The Stacks project

62.10 Properties of derived lower shriek

Here are some properties of derived lower shriek.

Lemma 62.10.1. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a ring.

  1. Let $K_ i \in D^+_{tors}(X_{\acute{e}tale}, \Lambda )$, $i \in I$ be a family of objects. Assume given $a \in \mathbf{Z}$ such that $H^ n(K_ i) = 0$ for $n < a$ and $i \in I$. Then $Rf_!(\bigoplus _ i K_ i) = \bigoplus _ i Rf_!K_ i$.

  2. If $\Lambda $ is torsion, then the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ commutes with direct sums.

Proof. By construction it suffices to prove this when $f$ is an open immersion and when $f$ is a proper morphism. For any open immersion $j : U \to X$ of schemes, the functor $j_! : D(U_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ is a left adjoint to pullback $j^{-1} : D(X_{\acute{e}tale}) \to D(U_{\acute{e}tale})$ and hence commutes with direct sums, see Cohomology on Sites, Lemma 21.20.8. In the proper case we have $Rf_! = Rf_*$ and we get the result from Étale Cohomology, Lemma 59.52.6 in the bounded belo case and from Étale Cohomology, Lemma 59.96.4 in the case that our coefficient ring $\Lambda $ is a torsion ring. $\square$

Lemma 62.10.2. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a ring. The functors $Rf_!$ constructed in Section 62.9 are bounded in the following sense: There exists an integer $N$ such that for $E \in D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or $E \in D(X_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion, we have

  1. $H^ i(Rf_!(\tau _{\leq a}E) \to H^ i(Rf_!(E))$ is an isomorphism for $i \leq a$,

  2. $H^ i(Rf_!(E)) \to H^ i(Rf_!(\tau _{\geq b - N}E))$ is an isomorphism for $i \geq b$,

  3. if $H^ i(E) = 0$ for $i \not\in [a, b]$ for some $-\infty \leq a \leq b \leq \infty $, then $H^ i(Rf_!(E)) = 0$ for $i \not\in [a, b + N]$.

Proof. Assume $\Lambda $ is torsion and consider the functor $Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$. By construction it suffices to prove this when $f$ is an open immersion and when $f$ is a proper morphism. For any open immersion $j : U \to X$ of schemes, the functor $j_! : D(U_{\acute{e}tale}) \to D(X_{\acute{e}tale})$ is exact and hence the statement holds with $N = 0$ in this case. If $f$ is proper then $Rf_! = Rf_*$, i.e., it is a right derived functor. Hence the bound on the left by Derived Categories, Lemma 13.16.1. Moreover in this case $f_* : \textit{Mod}(X_{\acute{e}tale}, \Lambda ) \to \textit{Mod}(Y_{\acute{e}tale}, \Lambda )$ has bounded cohomological dimension by Morphisms, Lemma 29.28.5 and Étale Cohomology, Lemma 59.92.2. Thus we conclude by Derived Categories, Lemma 13.32.2.

Next, assume $\Lambda $ is arbitrary and let us consider the functor $Rf_! : D^+_{tors}(X_{\acute{e}tale}, \Lambda ) \to D^+_{tors}(Y_{\acute{e}tale}, \Lambda )$. Again we immediately reduce to the case where $f$ is proper and $Rf_! = Rf_*$. Again part (1) is immediate. To show part (3) we can use induction on $b - a$, the distinguished triangles of trunctions, and Étale Cohomology, Lemma 59.92.2. Part (2) follows from (3). Details omitted. $\square$

Lemma 62.10.3. Let $f : X \to Y$ be a quasi-finite separated morphism of quasi-compact and quasi-separated schemes. Then the functors $Rf_!$ constructed in Section 62.9 agree with the restriction of the functor $f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ constructed in Section 62.7 to their common domains of definition.

Proof. By Zariski's main theorem (More on Morphisms, Lemma 37.42.3) we can find an open immersion $j : X \to \overline{X}$ and a finite morphism $\overline{f} : \overline{X} \to Y$ with $f = \overline{f} \circ j$. By construction we have $Rf_! = R\overline{f}_* \circ j_!$. Since $\overline{f}$ is finite, we have $R\overline{f}_* = \overline{f}_*$ by Étale Cohomology, Proposition 59.55.2. The lemma follows because $\overline{f}_* \circ j_! = f_!$ for example by Lemma 62.3.6. $\square$

Lemma 62.10.4. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $U$ and $V$ be quasi-compact opens of $X$ such that $X = U \cup V$. Denote $a : U \to Y$, $b : V \to Y$ and $c : U \cap V \to Y$ the restrictions of $f$. Let $\Lambda $ be a ring. For $K$ in $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or $K \in D(X_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion, we have a distinguished triangle

\[ Rc_!(K|_{U \cap V}) \to Ra_!(K|_ U) \oplus Rb_!(K|_ V) \to Rf_!K \to Rc_!(K|_{U \cap V})[1] \]

in $D(Y_{\acute{e}tale}, \Lambda )$.

Proof. This follows from Lemma 62.7.3, the fact that $Rf_! \circ Rj_{U!} = Ra_!$ by Lemma 62.9.2, and the fact that $Rj_{U!} = j_{U!}$ by Lemma 62.10.3. $\square$

Lemma 62.10.5. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $U$ be a quasi-compact open of $X$ with complement $Z \subset X$. Denote $g : U \to Y$ and $h : Z \to Y$ the restrictions of $f$. Let $\Lambda $ be a ring. For $K$ in $D^+_{tors}(X_{\acute{e}tale}, \Lambda )$ or $K \in D(X_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion, we have a distinguished triangle

\[ Rg_!(K|_ U) \to Rf_!K \to Rh_!(K|_ Z) \to Rg_!(K|_ U)[1] \]

in $D(Y_{\acute{e}tale}, \Lambda )$.

Proof. This follows from Lemma 62.7.4, the fact that $Rf_! \circ Rj_! = Rg_!$ and $Rf_! \circ Ri_!$ by Lemma 62.9.2, and the fact that $Rj_! = j_!$ and $Ri_! = i_! = i_*$ by Lemma 62.10.3. $\square$

Lemma 62.10.6. Let $f' : X' \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $i : X \to X'$ be a thickening and denote $f = f' \circ i$. Let $\Lambda $ be a ring. For $K'$ in $D^+_{tors}(X'_{\acute{e}tale}, \Lambda )$ or $K' \in D(X'_{\acute{e}tale}, \Lambda )$ if $\Lambda $ is torsion, we have $Rf_!i^{-1}K' = Rf'_!K'$.

Proof. This is true because $i^{-1}$ and $i_* = i_!$ inverse equivalences of categories by the topological invariance of the small étale topos (Étale Cohomology, Theorem 59.45.2) and we can apply Lemma 62.9.2. $\square$

Lemma 62.10.7. 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 $E \in D(X_{\acute{e}tale}, \Lambda )$ and $K \in D(Y_{\acute{e}tale}, \Lambda )$. Then

\[ Rf_!E \otimes _\Lambda ^\mathbf {L} K = Rf_!(E \otimes _\Lambda ^\mathbf {L} f^{-1}K) \]

in $D(Y_{\acute{e}tale}, \Lambda )$.

Proof. Choose $j : X \to \overline{X}$ and $\overline{f} : \overline{X} \to Y$ as in the construction of $Rf_!$. We have $j_!E \otimes _\Lambda ^\mathbf {L} \overline{f}^{-1}K = j_!(E \otimes _\Lambda ^\mathbf {L} f^{-1}K)$ by Cohomology on Sites, Lemma 21.20.9. Then we get the result by applying Étale Cohomology, Lemma 59.96.6 and using that $f^{-1} = j^{-1} \circ \overline{f}^{-1}$ and $Rf_! = R\overline{f}_*j_!$. $\square$

Remark 62.10.8. Let $\Lambda _1 \to \Lambda _2$ be a homomorphism of torsion rings. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. The diagram

\[ \xymatrix{ D(X_{\acute{e}tale}, \Lambda _2) \ar[r]_{res} \ar[d]_{Rf_!} & D(X_{\acute{e}tale}, \Lambda _1) \ar[d]^{Rf_!} \\ D(Y_{\acute{e}tale}, \Lambda _2) \ar[r]^{res} & D(Y_{\acute{e}tale}, \Lambda _1) } \]

commutes where $res$ is the “restriction” functor which turns a $\Lambda _2$-module into a $\Lambda _1$-module using the given ring map. Writing $Rf_! = R\overline{f}_* \circ j_!$ for a factorization $f = \overline{f} \circ j$ as in Section 62.9, we see that the result holds for $j_!$ by inspection and for $R\overline{f}_*$ by Cohomology on Sites, Lemma 21.20.7. On the other hand, also the diagram

\[ \xymatrix{ D(X_{\acute{e}tale}, \Lambda _1) \ar[r]_{- \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2} \ar[d]_{Rf_!} & D(X_{\acute{e}tale}, \Lambda _2) \ar[d]^{Rf_!} \\ D(Y_{\acute{e}tale}, \Lambda _1) \ar[r]^{- \otimes _{\Lambda _1}^\mathbf {L} \Lambda _2} & D(Y_{\acute{e}tale}, \Lambda _2) } \]

is commutative as follows from Lemma 62.10.7.

Remark 62.10.9. Let $f : X \to Y$ be a separated finite type morphism of quasi-compact and quasi-separated schemes. Let $\Lambda $ be a torsion coefficient ring and let $K$ and $L$ be objects of $D(X_{\acute{e}tale}, \Lambda )$. We claim there is a canonical map

\[ \alpha : Rf_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (Rf_!K, Rf_!L) \]

functorial in $K$ and $L$. Namely, choose $j : X \to \overline{X}$ and $\overline{f} : \overline{X} \to Y$ as in the construction of $Rf_!$. We first define a map

\[ \beta : Rj_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (j_!K, j_!L) \]

By the construction of internal hom in the derived category, this is the same thing as defining a map

\[ \beta ' : Rj_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \otimes _\Lambda ^\mathbf {L} j_!K \longrightarrow j_!L \]

See Cohomology on Sites, Section 21.34. The source of $\beta '$ is equal to

\[ j_!\left(R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \otimes _\Lambda ^\mathbf {L} K\right) \]

by Cohomology on Sites, Lemma 21.20.9. Hence we can set $\beta ' = j_!\beta ''$ where $\beta '' : R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L) \otimes _\Lambda ^\mathbf {L} K \to L$ corresponds to the identity on $R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (K, L)$ via the universal property of internal hom mentioned above. By Cohomology on Sites, Remark 21.34.10 we have a canonical map

\[ \gamma : R\overline{f}_*R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (j_!K, j_!L) \longrightarrow R\mathop{\mathcal{H}\! \mathit{om}}\nolimits _\Lambda (R\overline{f}_*j_!K, R\overline{f}_*j_!L) \]

Since $Rf_! = R\overline{f}_*j_!$ and $Rf_* = R\overline{f}_* Rj_*$ (by Leray) we obtain the desired map $\alpha = \gamma \circ R\overline{f}_*\beta $.


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 0G28. Beware of the difference between the letter 'O' and the digit '0'.