= Determinant line bundle
{title2=$\det E=\bigwedge^{\operatorname{rank}E}E$}
The top <exterior power> of a rank-$r$ <vector bundle> is its <determinant line bundle>. A <short exact sequence> of <vector bundles> gives a canonical isomorphism of the middle <determinant> with the <tensor product> of the other two. Locally, a subbundle frame followed by lifts of a quotient frame defines it; changing lifts only adds off-diagonal blocks. This is the <determinant> step in <adjunction for a smooth submanifold>.
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