# Adjuction Formula: Ideal sheafs and tensor products

1. Sep 30, 2008

### cduston

Hey everyone, I'm a physicist trying to learn some topology. I'm now working on figuring out the Adjunction theorem, and I barely understand the proof. Here are the first two (and almost only steps)

There exists an exact sequence on a nonsingular n-fold X with codimension 1 subvariety Y:
$$0\to\mathcal{I}_Y\to\mathcal{O}_X\to\mathcal{O}_Y\to 0$$

where I_Y is the ideal sheaf defining Y and O_{X,Y} is the structure sheaf (I think) of X and Y. Tensoring this equation with O_{X}(Y) gives

$$0\to\mathcal{O}_X\to\mathcal{O}_X(Y)\to\mathcal{O}_Y(Y)\to 0$$

Which is the first part of the proof. So I understand structure sheafs as "The set of all regular functions on X", with transition functions similar to bundles. So I would understand the first exact sequence if i just knew what an IDEAL sheaf is. So the first question is what is an ideal sheaf?

The second question is how to take the tensor product of the sequence. I might be able to actually work it out if someone could possibly help explain what O_{X}(Y) might be...I think it's called the divisorial sheaf?

Any help appreciated, even if it's incomplete. Thanks!

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