Originally Posted by

**cduston** Hey everyone, I'm studying the Adjunction formula and part of the proof is confusing me. I have two exact sequences on a subvariety Y of codimension 1 in variety X:

$\displaystyle 0\to\mathcal{I}_Y\to\mathcal{O}_X\to\mathcal{O}_Y\ to 0 $

where the $\displaystyle \mathcal{O}_X$ are structure sheafs and (I think) $\displaystyle \mathcal{O}_{X}(Y)$ are divisorial sheafs. We also have

$\displaystyle 0\to T_Y \to T_{X}\mid Y \to N_{X\mid Y}\to 0$

Where these are tangent and normal bundles. The next line says "Taking determinants gives:"

$\displaystyle det T_X \mid Y=det T_Y\otimes N_{X\mid Y}$

which is the Adjunction formula. So can someone give me some insight into how one thinks about taking the determinants of something like an exact sequence?

Thanks!