A subscheme of a scheme 
is a scheme 
together with a scheme immersion 
. It is an open
subscheme or closed subscheme when 
is an open immersion or
closed immersion, respectively.
A closed subscheme is determined by an ideal sheaf 
that is a quasi-coherent
sheaf on 
.
On an affine scheme 
, the closed subscheme defined by an ideal 
of 
is 
(Stacks Project 2026). The underlying closed set does
not in general determine the subscheme, because different ideal sheaves can endow
the same closed set with different nilpotent structure.
