The Stacks project

62.16 More on derived upper shriek

Let $\Lambda $ be a torsion ring. Consider a commutative diagram

\[ \xymatrix{ U \ar[rr]_ j \ar[rd]_ g & & U' \ar[ld]^{g'} \\ & Y } \]

of quasi-compact and quasi-separated schemes with $g$ and $g'$ separated and of finite type and with $j$ étale. This induces a canonical map

\[ Rg_!\Lambda \longrightarrow Rg'_!\Lambda \]

in $D(Y_{\acute{e}tale}, \Lambda )$. Namely, by Lemmas 62.9.2 and 62.10.3 we have $Rg_! = Rg'_! \circ j_!$. On the other hand, since $j_!$ is left adjoint to $j^{-1}$ we have the counit $\text{Tr}_ j : j_!\Lambda = j_!j^{-1}\Lambda \to \Lambda $; we also call this the trace map for $j$, see Remark 62.5.6. The map above is constructed as the composition

\[ Rg_!\Lambda = Rg'_!j_!\Lambda \xrightarrow {Rg'_! \text{Tr}_ j} Rg'_!\Lambda \]

Given a second étale morphism $j' : U' \to U''$ for some $g'' : U'' \to Y$ separated and of finite type the composition

\[ Rg_!\Lambda \longrightarrow Rg'_!\Lambda \longrightarrow Rg''_!\Lambda \]

of the maps for $j$ and $j'$ is equal to the map $Rg_!\Lambda \longrightarrow Rg''_!\Lambda $ constructed for $j' \circ j$. This follows from the corresponding statement on trace maps, see Lemma 62.5.4 for a more general case.

Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. Then we obtain a functor

\[ X_{affine, {\acute{e}tale}} \longrightarrow \left\{ \begin{matrix} \text{schemes separated of finite type over }Y \\ \text{with étale morphisms between them} \end{matrix} \right\} \]

Thus the construction above determines a functor $X_{affine, {\acute{e}tale}}^{opp} \to D(Y_{\acute{e}tale}, \Lambda )$ sending $U$ to $R(U \to Y)_!\Lambda $.

Lemma 62.16.1. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a torsion ring. Let $K \in D(Y_{\acute{e}tale}, \Lambda )$. For $n \in \mathbf{Z}$ the cohomology sheaf $H^ n(Rf^!K)$ restricted to $X_{affine, {\acute{e}tale}}$ is the sheaf associated to the presheaf

\[ U \longmapsto \mathop{\mathrm{Hom}}\nolimits _ Y(R(U \to Y)_!\Lambda , K[n]) \]

See discussion above for the functorial nature of $R(U \to Y)_!\Lambda $.

Proof. Let $j : U \to X$ be an object of $X_{affine, {\acute{e}tale}}$ and set $g = f \circ j$. Recall that $\mathop{\mathrm{Hom}}\nolimits _ X(j_!\Lambda , M[n]) = H^ n(U, M)$ for any $M$ in $D(X_{\acute{e}tale}, \Lambda )$. Then $H^ n(Rf^!K)$ is the sheaf associated to the presheaf

\[ U \mapsto H^ n(U, Rf^!K) = \mathop{\mathrm{Hom}}\nolimits _ X(j_!\Lambda , Rf^!K[n]) = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!j_!\Lambda , K[n] = \mathop{\mathrm{Hom}}\nolimits _ Y(Rg_!\Lambda , K[n]) \]

We omit the verification that the transition maps are given by the transition maps between the objects $Rg_!\Lambda = R(U \to Y)_!\Lambda $ we constructed above. $\square$


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