Inverse image ideal sheaf, blow-up, and exceptional divisor

If we have morphism , and a sheaf of ideal on , then the inverse image sheaf is therefore a subsheaf of , whose action on makes it an algebra. Thus also acts on , and thus we can define the inverse image ideal sheaf of by .
This sheaf of ideals is exactly the image of the pull back under the natural morphism :

Now let’s look at the situation where we blow up a scheme along a coherent sheaf of ideals . By definition

We shall see that the inverse image ideal sheaf of is exactly the canonical sheaf , hence is invertible, therefore determines a Cartier divisor on , which we would call the exceptional divisor.

Now we can look at the local picture. Locally over , the blow up along an ideal is , and the canonical invertible sheaf corresponds to the homogenous ideal ( graded module ) , which is nothing but the homogenous ideal generated by in . Thus the canonical sheaf coincides with the inverse image ideal sheaf of in .
This ideal sheaf defines a Cartier divisor, which is called exceptional divisor. This is the same as the inverse image scheme where is the closed subscheme of defined by .