_p\) is surjective (since \(O_{X,p}=k\oplus m_p\)); by Nakayama, that \(m_q/m_q^2 \to m_p/m_p^2\) is surjective, which means by dualizing that \(T_pX \to T_qY\) is injective.

Proof DV Let Z be the image of X; as before Z is closed in Y and \(f:X \to Z\) is a homeomorphism. It is enough to show Z is a manifold of the same dimension of X, since condition 2 then implies that f is locally a diffeomorphism. By assumption, f is locally a closed embedding of manifolds, that is we can cover X by opens U such that there exists a V in Y open with \(f:U \to V\) a closed embedding.
Let X' be the complement of U in X; it is closed, so its image Z' in Y is closed, thus Z' is closed in Z. Let \(W:=V\cap Z\). Since f is bijective, the inverse image of Z' is X', so the inverse image of W is U, and \(f:U \to W\) is a diffeomorphism. By assumption W is a closed submanifold of V. We can cover Y by such opens V, plus the complement of Z; being a closed submanifold is local, this concludes the proof.

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