Closed immersion
The following are equivalent:
#formula_1 is a closed immersion. #For every open affine formula_6, there exists an ideal formula_7 such that formula_8 as schemes over "U". #There exists an open affine covering formula_9 and for each "j" there exists an ideal formula_10 such that formula_11 as schemes over formula_12.
#There is a quasi-coherent sheaf of ideals formula_13 on "X" such that formula_14 and "f" is an isomorphism of "Z" onto the global Spec of formula_15 over "X". In the case of locally ringed spaces a morphism formula_16 is a closed immersion if a similar list of criterion is satisfied
# The map formula_17 is a homeomorphism of formula_18 onto its image
# The associated sheaf map formula_19 is surjective with kernel formula_13
# The kernel formula_13 is locally generated by sections as an formula_22-module
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. 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.
Other characterizations
The latter condition can be formalized by saying that formula_2 is surjective. An example is the inclusion map