Locally closed immersion

A locally closed immersion factors as a closed immersion followed by an open immersion. Equivalently, it identifies its source with a closed subscheme of an open subscheme of the target.

An open immersion is a morphism of schemes that identifies its source with an open subscheme of its target. It is an isomorphism onto that open subscheme as a locally ringed space.

The diagonal morphism sends a point to the pair consisting of that point twice. Every diagonal of schemes is a locally closed immersion, and a morphism is separated exactly when its diagonal is a closed immersion.

For morphisms $f,g:X\to Y$ over a base $S$, their scheme-theoretic coincidence locus is the fibre productIt is the equalizer of $f$ and $g$, hence is universal among subschemes on which they agree. It is locally closed because every diagonal is an immersion, and closed when $Y\to S$ is separated.