nLab
tensor product of algebras

Idea

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

$$\mathsf{Alg}_R = {R \downarrow \mathsf{Ring}}$$

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

$$\mathsf{Alg}_R = {R \downarrow \mathsf{Rig}} .$$

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 $(a,b) \in A \times B \stackrel{\otimes}{\to} A \otimes_R B$ the algebra structure is given by

$$(a_1, b_1) \cdot (a_2, b_2) = (a_1 \cdot a_2, b_1 \cdot b_2) \,.$$

We write also $A \otimes_R B$ for the tensor product of algebras.

For commutative $R$-algebras, the tensor product is the coproduct in $Comm Alg_R$:

$$A \otimes_R B \simeq A \coprod B \in Comm Alg_R = Comm R \downarrow \mathsf{Rig} ;$$

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

$$\array{ && R \\ & \swarrow && \searrow \\ A &&&& B \\ & \searrow && \swarrow \\ && A \otimes_R B } \,.$$

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 $\boxtimes \colon Ab \times Ab \to Ab$:

$$(A \otimes_k B) Mod \simeq (A Mod) \otimes (B Mod) \,.$$

See at tensor product of abelian categories for more.

Related concepts

• tensor product

• Hochschild cohomology