Editing Exterior algebra (section)

== Functoriality ==
Suppose that <math>V</math> and <math>W</math> are a pair of vector spaces and <math>f : V \to W</math> is a [[linear map]].  Then, by the universal property, there exists a unique homomorphism of graded algebras
: <math>{\textstyle\bigwedge}(f) : {\textstyle\bigwedge}(V)\rightarrow {\textstyle\bigwedge}(W)</math>
such that
: <math>
{\textstyle\bigwedge}(f)\left|_{{\textstyle\bigwedge}^{\!1}(V)}\right.
= f : V={\textstyle\bigwedge}^{\!1}(V)\rightarrow W={\textstyle\bigwedge}^{\!1}(W).
</math>

In particular, <math>{\textstyle\bigwedge}(f)</math> preserves homogeneous degree.  The {{math|''k''}}-graded components of <math display=inline> \bigwedge\left(f\right) </math> are given on decomposable elements by
: <math>
{\textstyle\bigwedge}(f)(x_1 \wedge \cdots \wedge x_k)
= f(x_1) \wedge \cdots \wedge f(x_k).
</math>

Let
: <math>
{\textstyle\bigwedge}^{\!k}(f)
= {\textstyle\bigwedge} (f)\left|_{{\textstyle\bigwedge}^{\!k}(V)}\right.
: {\textstyle\bigwedge}^{\!k}(V) \rightarrow {\textstyle\bigwedge}^{\!k}(W).
</math>

The components of the transformation {{tmath|{\textstyle\bigwedge}^{\!k}(f)}} relative to a basis of <math>V</math> and <math>W</math> is the matrix of <math>k \times k</math> minors of {{tmath|f}}.  In particular, if <math>V = W</math> and <math>V</math> is of finite dimension {{tmath|n}}, then <math>{\textstyle\bigwedge}^{\!n}(f) </math> is a mapping of a one-dimensional vector space <math>{\textstyle\bigwedge}^{\!n}(V)</math> to itself, and is therefore given by a scalar: the [[determinant]] of {{tmath|f}}.

=== Exactness ===
If <math> 0 \to U \to V \to W \to 0 </math> is a [[short exact sequence]] of vector spaces, then
: <math>
0 \to {\textstyle\bigwedge}^{\!1}(U) \wedge {\textstyle\bigwedge}(V)
\to {\textstyle\bigwedge}(V) \to {\textstyle\bigwedge}(W) \to 0
</math>

is an exact sequence of graded vector spaces,<ref>This part of the statement also holds in greater generality if <math>V</math> and <math>W</math> are modules over a commutative ring: That <math>{\textstyle\bigwedge}</math> converts epimorphisms to epimorphisms.  See {{harvtxt|Bourbaki|1989|loc=Proposition 3, §III.7.2}}.</ref> as is
: <math>0 \to {\textstyle\bigwedge}(U) \to {\textstyle\bigwedge}(V).
</math><ref>This statement generalizes only to the case where {{math|''V''}} and {{math|''W''}} are projective modules over a commutative ring.  Otherwise, it is generally not the case that <math>{\textstyle\bigwedge}</math> converts monomorphisms to monomorphisms.  See {{harvtxt|Bourbaki|1989|loc=Corollary to Proposition 12, §III.7.9}}.</ref>

=== Direct sums ===
In particular, the exterior algebra of a direct sum is isomorphic to the tensor product of the exterior algebras:
: <math>
{\textstyle\bigwedge}(V \oplus W)
\cong {\textstyle\bigwedge}(V) \otimes {\textstyle\bigwedge}(W).
</math>

This is a graded isomorphism; i.e.,
: <math>
{\textstyle\bigwedge}^{\!k}(V \oplus W)
\cong \bigoplus_{p+q=k} {\textstyle\bigwedge}^{\!p}(V) \otimes {\textstyle\bigwedge}^{\!q}(W).
</math>

In greater generality, for a short exact sequence of vector spaces <math display=inline> 0 \to U \mathrel{\overset{f}\to} V \mathrel{\overset{g}\to} W \to 0, </math> there is a natural [[Filtration (mathematics)|filtration]] 
: <math>
0 = F^0 \subseteq F^1 \subseteq \cdots \subseteq F^k \subseteq F^{k+1}
= {\textstyle\bigwedge}^{\!k}(V)
</math>
where <math> F^p </math> for <math> p \geq 1 </math> is spanned by elements of the form <math> u_1 \wedge \ldots \wedge u_{k + 1-p} \wedge v_1 \wedge \ldots v_{p - 1} </math> for <math> u_i \in U </math> and <math> v_i \in V. </math>
The corresponding quotients admit a natural isomorphism
: <math>
F^{p+1}/F^p \cong {\textstyle\bigwedge}^{\!k-p}(U) \otimes {\textstyle\bigwedge}^{\!p}(W)
</math> given by <math>
u_1 \wedge \ldots \wedge u_{k -p} \wedge v_1 \wedge \ldots \wedge v_{p} 
\mapsto u_1 \wedge \ldots \wedge u_{k-p} \otimes g(v_1) \wedge \ldots \wedge g(v_{p}).
</math>

In particular, if ''U'' is 1-dimensional then
: <math>
0 \to U \otimes {\textstyle\bigwedge}^{\!k-1}(W)
\to {\textstyle\bigwedge}^{\!k}(V)
\to {\textstyle\bigwedge}^{\!k}(W) \to 0
</math>
is exact, and if ''W'' is 1-dimensional then
: <math>
0 \to {\textstyle\bigwedge}^k(U)
\to {\textstyle\bigwedge}^{\!k}(V)
\to {\textstyle\bigwedge}^{\!k-1}(U) \otimes W \to 0
</math>
is exact.<ref>Such a filtration also holds for [[vector bundle]]s, and projective modules over a commutative ring.  This is thus more general than the result quoted above for direct sums, since not every short exact sequence splits in other [[abelian category|abelian categories]].</ref>