Remark 48.32.9. In Remarks 48.32.7 and 48.32.8 we have seen that the construction of compactly supported cohomology is covariant with respect to open immersions and étale morphisms. In fact, the correct generality is that given a commutative diagram

of $\textit{FTS}_ S$ with $h$ flat and quasi-finite there exists a canonical transformation

As in Remark 48.32.8 this map can be constructed using a transformation of functors $h^* \to h^!$ on $D^+_{\textit{Coh}}(\mathcal{O}_ X)$. Recall that $h^!K = h^*K \otimes \omega _{U/X}$ where $\omega _{U/X} = h^!\mathcal{O}_ X$ is the relative dualizing sheaf of the flat quasi-finite morphism $h$ (see Lemmas 48.17.9 and 48.21.6). Recall that $\omega _{U/X}$ is the same as the relative dualizing module which will be constructed in Discriminants, Remark 49.2.11 by Discriminants, Lemma 49.15.1. Thus we can use the trace element $\tau _{U/X} : \mathcal{O}_ U \to \omega _{U/X}$ which will be constructed in Discriminants, Remark 49.4.7 to define our transformation. If we ever need this, we will precisely formulate and prove the result here.

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