Infinite tensor product

In ring theory, the (possibly) infinite tensor product generalizes the tensor product to possibly infinite families of modules or associative algebras.

Definition

The tensor product of a (finite or infinite) family of modules ${\displaystyle (M_{i})_{i\in I}}$ over a commutative ring ${\displaystyle K}$ is the ${\displaystyle K}$-module given by

${\displaystyle \bigotimes _{i\in I}M_{i}=\varinjlim _{J\subseteq I \atop |J|<\aleph _{0}}\bigotimes _{i\in J}M_{i}.}$

The limit on the right side is the inductive limit in the category of ${\displaystyle K}$-modules. So, the infinite tensor product is defined as the inductive limit of the finite tensor products;^: 329 the finite tensor products are well-defined, by the associativity of the tensor product up to isomorphism.

The tensor product of a (finite or infinite) family of algebras ${\displaystyle (A_{i})_{i\in I}}$ over a commutative ring ${\displaystyle K}$ is the ${\displaystyle K}$-algebra^: 272

${\displaystyle \bigotimes _{i\in I}A_{i}=\varinjlim _{J\subseteq I \atop |J|<\aleph _{0}}\bigotimes _{i\in J}A_{i}.}$

The limit on the right side is the inductive limit in the category of ${\displaystyle K}$-algebras. This is essentially the infinite tensor product of ${\displaystyle K}$-modules endowed with a natural ${\displaystyle K}$-algebra structure.

The possibly infinite tensor product of ${\displaystyle K}$-commutative algebras is again a ${\displaystyle K}$-commutative algebra. In this case, the possibly infinite tensor products are just the coproducts in the category of ${\displaystyle K}$-commutative algebras. The universal property for the possibly infinite tensor product of ${\displaystyle K}$-algebras is less elegant in that it requires that the images of the canonical mappings commute.

References

External links

• Infinitary tensor product at the nLab