LEC ALG4 13

The tensor algebra

Recall what adjoint functors are; we’ve seen that tensor and $\text{Hom}$ are adjoint.

Example 419.1(Example).

Here’s another example: Let $\text{D}=R\text{-Mod}$, $\text{C}=\text{Set}$. Let $\mathcal{G}:\text{D}\to \text{C}$ be the forgetful functor. Then, $\mathcal{G}$ has a left adjoint, namely the free functor $\mathcal{F}:\text{C}\to \text{D}$ defined by $S\mapsto F(S)$, where $F^R(S)$ is the free module on $S$.

$${\text{Hom}}_{R\text{-Mod}}(F^R(S),M)\cong {\text{Hom}}_{\text{Set}}(S,M).$$

Now, consider, $R\text{-AAlg}$, the category of associative $R$-algebras. It is a subcategory of $R\text{-Mod}$. Suppose that $A\in R\text{-AAlg}$, $M\in R\text{-Mod}$. Does there exist a left adjoint $T:R\text{-Mod}\to R\text{-AAlg}$ to the forgetful functor $R\text{-AAlg}\to R\text{-Mod}$, so that

$${\text{Hom}}_{R\text{-AAlg}}(T(M),A)\cong {\text{Hom}}_{R\text{-Mod}}(M,A)?$$

Definition 419.2(Tensor algebra).

Define

$$T^n(M):=\{\begin{array}{ll}R& n=0\\ M& n=1\\ \underset{n\text{ fold}}{\underbrace{M\otimes \cdots \otimes M}}& n⩾2\end{array}$$ $$T(M)=\underset{n⩾0}{⨁}{T}^n(M).$$

Proposition 419.3(Proposition).

With addition inherited from the $R$-module structure of $T^n(M)$ for $n⩾0$ and multiplication defined by

$$\underset{\in T^n(M)}{\underbrace{(x_1\otimes \cdots \otimes x_n)}}\cdot \underset{\in T^m(M)}{\underbrace{(x_{n+1}\otimes \cdots \otimes x_{n+m})}}=\underset{\in T^{n+m}(M)}{\underbrace{x_1\otimes \cdots \otimes x_{n+m}}}$$

and extended $R$-linearly, $T(M)$ is an associative $R$-algebra with structure map $R=T^0(M)\subseteq T(M)$.

Proposition 419.4(Universal property of tensor algebras).

Let $A\in R\text{-AAlg}$, $M\in R\text{-Mod}$, $\phi \in {\text{Hom}}_{R\text{-Mod}}(M,A)$. Then there exists unique $\Phi \in {\text{Hom}}_{R\text{-AAlg}}(T(M),A)$ such that the following diagram commutes: