Closed immersion

In algebraic geometry, a closed immersion of schemes is a morphism of schemes ${\displaystyle f:Z\to X}$ that identifies Z as a closed subset of X such that locally, regular functions on Z can be extended to X.^[1] The latter condition can be formalized by saying that ${\displaystyle f^{\#}:{\mathcal {O}}_{X}\rightarrow f_{\ast }{\mathcal {O}}_{Z}}$ is surjective.^[2]

An example is the inclusion map ${\displaystyle \operatorname {Spec} (R/I)\to \operatorname {Spec} (R)}$ induced by the canonical map ${\displaystyle R\to R/I}$.