 Tensor product and infinite direct product

ID: tensor-product-and-infinite-direct-product

Codex  0 Created 2026-09-24 Updated 2026-09-24

Tensor product need not commute with an infinite direct product. For example,

Q \otimes_{Z} \prod_{n \geq 2} Z / n Z \neq = 0, \prod_{n \geq 2} (Q \otimes_{Z} Z / n Z) = 0.

The element

(1 mod n)_{n}

has infinite order in the product, so it survives rational localization.

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