Lemma 48.18.3 (Makeshift base change). In Situation 48.16.1 let

\[ \xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y } \]

be a cartesian diagram of $\textit{FTS}_ S$. Let $E \in D^+_\mathit{QCoh}(\mathcal{O}_ Y)$. If $f$ is flat, then $L(g')^*f^!E$ and $(f')^!Lg^*E$ restrict to isomorphic objects of $D(\mathcal{O}_{U'})$ for $U' \subset X'$ affine open mapping into affine opens of $Y$, $Y'$, and $X$.

**Proof.**
By our assumptions we immediately reduce to the case where $X$, $Y$, $Y'$, and $X'$ are affine. Say $Y = \mathop{\mathrm{Spec}}(R)$, $Y' = \mathop{\mathrm{Spec}}(R')$, $X = \mathop{\mathrm{Spec}}(A)$, and $X' = \mathop{\mathrm{Spec}}(A')$. Then $A' = A \otimes _ R R'$. Let $E$ correspond to $K \in D^+(R)$. Denoting $\varphi : R \to A$ and $\varphi ' : R' \to A'$ the given maps we see from Remark 48.17.4 that $L(g')^*f^!E$ and $(f')^!Lg^*E$ correspond to $\varphi ^!(K) \otimes _ A^\mathbf {L} A'$ and $(\varphi ')^!(K \otimes _ R^\mathbf {L} R')$ where $\varphi ^!$ and $(\varphi ')^!$ are the functors from Dualizing Complexes, Section 47.24. The result follows from Dualizing Complexes, Lemma 47.24.6.
$\square$

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