Editing Line bundle (section)

==Determinant bundles==
{{See also|Quillen metric#Determinant line bundle of a family of operators}}
In general if <math>V</math> is a vector bundle on a space <math>X</math>, with constant fibre dimension <math>n</math>, the <math>n</math>-th [[exterior power]] of <math>V</math> taken fibre-by-fibre is a line bundle, called the '''determinant line bundle'''. This construction is in particular applied to the [[cotangent bundle]] of a [[smooth manifold]]. The resulting determinant bundle is responsible for the phenomenon of [[tensor density|tensor densities]], in the sense that for an [[orientable manifold]] it has a nonvanishing global section, and its tensor powers with any real exponent may be defined and used to 'twist' any vector bundle by [[tensor product]].

The same construction (taking the top exterior power) applies to a [[finitely generated module|finitely generated]] [[projective module]] <math>M</math> over a Noetherian domain and the resulting invertible module is called the '''determinant module''' of <math>M</math>.