The Stacks project

Lemma 59.71.9. Let $X$ be a scheme. Let $\Lambda $ be a Noetherian ring. The tensor product of two constructible sheaves of $\Lambda $-modules on $X_{\acute{e}tale}$ is a constructible sheaf of $\Lambda $-modules.

Proof. The question immediately reduces to the case where $X$ is affine. Since any two partitions of $X$ with constructible locally closed strata have a common refinement of the same type and since pullbacks commute with tensor product we reduce to Lemma 59.64.7. $\square$

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