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 give structure sheaves with different
nilpotent elements on the same closed set.
