Editing Exterior algebra (section)

=== Graded structure ===
The exterior product of a {{math|''k''}}-vector with a {{math|''p''}}-vector is a <math>(k + p)</math>-vector, once again invoking bilinearity.  As a consequence, the direct sum decomposition of the preceding section
: <math>
{\textstyle\bigwedge}(V) = {\textstyle\bigwedge}^{\!0}(V) \oplus {\textstyle\bigwedge}^{\!1}(V) \oplus {\textstyle\bigwedge}^{\!2}(V) \oplus \cdots \oplus {\textstyle\bigwedge}^{\!n}(V)
</math>
gives the exterior algebra the additional structure of a [[graded algebra]], that is
: <math>
{\textstyle\bigwedge}^{\!k}(V)
\wedge {\textstyle\bigwedge}^{\!p}(V)
\sub {\textstyle\bigwedge}^{\!k+p}(V).
</math>

Moreover, if {{math|''K''}} is the base field, we have
: <math>
{\textstyle\bigwedge}^{\!0}(V) = K
</math> and <math>
{\textstyle\bigwedge}^{\!1}(V) = V.
</math>

The exterior product is graded anticommutative, meaning that if <math>\alpha \in {\textstyle\bigwedge}^{\!k}(V) </math> and {{tmath|\beta \in {\textstyle\bigwedge}^{\!p}(V)}}, then
: <math> \alpha \wedge \beta = (-1)^{kp}\beta \wedge \alpha. </math>

In addition to studying the graded structure on the exterior algebra, {{harvtxt|Bourbaki|1989}} studies additional graded structures on exterior algebras, such as those on the exterior algebra of a [[graded module]] (a module that already carries its own gradation).