Closed immersion
In algebraic geometry, a closed immersion of schemes is a morphism of schemes formula_1 that identifies "Z" as a closed subset of "X" such that locally, regular functions on "Z" can be extended to "X". The latter condition can be formalized by saying that formula_2 is surjective. An example is the inclusion map formula_3 induced by the canonical map formula_4. Other characterizations. The following are equivalent: Definition for locally ringed spaces. In the case of locally ringed spaces a morphism formula_16 is a closed immersion if a similar list of criterion is satisfied The only varying condition is  the third. It is instructive to look at a counter-example to get a feel for what the third condition yields by looking at a map which is not a closed immersion, formula_23 whereformula_24If we look at the stalk of formula_25 at formula_26 then there are no sections. This implies for any open subscheme formula_27 containing formula_28 the sheaf has no sections. This violates the third condition since at least one open subscheme formula_29 covering formula_30 contains formula_28. Properties. A closed immersion is finite and radicial (universally injective). In particular, a closed immersion is universally closed. A closed immersion is  stable under base change and composition. The notion of a closed immersion is local in the sense that "f" is a closed immersion if and only if for some (equivalently every) open covering formula_32 the induced map formula_33 is a closed immersion. If the composition formula_34 is a closed immersion and formula_35 is separated, then formula_36 is a closed immersion. If "X" is a separated "S"-scheme, then every "S"-section of "X" is a closed immersion. If formula_37 is a closed immersion and formula_38 is the quasi-coherent sheaf of ideals cutting out "Z", then the direct image formula_39 from the  category of quasi-coherent sheaves over "Z" to the category of quasi-coherent sheaves over "X" is exact, fully faithful with the essential image consisting of formula_40 such that formula_41. A flat closed immersion of finite presentation is the open immersion of an open closed subscheme.