# nLab tensor product of algebras

### Context

#### Algebra

higher algebra

universal algebra

# Contents

## Idea

Let $R$ be a commutative ring. The category of associative algebras over $R$ is the category

${\mathrm{Alg}}_{R}={\mathrm{Ring}}^{R/}$Alg_R = Ring^{R/}

of rings under $R$. If $R$ is a commutative rig, we can do the same with

${\mathrm{Alg}}_{R}={\mathrm{Rig}}^{R/}.$Alg_R = Rig^{R/} .

The tensor product of $R$-algebras has as underlying $R$-module just the tensor product of modules of the underlying modules, $A{\otimes }_{R}B$. On homogeneous elements $\left(a,b\right)\in A×B\stackrel{\otimes }{\to }A{\otimes }_{R}B$ the algebra structure is given by

$\left({a}_{1},{b}_{1}\right)\cdot \left({a}_{2},{b}_{2}\right)=\left({a}_{1}\cdot {a}_{2},{b}_{1}\cdot {b}_{2}\right)\phantom{\rule{thinmathspace}{0ex}}.$(a_1, b_1) \cdot (a_2, b_2) = (a_1 \cdot a_2, b_1 \cdot b_2) \,.

We write also $A{\otimes }_{R}B$ for the tensor product of algebras.

For commutative $R$-algebras, the tensor product is the coproduct in $\mathrm{Comm}{\mathrm{Alg}}_{R}$:

$A{\otimes }_{R}B\simeq A\coprod B\in \mathrm{Comm}{\mathrm{Alg}}_{R}=\mathrm{Comm}{\mathrm{Rig}}^{R/};$A \otimes_R B \simeq A \coprod B \in Comm Alg_R = Comm Rig^{R/} ;

hence the pushout in Comm Ring? (or Comm Rig?)

$\begin{array}{ccc}& & R\\ & ↙& & ↘\\ A& & & & B\\ & ↘& & ↙\\ & & A{\otimes }_{R}B\end{array}\phantom{\rule{thinmathspace}{0ex}}.$\array{ && R \\ & \swarrow && \searrow \\ A &&&& B \\ & \searrow && \swarrow \\ && A \otimes_R B } \,.

## Properties

### Relation to tensor product of categories of modules

For $A$ an associative algebra over a field $k$, write $A$Mod for its category of modules of finite dimension. Then the tensor product of algebras corresponds to the Deligne tensor product of abelian categories $⊠:\mathrm{Ab}×\mathrm{Ab}\to \mathrm{Ab}$:

$\left(A{\otimes }_{k}B\right)\mathrm{Mod}\simeq \left(A\mathrm{Mod}\right)\otimes \left(B\mathrm{Mod}\right)\phantom{\rule{thinmathspace}{0ex}}.$(A \otimes_k B) Mod \simeq (A Mod) \otimes (B Mod) \,.

See at tensor product of abelian categories for more.

Revised on January 17, 2013 01:39:57 by Urs Schreiber (203.116.137.162)