The Stacks project

Lemma 47.9.7. Let $A \to B$ be a flat ring map and let $I \subset A$ be a finitely generated ideal such that $A/I = B/IB$. Then base change and restriction induce quasi-inverse equivalences $D_{I^\infty \text{-torsion}}(A) = D_{(IB)^\infty \text{-torsion}}(B)$.

Proof. More precisely the functors are $K \mapsto K \otimes _ A^\mathbf {L} B$ for $K$ in $D_{I^\infty \text{-torsion}}(A)$ and $M \mapsto M_ A$ for $M$ in $D_{(IB)^\infty \text{-torsion}}(B)$. The reason this works is that $H^ i(K \otimes _ A^\mathbf {L} B) = H^ i(K) \otimes _ A B = H^ i(K)$. The first equality holds as $A \to B$ is flat and the second by More on Algebra, Lemma 15.89.2. $\square$

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