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)(adsbygoogle = window.adsbygoogle || []).push({});

There exists an exact sequence on a nonsingular n-fold X with codimension 1 subvariety Y:

[tex]0\to\mathcal{I}_Y\to\mathcal{O}_X\to\mathcal{O}_Y\to 0 [/tex]

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

[tex]0\to\mathcal{O}_X\to\mathcal{O}_X(Y)\to\mathcal{O}_Y(Y)\to 0 [/tex]

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!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Adjuction Formula: Ideal sheafs and tensor products

Can you offer guidance or do you also need help?

**Physics Forums | Science Articles, Homework Help, Discussion**