An ideal sheaf on a scheme 
is a subsheaf 
of the structure sheaf 
such that 
is an ideal of 
for every open set 
of 
.
A quasi-coherent sheaf of ideals determines
a closed subscheme 
of 
with structure sheaf 
. On an affine open set 
, quasi-coherence means that 
comes from an ideal 
of 
,
and the corresponding closed subscheme is 
.
Conversely, if 
is a closed immersion, its ideal sheaf is the
kernel sheaf of the surjection 
(Stacks Project 2026).
Thus quasi-coherent ideal sheaves and closed subschemes
carry the same defining information.
