Lemma 90.4.4. If $A \to B$ is a ring map, then $L\pi _!(\pi ^{-1}M) = M$ with $\pi $ as in (90.4.0.1).

**Proof.**
This follows from Lemma 90.4.1 which tells us $L\pi _!(\pi ^{-1}M)$ is computed by $(\pi ^{-1}M)(P_\bullet , \epsilon )$ which is the constant simplicial object on $M$.
$\square$

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