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 
.
Conversely, if 
is a closed immersion, its ideal sheaf is the
kernel of the surjection 
(Stacks Project 2026).
Thus quasi-coherent ideal sheaves and closed subschemes
carry the same defining information.
