The Stacks project

Remark 74.23.4. The pseudo-coherent complex $L$ of part (B) of Lemma 74.23.3 is canonically associated to the situation. For example, formation of $L$ as in (B) is compatible with base change. In other words, given a cartesian diagram

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

of schemes we have canonical functorial isomorphisms

\[ \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_{Y'}}(Lg^*L, \mathcal{F}') \longrightarrow \mathop{\mathrm{Ext}}\nolimits ^ i_{\mathcal{O}_ X}(L(g')^*E, (g')^*\mathcal{G}^\bullet \otimes _{\mathcal{O}_{X'}} (f')^*\mathcal{F}') \]

for $\mathcal{F}'$ quasi-coherent on $Y'$. Obsere that we do not use derived pullback on $\mathcal{G}^\bullet $ on the right hand side. If we ever need this, we will formulate a precise result here and give a detailed proof.


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