nLab tensor product of algebras




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

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

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

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

The tensor product of RR-algebras has as underlying RR-module just the tensor product of modules of the underlying modules, A RBA \otimes_R B. On homogeneous elements (a,b)A×BA RB(a,b) \in A \times B \stackrel{\otimes}{\to} A \otimes_R B the algebra structure is given by

(a 1,b 1)(a 2,b 2)=(a 1a 2,b 1b 2). (a_1, b_1) \cdot (a_2, b_2) = (a_1 \cdot a_2, b_1 \cdot b_2) \,.

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

For commutative RR-algebras, the tensor product is the coproduct in CommAlg RComm Alg_R:

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

hence the pushout in CRing

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

See at pushouts of commutative monoids.


Relation to tensor product of categories of modules

For AA an associative algebra over a field kk, write AAMod for its category of modules of finite dimension. Then the tensor product of algebras corresponds to the Deligne tensor product of abelian categories :Ab×AbAb\boxtimes \colon Ab \times Ab \to Ab:

(A kB)Mod(AMod)(BMod). (A \otimes_k B) Mod \simeq (A Mod) \otimes (B Mod) \,.

See at tensor product of abelian categories for more.

Last revised on August 24, 2020 at 12:23:04. See the history of this page for a list of all contributions to it.