Immersion of schemes

A morphism of schemes $f:Z\to X$ is an immersion, also called a locally closed immersion, if there exist an open subscheme $j:U\hookrightarrow X$ (Open immersions of schemes) and a closed immersion $i:Z\to U$ (Closed immersions of schemes) with $f=j\circ i$. A factorization is a device exhibiting $f$, not extra data attached to it: whether $f$ is an immersion is a property of the morphism $f$ alone, and closed immersions ($U=X$) and open immersions (with $i$ an isomorphism onto $U$) are immersions.

For such a factorization the image $f(Z)=j(i(Z))$ is closed in the open subscheme $U$, hence is a locally closed subset of $X$, and $i$ identifies $Z$ with a closed subscheme of $U$; one says that $Z$ carries the induced locally closed subscheme structure on that image. Since $j$ is a homeomorphism onto $U$, the induced structure is unchanged when the ambient open subscheme is replaced by a smaller one containing $f(Z)$, so the property may be checked after restricting the target to any open subscheme containing the image.