The Stacks project

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 ├ętale morphisms. In fact, the correct generality is that given a commutative diagram

\[ \xymatrix{ U \ar[rr]_ h \ar[rd]_ g & & X \ar[ld]^ f \\ & Y } \]

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

\[ Rg_! \circ h^* \longrightarrow Rf_! \]

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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