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A basic result in commutative algebra asserts that direct limits commute with tensor products. My favourite counterexample to the statement obtained by replacing "direct" with "inverse" is the following. Let $p$ be a prime number; then

$\bigl(\varprojlim_n\mathbb Z/p^n\mathbb Z\bigr)\otimes_{\mathbb Z}\mathbb Q\cong\mathbb Q_p$,

the field of $p$-adic integers numbers (completion of $\mathbb Q$ with respect to the metric induced by the $p$-adic valuation), while

$\varprojlim_n\bigl((\mathbb Z/p^n\mathbb Z)\otimes_{\mathbb Z}\mathbb Q\bigr)=0$,

since every $\mathbb Z/p^n\mathbb Z$ is torsion and $\mathbb Q$ is divisible.

A basic result in commutative algebra asserts that direct limits commute with tensor products. My favourite counterexample to the statement obtained by replacing "direct" with "inverse" is the following. Let $p$ be a prime number; then
$\bigl(\varprojlim_n\mathbb Z/p^n\mathbb Z\bigr)\otimes_{\mathbb Z}\mathbb Q\cong\mathbb Q_p$,
the field of $p$-adic integers (completion of $\mathbb Q$ with respect to the metric induced by the $p$-adic valuation), while
$\varprojlim_n\bigl((\mathbb Z/p^n\mathbb Z)\otimes_{\mathbb Z}\mathbb Q\bigr)=0$,
since every $\mathbb Z/p^n\mathbb Z$ is torsion and $\mathbb Q$ is divisible.