Definition:Closed Subscheme defined by Sheaf of Ideals

Definition
Let $\struct {X, \OO_X}$ be a scheme.

Let $\II$ be a quasi-coherent sheaf of ideals of $\OO_X$.

Let $Y$ be the closed subset of $X$ defined by $\II$.

Let $\OO_Y := \OO_X / \II$ be the quotient sheaf of rings of $\OO_Y$ by $\II$.

Then the closed subscheme of $\struct {X, \OO_X}$ defined by $\II$ is:
 * $\struct {Y, \OO_Y}$

Also see

 * Closed Subscheme defined by Sheaf of Ideals is Scheme
 * Closed Subscheme defined by Sheaf of Ideals is Closed Subscheme