The Stacks project

Proof. Without any assumptions there is a map $Rf_*(E) \otimes _{\mathcal{O}_ Y}^\mathbf {L} K \to Rf_*(E \otimes _{\mathcal{O}_ X}^\mathbf {L} Lf^*K)$. Namely, it is the adjoint to the canonical map

coming from the map $Lf^*Rf_*E \to E$. See Cohomology on Sites, Lemmas 21.18.4 and 21.19.1. To check it is an isomorphism we may work étale locally on $Y$. Hence we reduce to the case that $Y$ is an affine scheme.

Suppose that $K = \bigoplus K_ i$ is a direct sum of some complexes $K_ i \in D_\mathit{QCoh}(\mathcal{O}_ Y)$. If the statement holds for each $K_ i$, then it holds for $K$. Namely, the functors $Lf^*$ and $\otimes ^\mathbf {L}$ preserve direct sums by construction and $Rf_*$ commutes with direct sums (for complexes with quasi-coherent cohomology sheaves) by Lemma 75.6.2. Moreover, suppose that $K \to L \to M \to K[1]$ is a distinguished triangle in $D_\mathit{QCoh}(Y)$. Then if the statement of the lemma holds for two of $K, L, M$, then it holds for the third (as the functors involved are exact functors of triangulated categories).

Assume $Y$ affine, say $Y = \mathop{\mathrm{Spec}}(A)$. The functor $\widetilde{\ } : D(A) \to D_\mathit{QCoh}(\mathcal{O}_ Y)$ is an equivalence by Lemma 75.4.2 and Derived Categories of Schemes, Lemma 36.3.5. Let $T$ be the property for $K \in D(A)$ that the statement of the lemma holds for $\widetilde{K}$. The discussion above and More on Algebra, Remark 15.59.11 shows that it suffices to prove $T$ holds for $A[k]$. This finishes the proof, as the statement of the lemma is clear for shifts of the structure sheaf. $\square$