Exterior algebra - Knowlpedia

Definition

Let $M$ be a module over a commutative ring $R$ . Its exterior algebra, also called its Grassmann algebra, is

\bigwedge_R M =T_R(M)\big/\langle x\otimes x:x\in M\rangle,

where $T_R(M)$ is the tensor algebra. It is graded as $\bigwedge_R M=\bigoplus_{d\geq0}\bigwedge^d_RM$ , with $M$ in degree one.

Alternating universal property

The map $M^d\to\bigwedge^d_RM$ given by

(x_1,\ldots,x_d)\longmapsto x_1\wedge\cdots\wedge x_d

is alternating and universal among alternating $R$ -multilinear maps out of $M^d$ .

Signs

The defining relation implies

x\wedge y=-y\wedge x

for degree-one elements over every commutative base ring. More generally, homogeneous elements satisfy $u\wedge v=(-1)^{pq}v\wedge u$ for degrees $p,q$ . In characteristic $2$ , graded commutativity alone does not imply $x\wedge x=0$ , so the alternating relation remains part of the definition.

Finite free case

If $M$ is free of rank $n$ with basis $e_1,\ldots,e_n$ , then the products $e_{i_1}\wedge\cdots\wedge e_{i_d}$ with $i_1<\cdots<i_d$ form a basis of $\bigwedge^d_RM$ . Hence $\bigwedge_RM$ has rank $2^n$ .

References

Nicolas Bourbaki, Algebra I: Chapters 1–3, Springer, 1989. DOI record. Relevant: Chapter III, exterior algebras.