Symmetric algebra - Knowlpedia

Definition

Let $M$ be a module over a commutative ring $R$ . Its symmetric algebra is

\operatorname{Sym}_R(M) =T_R(M)\big/\langle x\otimes y-y\otimes x:x,y\in M\rangle,

where $T_R(M)$ is the tensor algebra. It is a commutative $R$ -algebra graded by tensor degree, and its degree-one part receives the canonical map $M\to\operatorname{Sym}_R(M)$ .

Universal property

For every commutative unital $R$ -algebra $A$ and every $R$ -linear map $f:M\to A$ , there is a unique unital $R$ -algebra homomorphism

\widetilde f:\operatorname{Sym}_R(M)\longrightarrow A

extending $f$ . This characterizes $\operatorname{Sym}_R(M)$ as the free commutative $R$ -algebra on $M$ .

Examples

If $M$ is free with basis $x_1,\ldots,x_n$ , then

\operatorname{Sym}_R(M)\cong R[x_1,\ldots,x_n].

For a vector space $V$ , the degree- $d$ summand is its $d$ th symmetric power $\operatorname{Sym}^d(V)$ .

References

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