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Let $f:X\to Y$ be a morphism of locally ringed spaces. In this MSE answer, the first definition below is suggested.

  1. Say $f:X\to Y$ is an $R$-immersion of locally ringed spaces if it's a topological embedding and for all $x\in X$, the canonical homomorphism $O_{Y, f(x)}\to O_{X,x}$ is surjective.

  2. Görtz & Wedhorn define $f$ to be an immersion (of schemes, but still) if's a topological embedding with locally closed image and for all $x\in X$, the canonical homomorphism $O_{Y, f(x)}\to O_{X,x}$ is surjective.

  3. Stacks defines $f$ to be an immersion if it's locally closed, i.e a closed immersion followed by an open immersion.

Following the stacks project's definition of immersion (3), the linked answer, this comment, and this comment, make the following assertions.

  1. An $R$-immersion is an immersion iff it's locally of finite type.
  2. $R$-immersions are stable under base change.
  3. An $R$-immersion is an immersion iff its image is locally closed.

Question. The proof of the second assertion is said to be 'quite long', but what about the other two? How to prove them?

I ask because in the $C^p$ category, the fact $C^p$ embeddings have locally closed image seems to follow from the local structure of constant rank maps, and for nice schemes I thought this follows from ZMT.

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