# Symmetric algebra

In this post, I will introduce yet another “universal” construction over an -module for a commutative ring. Recall that the tensor algebra can be seen as the free algebra over a given module. It is therefore natural to expect important constructions to be given by quotients of this algebra. Here is a first example : the symmetric algebra.

**Definition:** The symmetric algebra of an -module is the quotient of the tensor algebra by the ideal generated by the elements of the form for all It is denoted by

Let’s see what we got here. Denoting by the set of all finite linear combinations of elements in of the form

we have Notice that by expanding this expression, we see that is generated by the elements of which are the difference of two k-tensors that differ only by a transposition of two adjacent entries (the i’th and (i+1)’th entries are swapped). Therefore, quotienting by result in giving us the right to swap two adjacent entries, ie:

(modulo But since for any adjacent transpositions in generate all of the permutation group tensors in which can be obtained from each others by permuting their entries are all identified in In other words, the k’th symmetric power denoted is equal to modulo the submodule generated by all elements of the form

for all and all

In particular, is a commutative algebra. In fact, it satisfies the universal property with respect to maps from a module to a commutative algebra. It is also universal with respect to symmetric multilinear maps. Recall that a k-multilinear map is symmetric if

Theorem:

- (
Universal Property for Symmetric Multilinear Maps)Let be a symmetric k-multilinear map with respect to then there is a unique -module homomorphism such that where is the canonical inclusion map from to modulo(Universal Property for maps to commutative -algebras)Let be a commutative -algebra and an -module homomorphism, then there is a unique -algebra homomorphism such that

**Proof:** The proof of both assertions is very similar to the proof of the universal property of tensor algebras. For 1, the map is multilinear so it induces a unique homomorphism such that

But then is symmetric thus is contained in the kernel of Hence, the -module homomorphism defined by

is well defined and has the desired properties.

For 2, we proceed as in the proof linked above. For we define by

which is well-defined since is commutative.

QED

**Reference:** D. S. Dummit, R. M. Foote, Abstract Algebra, 3rd ed.

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