The graded exterior algebra $\mathrm{\Lambda }V$

The exterior algebra of $V$ is the graded vector space

$\mathrm{\Lambda }V:={⨁}_{k\ge 0}{\mathrm{\Lambda }}^{k}V={\mathrm{\Lambda }}^{0}V\oplus {\mathrm{\Lambda }}^{1}V\oplus {\mathrm{\Lambda }}^{2}V\oplus \cdots ,$

whose homogeneous piece of degree $k$ is the exterior power of The $k$th exterior power as the tensor-power quotient by repeated-vector relations. On decomposables of Decomposable $k$-vectors and the basic wedge product define the wedge product

$(v_1\wedge \cdots \wedge v_k)\wedge (w_1\wedge \cdots \wedge w_{\mathrm{\ell }}):=v_1\wedge \cdots \wedge v_k\wedge w_1\wedge \cdots \wedge w_{\mathrm{\ell }}\in {\mathrm{\Lambda }}^{k+\mathrm{\ell }}V,$

and extend bilinearly to all of ${\mathrm{\Lambda }}^{k}V\times {\mathrm{\Lambda }}^{\mathrm{\ell }}V$. The product of two homogeneous elements is homogeneous of the sum of the degrees, so $\mathrm{\Lambda }V$ is a graded algebra with unit $1\in F={\mathrm{\Lambda }}^{0}V$.

The product is well defined. Each generator $v_1\otimes \cdots \otimes v_k$ of $W_k$ contains a repeated pair, so its concatenation with any pure tensor of degree $\mathrm{\ell }$ lies in $W_{k+\mathrm{\ell }}$. By linearity, concatenation sends $W_k\times V^{\otimes \mathrm{\ell }}$ and $V^{\otimes k}\times W_{\mathrm{\ell }}$ into $W_{k+\mathrm{\ell }}$. It therefore descends to a bilinear map on the two quotient spaces, and that descended map is exactly the displayed wedge product. Associativity and the graded commutation law are proved in Exterior multiplication is well defined, graded, associative, unital, and graded-commutative.