Lemma 63.9.1. Let f : X \to Y be a finite type separated morphism of quasi-compact and quasi-separated schemes. The functors Rf_! constructed above are, up to canonical isomorphism, independent of the choice of the compactification.
63.9 Derived lower shriek via compactifications
Let f : X \to Y be a finite type separated morphism of schemes with Y quasi-compact and quasi-separated. Choose a compactification j : X \to \overline{X} over Y, see More on Flatness, Theorem 38.33.8. Let \Lambda be a ring. Denote D^+_{tors}(X_{\acute{e}tale}, \Lambda ) the strictly full saturated triangulated subcategory of D(X_{\acute{e}tale}, \Lambda ) consisting of objects K which are bounded below and whose cohomology sheaves are torsion. We will consider the functor
Rf_! = R\overline{f}_* \circ j_! : D^+_{tors}(X_{\acute{e}tale}, \Lambda ) \longrightarrow D^+_{tors}(Y_{\acute{e}tale}, \Lambda )
where \overline{f} : \overline{X} \to Y is the structure morphism. This makes sense: the functor j_! sends D^+_{tors}(X_{\acute{e}tale}, \Lambda ) into D^+_{tors}(\overline{X}_{\acute{e}tale}, \Lambda ) by Remark 63.7.2 and R\overline{f}_* sends D^+_{tors}(\overline{X}_{\acute{e}tale}, \Lambda ) into D^+_{tors}(Y_{\acute{e}tale}, \Lambda ) by Étale Cohomology, Lemma 59.78.2. If \Lambda is a torsion ring, then we define
Rf_! = R\overline{f}_* \circ j_! : D(X_{\acute{e}tale}, \Lambda ) \longrightarrow D(Y_{\acute{e}tale}, \Lambda )
Here is the obligatory lemma.
Proof. We will prove this for the functor Rf_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ) when \Lambda is a torsion ring; the case of the functor Rf_! : D^+_{tors}(X_{\acute{e}tale}, \Lambda ) \to D^+_{tors}(Y_{\acute{e}tale}, \Lambda ) is proved in exactly the same way.
Consider the category of compactifications of X over Y, which is cofiltered according to More on Flatness, Theorem 38.33.8 and Lemmas 38.32.1 and 38.32.2. To every choice of a compactification
j : X \to \overline{X},\quad \overline{f} : \overline{X} \to Y
the construction above associates the functor R\overline{f}_* \circ j_! : D(X_{\acute{e}tale}, \Lambda ) \to D(Y_{\acute{e}tale}, \Lambda ). Let's be a little more explicit. Given a complex \mathcal{K}^\bullet of sheaves of \Lambda -modules on X_{\acute{e}tale}, we choose a quasi-isomorphism j_!\mathcal{K}^\bullet \to \mathcal{I}^\bullet into a K-injective complex of sheaves of \Lambda -modules on \overline{X}_{\acute{e}tale}. Then our functor sends \mathcal{K}^\bullet to \overline{f}_*\mathcal{I}^\bullet .
Suppose given a morphism g : \overline{X}_1 \to \overline{X}_2 between compactifications j_ i : X \to \overline{X}_ i over Y. Then we get an isomorphism
R\overline{f}_{2, *} \circ j_{2, !} = R\overline{f}_{2, *} \circ Rg_* \circ j_{1, !} = R\overline{f}_{1, *} \circ j_{1, !}
using Lemma 63.8.6 in the first equality.
To finish the proof, since the category of compactifications of X over Y is cofiltered, it suffices to show compositions of morphisms of compactifications of X over Y are turned into compositions of isomorphisms of functors1. To do this, suppose that j_3 : X \to \overline{X}_3 is a third compactification and that h : \overline{X}_2 \to \overline{X}_3 is a morphism of compactifications. Then we have to show that the composition
R\overline{f}_{3, *} \circ j_{3, !} = R\overline{f}_{3, *} \circ Rh_* \circ j_{2, !} = R\overline{f}_{2, *} \circ j_{2, !} = R\overline{f}_{2, *} \circ Rg_* \circ j_{1, !} = R\overline{f}_{1, *} \circ j_{1, !}
is equal to the isomorphism of functors constructed using simply j_3, g \circ h, and j_1. A calculation shows that it suffices to prove that the composition of the maps
j_{3, !} \to Rh_* \circ j_{2, !} \to Rh_* \circ Rg_* \circ j_{1, !}
of Lemma 63.8.6 agrees with the corresponding map j_{3, !} \to R(h \circ g)_* \circ j_{1, !} via the identification R(h \circ g)_* = Rh_* \circ Rg_*. Since the map of Lemma 63.8.6 is a special case of the map of Lemma 63.8.1 (as j_1 and j_2 are separated) this follows immediately from Lemma 63.8.2. \square
Lemma 63.9.2. Let f : X \to Y and g : Y \to Z be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then there is a canonical isomorphism Rg_! \circ Rf_! \to R(g \circ f)_!.
Proof. Choose a compactification i : Y \to \overline{Y} of Y over Z. Choose a compactification X \to \overline{X} of X over \overline{Y}. This uses More on Flatness, Theorem 38.33.8 and Lemma 38.32.2 twice. Let U be the inverse image of Y in \overline{X} so that we get the commutative diagram
\xymatrix{ X \ar[r]_ j \ar[d]_ f & U \ar[dl]^{f'} \ar[r]_{j'} & \overline{X} \ar[dl]^{\overline{f}} \\ Y \ar[r]_ i \ar[d]_ g & \overline{Y} \ar[dl]^{\overline{g}} \\ Z }
Then we have
\begin{align*} R(g \circ f)_! & = R(\overline{g} \circ \overline{f})_* \circ (j' \circ j)_! \\ & = R\overline{g}_* \circ R\overline{f}_* \circ j'_! \circ j_! \\ & = R\overline{g}_* \circ i_! \circ Rf'_* \circ j_! \\ & = Rg_! \circ Rf_! \end{align*}
The first equality is the definition of R(g \circ f)_!. The second equality uses the identifications R(\overline{g} \circ \overline{f})_* = R\overline{g}_* \circ R\overline{f}_* and (j' \circ j)_! = j'_! \circ j_! of Lemma 63.3.13. The identification i_! \circ Rf'_* \to R\overline{f}_* \circ j_! used in the third equality is Lemma 63.8.1. The final fourth equality is the definition of Rg_! and Rf_!. To finish the proof we show that this isomorphism is independent of choices made.
Suppose we have two diagrams
\vcenter { \xymatrix{ X \ar[r]_{j_1} \ar[d] & U_1 \ar[dl]^{f_1} \ar[r]_{j'_1} & \overline{X}_1 \ar[dl]^{\overline{f}_1} \\ Y \ar[r]_{i_1} \ar[d] & \overline{Y}_1 \ar[dl]^{\overline{g}_1} \\ Z } } \quad \text{and}\quad \vcenter { \xymatrix{ X \ar[r]_{j_2} \ar[d] & U_2 \ar[dl]^{f_2} \ar[r]_{j'_2} & \overline{X}_2 \ar[dl]^{\overline{f}_2} \\ Y \ar[r]_{i_2} \ar[d] & \overline{Y}_2 \ar[dl]^{\overline{g}_2} \\ Z } }
We can first choose a compactification i : Y \to \overline{Y} of Y over Z which dominates both \overline{Y}_1 and \overline{Y}_2, see More on Flatness, Lemma 38.32.1. By More on Flatness, Lemma 38.32.3 and Categories, Lemmas 4.27.13 and 4.27.14 we can choose a compactification X \to \overline{X} of X over \overline{Y} with morphisms \overline{X} \to \overline{X}_1 and \overline{X} \to \overline{X}_2 and such that the composition \overline{X} \to \overline{Y} \to \overline{Y}_1 is equal to the composition \overline{X} \to \overline{X}_1 \to \overline{Y}_1 and such that the composition \overline{X} \to \overline{Y} \to \overline{Y}_2 is equal to the composition \overline{X} \to \overline{X}_2 \to \overline{Y}_2. Thus we see that it suffices to compare the maps determined by our diagrams when we have a commutative diagram as follows
\xymatrix{ X \ar[rr]_{j_1} \ar@{=}[d] & & U_1 \ar[d]^{h'} \ar[ddll] \ar[rr]_{j'_1} & & \overline{X}_1 \ar[d]^ h \ar[ddll] \\ X \ar '[r][rr]^-{j_2} \ar[d] & & U_2 \ar '[dl][ddll] \ar '[r][rr]^-{j'_2} & & \overline{X}_2 \ar[ddll] \\ Y \ar[rr]^{i_1} \ar@{=}[d] & & \overline{Y}_1 \ar[d]^ k \\ Y \ar[rr]^{i_2} \ar[d] & & \overline{Y}_2 \ar[dll] \\ Z }
Each of the squares
\xymatrix{ X \ar[r]_{j_1} \ar[d]_{\text{id}} \ar@{}[dr]|A & U_1 \ar[d]^{h'} \\ X \ar[r]^{j_2} & U_2 } \quad \xymatrix{ U_2 \ar[r]_{j_2'} \ar[d]_{f_2} \ar@{}[dr]|B & \overline{X}_2 \ar[d]^{\overline{f}_2} \\ Y \ar[r]^{i_2} & \overline{Y}_2 } \quad \xymatrix{ U_1 \ar[r]_{j_1'} \ar[d]_{f_1} \ar@{}[dr]|C & \overline{X}_1 \ar[d]^{\overline{f}_1} \\ Y \ar[r]^{i_1} & \overline{Y}_1 } \quad \xymatrix{ Y \ar[r]_{i_1} \ar[d]_{\text{id}} \ar@{}[dr]|D & \overline{Y}_1 \ar[d]^ k \\ Y \ar[r]^{i_2} & \overline{Y}_2 } \quad \xymatrix{ X \ar[r]_{j_1' \circ j_1} \ar[d]_{\text{id}} \ar@{}[dr]|E & \overline{X}_1 \ar[d]^ h \\ X \ar[r]^{j_2} & \overline{X}_2 }
gives rise to an isomorphism as follows
\begin{align*} \gamma _ A & : j_{2, !} \to Rh'_* \circ j_{1, !} \\ \gamma _ B & : i_{2, !} \circ Rf_{2, *} \to R\overline{f}_{2, *} \circ j'_{2, !} \\ \gamma _ C & : i_{1, !} \circ Rf_{1, *} \to R\overline{f}_{1, *} \circ j'_{1, !} \\ \gamma _ D & : i_{2, !} \to Rk_* \circ i_{1, !} \\ \gamma _ E & : j_{2, !} \to Rh_* \circ (j'_1 \circ j_1)_! \end{align*}
by applying the map from Lemma 63.8.1 (which is the same as the map in Lemma 63.8.6 in case the left vertical arrow is the identity). Let us write
\begin{align*} F_1 & = Rf_{1, *} \circ j_{1, !} \\ F_2 & = Rf_{2, *} \circ j_{2, !} \\ G_1 & = R\overline{g}_{1, *} \circ i_{1, !} \\ G_2 & = R\overline{g}_{2, *} \circ i_{2, !} \\ C_1 & = R(\overline{g}_1 \circ \overline{f}_1)_* \circ (j'_1 \circ j_1)_! \\ C_2 & = R(\overline{g}_2 \circ \overline{f}_2)_* \circ (j'_2 \circ j_2)_! \end{align*}
The construction given in the first paragraph of the proof and in Lemma 63.9.1 uses
\gamma _ C for the map G_1 \circ F_1 \to C_1,
\gamma _ B for the map G_2 \circ F_2 \to C_2 ,
\gamma _ A for the map F_2 \to F_1,
\gamma _ D for the map G_2 \to G_1, and
\gamma _ E for the map C_2 \to C_1.
This implies that we have to show that the diagram
\xymatrix{ C_2 \ar[rr]_{\gamma _ E} & & C_1 \\ G_2 \circ F_2 \ar[rr]^{\gamma _ D \circ \gamma _ A} \ar[u]^{\gamma _ B} & & G_1 \circ F_1 \ar[u]_{\gamma _ C} }
is commutative. We will use Lemmas 63.8.2 and 63.8.3 and with (abuse of) notation as in Remark 63.8.4 (in particular dropping \star products with identity transformations from the notation). We can write \gamma _ E = \gamma _ F \circ \gamma _ A where
\xymatrix{ U_1 \ar[r]_{j'_1} \ar[d]_{h'} \ar@{}[rd]|F & \overline{X}_1 \ar[d]^ h \\ U_2 \ar[r]^{j'_2} & \overline{X}_2 }
Thus we see that
\gamma _ E \circ \gamma _ B = \gamma _ F \circ \gamma _ A \circ \gamma _ B = \gamma _ F \circ \gamma _ B \circ \gamma _ A
the last equality because the two squares A and B only intersect in one point (similar to the last argument in Remark 63.8.4). Thus it suffices to prove that \gamma _ C \circ \gamma _ D = \gamma _ F \circ \gamma _ B. Since both of these are equal to the map for the square
\xymatrix{ U_1 \ar[r] \ar[d] & \overline{X}_1 \ar[d] \\ Y \ar[r] & \overline{Y}_2 }
we conclude. \square
Lemma 63.9.3. Let f : X \to Y, g : Y \to Z, h : Z \to T be separated morphisms of finite type of quasi-compact and quasi-separated schemes. Then the diagram
\xymatrix{ Rh_! \circ Rg_! \circ Rf_! \ar[r]_{\gamma _ C} \ar[d]^{\gamma _ A} & R(h \circ g)_! \circ Rf_! \ar[d]_{\gamma _{A + B}} \\ Rh_! \circ R(g \circ f)_! \ar[r]^{\gamma _{B + C}} & R(h \circ g \circ f)_! }
of isomorphisms of Lemma 63.9.2 commutes (for the meaning of the \gamma 's see proof).
Proof. To do this we choose a compactification \overline{Z} of Z over T, then a compactification \overline{Y} of Y over \overline{Z}, and then a compactification \overline{X} of X over \overline{Y}. This uses More on Flatness, Theorem 38.33.8 and Lemma 38.32.2. Let W \subset \overline{Y} be the inverse image of Z under \overline{Y} \to \overline{Z} and let U \subset V \subset \overline{X} be the inverse images of Y \subset W under \overline{X} \to \overline{Y}. This produces the following diagram
\xymatrix{ X \ar[d]_ f \ar[r] & U \ar[r] \ar[d] \ar@{}[dr]|A & V \ar[d] \ar[r] \ar@{}[rd]|B & \overline{X} \ar[d] \\ Y \ar[d]_ g \ar[r] & Y \ar[r] \ar[d] & W \ar[r] \ar[d] \ar@{}[rd]|C & \overline{Y} \ar[d] \\ Z \ar[d]_ h \ar[r] & Z \ar[d] \ar[r] & Z \ar[d] \ar[r] & \overline{Z} \ar[d] \\ T \ar[r] & T \ar[r] & T \ar[r] & T }
Without introducing tons of notation but arguing exactly as in the proof of Lemma 63.9.2 we see that the maps in the first displayed diagram use the maps of Lemma 63.8.1 for the rectangles A + B, B + C, A, and C as indicated in the diagram in the statement of the lemma. Since by Lemmas 63.8.2 and 63.8.3 we have \gamma _{A + B} = \gamma _ B \circ \gamma _ A and \gamma _{B + C} = \gamma _ B \circ \gamma _ C we conclude that the desired equality holds provided \gamma _ A \circ \gamma _ C = \gamma _ C \circ \gamma _ A. This is true because the two squares A and C only intersect in one point (similar to the last argument in Remark 63.8.4). \square
Lemma 63.9.4. Consider a cartesian square
\xymatrix{ X' \ar[r]_{g'} \ar[d]_{f'} & X \ar[d]^ f \\ Y' \ar[r]^ g & Y }
of quasi-compact and quasi-separated schemes with f separated and of finite type. Then there is a canonical isomorphism
g^{-1} \circ Rf_! \to Rf'_! \circ (g')^{-1}
Moreover, these isomorphisms are compatible with the isomorphisms of Lemma 63.9.2.
Proof. Choose a compactification j : X \to \overline{X} over Y and denote \overline{f} : \overline{X} \to Y the structure morphism. Let j' : X' \to \overline{X}' and \overline{f}' : \overline{X}' \to Y' denote the base changes of j and \overline{f}. Since Rf_! = R\overline{f}_* \circ j_! and Rf'_! = R\overline{f}'_* \circ j'_! the isomorphism can be constructed via
g^{-1} \circ R\overline{f}_* \circ j_! \to R\overline{f}'_* \circ (\overline{g}')^{-1} \circ j_! \to R\overline{f}'_* \circ j'_! \circ (g')^{-1}
where the first arrow is the isomorphism given to us by the proper base change theorem (Étale Cohomology, Lemma 59.91.12 in the bounded below torsion case and Étale Cohomology, Lemma 59.92.3 in the case that \Lambda is torsion) and the second arrow is the isomorphism of Lemma 63.3.12.
To finish the proof we have to show two things: first we have to show that the isomorphism of functors so obtained does not depend on the choice of the compactification and second we have to show that if we vertically stack two base change diagrams as in the lemma, then these base change isomorphisms are compatible with the isomorphisms of Lemma 63.9.2. A straightforward argument which we omit shows that both follow if we can show that the isomorphisms
Rg_* \circ Rf_* = R(g \circ f)_* for f : X \to Y and g : Y \to Z proper,
g_! \circ f_! = (g \circ f)_! for f : X \to Y and g : Y \to Z separated and quasi-finite, and
g_! \circ Rf'_* = Rf_* \circ g'_! for f : X \to Y and f' : X' \to Y' proper and g : Y' \to Y and g' : X' \to X separated and quasi-finite with f \circ g' = g \circ f'
are compatible with base change. This holds for (1) by Cohomology on Sites, Remark 21.19.4, for (2) by Remark 63.3.14, and (3) by Lemma 63.8.5. \square
Remark 63.9.5. Let f : X \to Y be a finite type separated morphism of schemes with Y quasi-compact and quasi-separated. Below we will construct a map
Rf_!K \longrightarrow Rf_*K
functorial for K in D^+_{tors}(X_{\acute{e}tale}, \Lambda ) or D(X_{\acute{e}tale}, \Lambda ) if \Lambda is torsion. This transformation of functors in both cases is compatible with
the isomorphism Rg_! \circ Rf_! \to R(g \circ f)_! of Lemma 63.9.2 and the isomorphism Rg_* \circ Rf_* \to R(g \circ f)_* of Cohomology on Sites, Lemma 21.19.2 and
the isomorphism g^{-1} \circ Rf_! \to Rf'_! \circ (g')^{-1} of Lemma 63.9.4 and the base change map of Cohomology on Sites, Remark 21.19.3.
Namely, choose a compactification j : X \to \overline{X} over Y and denote \overline{f} : \overline{X} \to Y the structure morphism. Since Rf_! = R\overline{f}_* \circ j_! and Rf_* = R\overline{f}_* \circ Rj_* it suffices to construct a transformation of functors j_! \to Rj_*. For this we use the canonical transformation j_! \to j_* of Étale Cohomology, Lemma 59.70.6. We omit the proof that the resulting transformation is independent of the choice of compactification and we omit the proof of the compatibilities (1) and (2).
[1] Namely, if \alpha , \beta : F \to G are morphisms of functors and \gamma : G \to H is an isomorphism of functors such that \gamma \circ \alpha = \gamma \circ \beta , then we conclude \alpha = \beta .
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