Editing Exterior algebra (section)

=== Universal property ===
Let {{math|''V''}} be a vector space over the field {{math|''K''}}.  Informally, multiplication in <math> {\textstyle\bigwedge}(V) </math> is performed by manipulating symbols and imposing a [[distributive law]], an [[associative law]], and using the identity <math> v \wedge v = 0 </math> for {{math|''v'' ∈ ''V''}}.  Formally, <math> {\textstyle\bigwedge}(V) </math> is the "most general" algebra in which these rules hold for the multiplication, in the sense that any unital associative {{math|''K''}}-algebra containing {{math|''V''}} with alternating multiplication on {{math|''V''}} must contain a homomorphic image of {{tmath|{\textstyle\bigwedge}(V)}}.  In other words, the exterior algebra has the following [[universal property]]:<ref>See {{harvtxt|Bourbaki|1989|loc=§III.7.1}}, and {{harvtxt|Mac Lane|Birkhoff|1999|loc=Theorem XVI.6.8}}.  More detail on universal properties in general can be found in {{harvtxt|Mac Lane|Birkhoff|1999|loc=Chapter VI}}, and throughout the works of Bourbaki.</ref>

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Given any unital associative {{math|''K''}}-algebra {{math|''A''}} and any {{math|''K''}}-[[linear map]] <math> j : V \to A </math> such that <math> j(v)j(v) = 0 </math> for every {{math|''v''}} in {{math|''V''}}, then there exists ''precisely one'' unital [[algebra homomorphism]] <math>f : {\textstyle\bigwedge}(V)\to A </math> such that {{math|1=''j''(''v'') = ''f''(''i''(''v''))}} for all {{math|''v''}} in {{math|''V''}} (here {{math|''i''}} is the natural inclusion of {{math|''V''}} in {{tmath|{\textstyle\bigwedge}(V)}}, see above).
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[[File:ExteriorAlgebra-01.svg|center|150px|Universal property of the exterior algebra]]

To construct the most general algebra that contains {{math|''V''}} and whose multiplication is alternating on {{math|''V''}}, it is natural to start with the most general associative algebra that contains {{math|''V''}}, the [[tensor algebra]] {{math|''T''(''V'')}}, and then enforce the alternating property by taking a suitable [[quotient ring|quotient]].  We thus take the two-sided [[ideal (ring theory)|ideal]] {{math|''I''}} in {{math|''T''(''V'')}} generated by all elements of the form {{math|''v'' ⊗ ''v''}} for {{math|''v''}} in {{math|''V''}}, and define <math> {\textstyle\bigwedge}(V) </math> as the quotient
: <math> {\textstyle\bigwedge}(V) = T(V)\,/\,I</math>
(and use {{math|∧}} as the symbol for multiplication in {{tmath|{\textstyle\bigwedge}(V)}}).  It is then straightforward to show that <math>{\textstyle\bigwedge}(V)</math> contains {{math|''V''}} and satisfies the above universal property.

As a consequence of this construction, the operation of assigning to a vector space {{math|''V''}} its exterior algebra <math>{\textstyle\bigwedge}(V)</math> is a [[functor]] from the [[category (mathematics)|category]] of vector spaces to the category of algebras.

Rather than defining <math>{\textstyle\bigwedge}(V)</math> first and then identifying the exterior powers <math>{\textstyle\bigwedge}^{\!k}(V)</math> as certain subspaces, one may alternatively define the spaces <math>{\textstyle\bigwedge}^{\!k}(V)</math> first and then combine them to form the algebra {{tmath|{\textstyle\bigwedge}(V)}}.  This approach is often used in differential geometry and is described in the next section.