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Tensor product of modules

Tensor product need not commute with an infinite direct product. For example,
$$
\mathbb Q\otimes_{\mathbb Z}\prod_{n\geq2}\mathbb Z/n\mathbb Z\ne0,
\qquad
\prod_{n\geq2}\left(\mathbb Q\otimes_{\mathbb Z}\mathbb Z/n\mathbb Z\right)=0.
$$
The element $(1\bmod n)_n$ has infinite order in the product, so it survives rational localization.


Tensor product and infinite direct product

Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 101 / 1 / iii / Solution

Tensor product and infinite direct product

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Tensor product and infinite direct product

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