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Symmetric algebra

April 19, 2012

In this post, I will introduce yet another “universal” construction over an R-module for R 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 R-module M is the quotient of the tensor algebra T(M) by the ideal C(M) generated by the elements of the form m_1 \otimes m_2 - m_2 \otimes m_1, for all m_1, m_2 \in M. It is denoted by S(M).

Let’s see what we got here. Denoting by C^k(M) the set of all finite linear combinations of elements in T^k(M) of the form

m_1 \otimes \cdots \otimes m_{i-1} \otimes (m_i \otimes m_{i+1} - m_{i+1} \otimes m_i) \otimes m_{i+2} \otimes \cdots \otimes m_k,

we have C(M) = \bigoplus_{k=2}^{\infty}C^k(M). Notice that by expanding this expression, we see that C^k(M) is generated by the elements of T^k(M) 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 T(M) by C(M) result in giving us the right to swap two adjacent entries, ie:

m_1 \otimes \cdots m_i \otimes m_{i+1} \otimes \cdots \otimes m_k \equiv m_1 \otimes \cdots \otimes m_{i+1} \otimes m_i \otimes \cdots \otimes m_k

(modulo C(M)). But since for any k, adjacent transpositions in S_k generate all of the permutation group S_k, tensors in T(M) which can be obtained from each others by permuting their entries are all identified in S(M). In other words, the k’th symmetric power T^k(M)/C^k(M), denoted S^k(M), is equal to T^k(M) modulo the submodule generated by all elements of the form

m_1 \otimes m_2 \otimes \cdots \otimes m_k - m_{\sigma(1)} \otimes m_{\sigma(2)} \otimes \cdots \otimes m_{\sigma(k)}

for all m_i \in M and all \sigma \in S_k.

In particular, S(M) 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 \varphi : M^k \to N is symmetric if \varphi(m_1, \ldots, m_k) = \varphi(m_{\sigma(1)}, \ldots \varphi(m_{\sigma(k)}.

Theorem:

  1. (Universal Property for Symmetric Multilinear Maps) Let \varphi : M^k \to N be a symmetric k-multilinear map with respect to R, then there is a unique R-module homomorphism \phi : S^k(M) \to N such that \varphi = \phi \circ \iota where \iota is the canonical inclusion map  from M^k to T^k(M) modulo C(M).
  2. (Universal Property for maps to commutative R-algebras) Let A be a commutative R-algebra and \varphi : M \to A an R-module homomorphism, then there is a unique R-algebra homomorphism \phi : S(M) \to A such that \phi|_M = \varphi.

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 \tilde{\phi}: T^k(M) \to N such that

\tilde{\phi}(m_1 \otimes \cdots \otimes m_k) = \varphi(m_1, \ldots, m_k).

But then \tilde{\phi} is symmetric thus C^k(M) is contained in the kernel of \tilde{\phi}. Hence, the R-module homomorphism \phi : S^k(M) \to N defined by

\phi (m_1 \otimes \cdots \otimes m_k \quad \text{mod }C(M)) = \tilde{\phi}(m_1 \otimes \cdots \otimes m_k)

is well defined and has the desired properties.

For 2, we proceed as in the proof linked above. For \varphi : M \to A, we define \phi : S(M) \to A by

 \phi(m_1 \otimes \cdots \otimes m_k \quad \text{mod }C(M)) = \varphi(m_1) \cdots \varphi(m_k)

which is well-defined since A is commutative.

QED

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

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