September | 2018 | Lovely little lemmas

An immersion (or locally closed immersion) of schemes is a morphism $f \colon X \to Y$ that can be factored as $X \to U \to Y$ , where $X \to U$ is a closed immersion and $U \to Y$ is an open immersion. If it is moreover an open morphism, it need not be an open immersion:

Example. Let $X$ be a nonreduced scheme, and let $X_{\text{red}} \to X$ be the reduction. This is a closed immersion, whose underlying set is the entire space. Thus, it is a homeomorphism, hence an open morphism. It is not an open immersion, for that would force it to be an isomorphism. $\qedsymbol$

Remark. However, every closed immersion is a closed immersion; see Tag 01IQ.

Lovely little lemmas

Or maybe ‘amusing observations’

Monthly Archives: September 2018

Not every open immersion is an open immersion