Introduction to tensor 2: Tensor Algebra

Let $A$$A$ be a ring, the centre of $A$$A$ is $Z(A)$$Z(A)$. The homomorphism $\varphi :R\to Z(A)$$\phi:R\to Z(A)$ define a $R-$$R-$ algebra over $A$$A$.

$A$$A$ is both ring and $R$$R$- module.

You can use $\mu :A\otimes A\to A$$\mu:A\otimes A\to A$ define the multiplication.

Tensor Algebra

Let $A$$A$ be a commutative ring, $M$$M$ be a $A-$$A-$ module. For $r\ge 0$$r\ge 0$, let

$$\begin{array}{}\text{(1)}& T^r(M):=M^{\otimes r}\end{array}$$

be the $r$$r$-$th$$th$ tensor power of $M$$M$. Then $T^0(M)=A$$T^0(M)=A$ and $T^1(M)=M$$T^1(M)=M$.

Proposition.1.1 $T^r:A-\mathrm{Mod}\to A-\mathrm{Mod}$$T^r:A-\mathrm{Mod}\to A-\mathrm{Mod}$ is a functor.

Proof. For the morphism, if $u:M\to N$$u:M\to N$ is a module homomorphism, then $T^r(u)(m_1\otimes ...\otimes m_n)=u(m_1)\otimes ...\otimes u(m_n)$$T^r(u)(m_1\otimes...\otimes m_n)=u(m_1)\otimes...\otimes u(m_n)$.

Define the tensor algebra of $M$$M$ be

$$\begin{array}{}\text{(2)}& T(M):\underset{r\ge 0}{⨁}{T}^r(M)\end{array}$$

Define the $A-$$A-$the algebra structure as follows.

Step 1. The ring structure over $T(M)$$T(M)$:

For $m=m_1\otimes ...\otimes m_r\in T^r(M)$$m=m_1\otimes...\otimes m_r\in T^r(M)$ and $n=n_1\otimes ...\otimes n_s\in T^s(M)$$n=n_1\otimes...\otimes n_s\in T^s(M)$

$$\begin{array}{}\text{(3)}& m\otimes n:=m_1\otimes ...\otimes m_r\otimes n_1\otimes ...\otimes n_s\in T^{r+s}(M)\end{array}$$

Step 2. $\iota :A\to T^0(M)$$\iota:A\to T^0(M)$.

Proposition 2.2. $T:A-\mathrm{Mod}\to A-\mathrm{Alg}$$T:A-\mathrm{Mod}\to A-\mathrm{Alg}$ is a functor.

Proof. For the morphism

$$\begin{array}{}\text{(4)}& T(u):=⨁T^r(u)\end{array}$$

Proposition 2.3. Tensor algebra is the left adjoint of forget functor.

Let $N$$N$ be a $R-$$R-$Algebra

$$\begin{array}{}\text{(5)}& {\mathrm{Hom}}_{R-\mathrm{Alg}}(T(M),N)\cong {\mathrm{Hom}}_{R-\mathrm{Mod}}(M,N)\end{array}$$

We already see that for $u\in {\mathrm{Hom}}_{R-\mathrm{Mod}}(M,N)$$u\in \mathrm{Hom}_{R-\mathrm{Mod}}(M,N)$, $\alpha :u⟼T(u)\in {\mathrm{Hom}}_{R-\mathrm{Alg}}(T(M),N)$$\alpha:u\longmapsto T(u)\in \mathrm{Hom}_{R-\mathrm{Alg}}(T(M),N)$ .

For $\psi \in {\mathrm{Hom}}_{R-\mathrm{Alg}}(T(M),N)$$\psi\in \mathrm{Hom}_{R-\mathrm{Alg}}(T(M),N)$, let ${\iota }^{\prime }:M\to T^1(M)$$\iota':M\to T^1(M)$, ${\iota }_{\ast }^{\prime }(\psi )=\psi \circ {\iota }^{\prime }\in {\mathrm{Hom}}_{R-\mathrm{Mod}}(M,N)$$\iota'_*(\psi)=\psi\circ\iota'\in \mathrm{Hom}_{R-\mathrm{Mod}}(M,N)$. Easy to see their pair of inverse.