The Stacks project

63.16 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 63.9.2 and 63.10.3 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 63.5.6. 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 63.5.4 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 63.16.1. 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. $\square$

Comments (0)

Post a comment

Your email address will not be published. Required fields are marked.

In your comment you can use Markdown and LaTeX style mathematics (enclose it like $\pi$). A preview option is available if you wish to see how it works out (just click on the eye in the toolbar).

Unfortunately JavaScript is disabled in your browser, so the comment preview function will not work.

All contributions are licensed under the GNU Free Documentation License.

In order to prevent bots from posting comments, we would like you to prove that you are human. You can do this by filling in the name of the current tag in the following input field. As a reminder, this is tag 0GLJ. Beware of the difference between the letter 'O' and the digit '0'.