Infinite tensor product

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

Definition

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

⨂_{i \in I} A_{i} = \underset{\binom{J \subseteq I}{| J | < ℵ_{0}}}{\lim_{\to}} ⨂_{i \in J} A_{i} .

The limit on the right side is the inductive limit in the category of $K$ -algebras. 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 coherent isomorphism.

The infinite tensor products of commutative algebras are commutative algebras. They are precisely the coproducts in the category of $K$ -commutative algebras. The universal property for the infinite tensor products of $K$ -algebras is less elegant in that it requires that the images of the canonical mappings commute.

References

↑ 李文威 (2019). 代数学方法. 第一卷, 基础架构. 现代数学基础 (in Chinese). Beijing: 高等教育出版社. ISBN 978-7-04-050725-6.
↑ Ng, Chi-Keung (2013). "On genuine infinite algebraic tensor products". Revista Matemática Iberoamericana. 29 (1): 329–356. arXiv:1112.3128. doi:10.4171/RMI/722. ISSN 0213-2230. MR 3010131. Zbl 1273.15033.

External links

Infinitary tensor product at the nLab

[李文威-1] 李文威 (2019). 代数学方法. 第一卷, 基础架构. 现代数学基础 (in Chinese). Beijing: 高等教育出版社. ISBN 978-7-04-050725-6.

[Ng-2] Ng, Chi-Keung (2013). "On genuine infinite algebraic tensor products". Revista Matemática Iberoamericana. 29 (1): 329–356. arXiv:1112.3128. doi:10.4171/RMI/722. ISSN 0213-2230. MR 3010131. Zbl 1273.15033.