Section 63.10 (0G28): Properties of derived lower shriek

63.10 Properties of derived lower shriek

Here are some properties of derived lower shriek.

Lemma 63.10.1. Let $f : X \to Y$ be a finite type separated morphism of quasi-compact and quasi-separated schemes. Let $\Lambda$ be a ring.

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 $Rf_!(\bigoplus _ i K_ i) = \bigoplus _ i Rf_!K_ i$ .
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 63.10.2. 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 63.9 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

$H^ i(Rf_!(\tau _{\leq a}E) \to H^ i(Rf_!(E))$ is an isomorphism for $i \leq a$ ,
$H^ i(Rf_!(E)) \to H^ i(Rf_!(\tau _{\geq b - N}E))$ is an isomorphism for $i \geq b$ ,
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 63.10.3. 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 63.9 agree with the restriction of the functor $f_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda )$ constructed in Section 63.7 to their common domains of definition.

Proof. By Zariski's main theorem (More on Morphisms, Lemma 37.43.3) 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 63.3.6. $\square$

Lemma 63.10.4. 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 63.7.3, the fact that $Rf_! \circ Rj_{U!} = Ra_!$ by Lemma 63.9.2, and the fact that $Rj_{U!} = j_{U!}$ by Lemma 63.10.3. $\square$

Lemma 63.10.5. 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 63.7.4, the fact that $Rf_! \circ Rj_! = Rg_!$ and $Rf_! \circ Ri_!$ by Lemma 63.9.2, and the fact that $Rj_! = j_!$ and $Ri_! = i_! = i_*$ by Lemma 63.10.3. $\square$

Lemma 63.10.6. 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 63.9.2. $\square$

Lemma 63.10.7. 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 63.10.8. 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 63.9, 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 63.10.7.

Remark 63.10.9. 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 21.35. 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 21.35.10 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 $\alpha = \gamma \circ R\overline{f}_*\beta$ .

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63.10 Properties of derived lower shriek

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