Thread: Determinant of an exact sequence sheafs

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:



where the are structure sheafs and (I think) are divisorial sheafs. We also have



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



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!

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:



where the are structure sheafs and (I think) are divisorial sheafs. We also have



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



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!

determinant here probably means the top exterior power, i.e. if is a vector space of dimension then: the reason that i think this might be the case is that

we know that if we have an exact sequence of vector spaces where then see if this helps!