Editing Exterior algebra (section)

=== Generalizations ===
Given a [[commutative ring]] <math>R</math> and an <math>R</math>-[[module (mathematics)|module]] {{tmath|M}}, we can define the exterior algebra <math>{\textstyle\bigwedge}(M)</math> just as above, as a suitable quotient of the tensor algebra {{tmath|\mathrm{T}(M)}}.  It will satisfy the analogous universal property.  Many of the properties of <math>{\textstyle\bigwedge}(M)</math> also require that <math>M</math> be a [[projective module]].  Where finite dimensionality is used, the properties further require that <math>M</math> be [[finitely generated module|finitely generated]] and projective.  Generalizations to the most common situations can be found in {{harvtxt|Bourbaki|1989}}.

Exterior algebras of [[vector bundle]]s are frequently considered in geometry and topology.  There are no essential differences between the algebraic properties of the exterior algebra of finite-dimensional vector bundles and those of the exterior algebra of finitely generated projective modules, by the [[Serre–Swan theorem]].  More general exterior algebras can be defined for [[sheaf (mathematics)|sheaves]] of modules.