Lemma 48.32.1. The functor Rf_! is, up to isomorphism, independent of the choice of the compactification.
48.32 Compactly supported cohomology for coherent modules
In Situation 48.16.1 given a morphism f : X \to Y in \textit{FTS}_ S, we will define a functor
Rf_! : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \longrightarrow \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)
Namely, we choose a compactification j : X \to \overline{X} over Y which is possible by More on Flatness, Theorem 38.33.8 and Lemma 38.32.2. Denote \overline{f} : \overline{X} \to Y the structure morphism. Then we set
Rf_!K = R\overline{f}_* Rj_! K
for K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X) where Rj_! is as in Remark 48.30.5.
In fact, the functor Rf_! will be characterized as a “left adjoint” to f^! which will determine it up to unique isomorphism.
Proof. 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 Rj_!. 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 Rj_{2, !} = R\overline{f}_{2, *} \circ Rg_* \circ j_{1, !} = R\overline{f}_{1, *} \circ Rj_{1, !}
using Lemma 48.31.2 in the first equality. In this way we see our functor is independent of the choice of compactification up to isomorphism. \square
Proposition 48.32.2. In Situation 48.16.1 let f : X \to Y be a morphism of \textit{FTS}_ S. Then the functors Rf_! and f^! are adjoint in the following sense: for all K \in D^ b_{\textit{Coh}}(\mathcal{O}_ X) and L \in D^+_{\textit{Coh}}(\mathcal{O}_ Y) we have
\mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L)
bifunctorially in K and L.
Proof. Choose a compactification j : X \to \overline{X} over Y and denote \overline{f} : \overline{X} \to Y the structure morphism. Then we have
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) & = \mathop{\mathrm{Hom}}\nolimits _ X(K, j^*\overline{f}{}^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_{\overline{X}})} (Rj_!K, \overline{f}{}^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_*Rj_!K, L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^+_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) \end{align*}
The first equality follows immediately from the construction of f^! in Section 48.16. By Lemma 48.17.6 we have \overline{f}{}^!L in D^+_{\textit{Coh}}(\mathcal{O}_{\overline{X}}) hence the second equality follows from Lemma 48.30.2. Since \overline{f} is proper the functor \overline{f}{}^! is the right adjoint of pushforward by construction. This is why we have the third equality. The fourth equality holds because Rf_! = Rf_* Rj_!. \square
Lemma 48.32.3. In Situation 48.16.1 let f : X \to Y be a morphism of \textit{FTS}_ S. Let
K \to L \to M \to K[1]
be a distinguished triangle of D^ b_{\textit{Coh}}(\mathcal{O}_ X). Then there exists an inverse system of distinguished triangles
K_ n \to L_ n \to M_ n \to K_ n[1]
in D^ b_{\textit{Coh}}(\mathcal{O}_ Y) such that the pro-systems (K_ n), (L_ n), and (M_ n) give Rf_!K, Rf_!L, and Rf_!M.
Proof. Choose a compactification j : X \to \overline{X} over Y and denote \overline{f} : \overline{X} \to Y the structure morphism. Choose an inverse system of distinguished triangles
\overline{K}_ n \to \overline{L}_ n \to \overline{M}_ n \to \overline{K}_ n[1]
in D^ b_{\textit{Coh}}(\mathcal{O}_{\overline{X}}) as in Lemma 48.30.4 corresponding to the open immersion j and the given distinguished triangle. Take K_ n = R\overline{f}_*\overline{K}_ n and similarly for L_ n and M_ n. This works by the very definition of Rf_!. \square
Remark 48.32.4. Let \mathcal{C} be a category. Suppose given an inverse system
\ldots \xrightarrow {\alpha _4} (M_{3, n}) \xrightarrow {\alpha _3} (M_{2, n}) \xrightarrow {\alpha _2} (M_{1, n})
of inverse systems in the category of pro-objects of \mathcal{C}. In other words, the arrows \alpha _ i are morphisms of pro-objects. By Categories, Example 4.22.6 we can represent each \alpha _ i by a pair (m_ i, a_ i) where m_ i : \mathbf{N} \to \mathbf{N} is an increasing function and a_{i, n} : M_{i, m_ i(n)} \to M_{i - 1, n} is a morphism of \mathcal{C} making the diagrams
\xymatrix{ \ldots \ar[r] & M_{i, m_ i(3)} \ar[d]^{a_{i, 3}} \ar[r] & M_{i, m_ i(2)} \ar[d]^{a_{i, 2}} \ar[r] & M_{i, m_ i(1)} \ar[d]^{a_{i, 1}} \\ \ldots \ar[r] & M_{i - 1, 3} \ar[r] & M_{i - 1, 2} \ar[r] & M_{i - 1, 1} }
commute. By replacing m_ i(n) by \max (n, m_ i(n)) and adjusting the morphisms a_ i(n) accordingly (as in the example referenced) we may assume that m_ i(n) \geq n. In this situation consider the inverse system
\ldots \to M_{4, m_4(m_3(m_2(4)))} \to M_{3, m_3(m_2(3))} \to M_{2, m_2(2)} \to M_{1, 1}
with general term
M_ k = M_{k, m_ k(m_{k - 1}(\ldots (m_2(k))\ldots ))}
For any object N of \mathcal{C} we have
\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_{i, n}, N) = \mathop{\mathrm{colim}}\nolimits _ k \mathop{\mathrm{Mor}}\nolimits _\mathcal {C}(M_ k, N)
We omit the details. In other words, we see that the inverse system (M_ k) has the property
\mathop{\mathrm{colim}}\nolimits _ i \mathop{\mathrm{Mor}}\nolimits _{\text{Pro-}\mathcal{C}}((M_{i, n}), N) = \mathop{\mathrm{Mor}}\nolimits _{\text{Pro-}\mathcal{C}}((M_ k), N)
This property determines the inverse system (M_ k) up to pro-isomorphism by the discussion in Categories, Remark 4.22.7. In this way we can turn certain inverse systems in \text{Pro-}\mathcal{C} into pro-objects with countable index categories.
Remark 48.32.5. In Situation 48.16.1 let f : X \to Y and g : Y \to Z be composable morphisms of \textit{FTS}_ S. Let us define the composition
Rg_! \circ Rf_! : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \longrightarrow \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Z)
Namely, by the very construction of Rf_! for K in D^ b_{\textit{Coh}}(\mathcal{O}_ X) the output Rf_!K is the pro-isomorphism class of an inverse system (M_ n) in D^ b_{\textit{Coh}}(\mathcal{O}_ Y). Then, since Rg_! is constructed similarly, we see that
\ldots \to Rg_!M_3 \to Rg_!M_2 \to Rg_!M_1
is an inverse system of \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y). By the discussion in Remark 48.32.4 there is a unique pro-isomorphism class, which we will denote Rg_! Rf_! K, of inverse systems in D^ b_{\textit{Coh}}(\mathcal{O}_ Z) such that
\mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Z)}(Rg_!Rf_!K, L) = \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Z)}(Rg_!M_ n, L)
We omit the discussion necessary to see that this construction is functorial in K as it will immediately follow from the next lemma.
Lemma 48.32.6. In Situation 48.16.1 let f : X \to Y and g : Y \to Z be composable morphisms of \textit{FTS}_ S. With notation as in Remark 48.32.5 we have Rg_! \circ Rf_! = R(g \circ f)_!.
Proof. By the discussion in Categories, Remark 4.22.7 it suffices to show that we obtain the same answer if we compute \mathop{\mathrm{Hom}}\nolimits into L in D^ b_{\textit{Coh}}(\mathcal{O}_ Z). To do this we compute, using the notation in Remark 48.32.5, as follows
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _ Z(Rg_!Rf_!K, L) & = \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _ Z(Rg_!M_ n, L) \\ & = \mathop{\mathrm{colim}}\nolimits _ n \mathop{\mathrm{Hom}}\nolimits _ Y(M_ n, g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Y(Rf_!K, g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _ X(K, (g \circ f)^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _ Z(R(g \circ f)_!K, L) \end{align*}
The first equality is the definition of Rg_!Rf_!K. The second equality is Proposition 48.32.2 for g. The third equality is the fact that Rf_!K is given by (M_ n). The fourth equality is Proposition 48.32.2 for f. The fifth equality is Lemma 48.16.3. The sixth is Proposition 48.32.2 for g \circ f. \square
Remark 48.32.7. In Situation 48.16.1 let f : X \to Y be a morphism of \textit{FTS}_ S and let U \subset X be an open. Set g = f|_ U : U \to Y. Then there is a canonical morphism
Rg_!(K|_ U) \longrightarrow Rf_!K
functorial in K in D^ b_{\textit{Coh}}(\mathcal{O}_ X) which can be defined in at least 3 ways.
Denote i : U \to X the inclusion morphism. We have Rg_! = Rf_! \circ Ri_! by Lemma 48.32.6 and we can use Rf_! applied to the map Ri_!(K|_ U) \to K which is a special case of Remark 48.31.3.
Choose a compactification j : X \to \overline{X} of X over Y with structure morphism \overline{f} : \overline{X} \to Y. Set j' = j \circ i : U \to \overline{X}. We can use that Rf_! = R\overline{f}_* \circ Rj_! and Rg_! = R\overline{f}_* \circ Rj'_! and we can use R\overline{f}_* applied to the map Rj'_!(K|_ U) \to Rj_!K of Remark 48.31.3.
We can use
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) & = \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) \\ & \to \mathop{\mathrm{Hom}}\nolimits _ U(K|_ U, f^!L|_ U) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(K|_ U, g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rg_!(K|_ U), L) \end{align*}functorial in L and K. Here we have used Proposition 48.32.2 twice and the construction of upper shriek functors which shows that g^! = i^* \circ f^!. The functoriality in L shows by Categories, Remark 4.22.7 that we obtain a canonical map Rg_!(K|_ U) \to Rf_!K in \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y) which is functorial in K by the functoriality of the arrow above in K.
Each of these three constructions gives the same arrow; we omit the details.
Remark 48.32.8. Let us generalize the covariance of compactly supported cohomology given in Remark 48.32.7 to étale morphisms. Namely, in Situation 48.16.1 suppose given a commutative diagram
\xymatrix{ U \ar[rr]_ h \ar[rd]_ g & & X \ar[ld]^ f \\ & Y }
of \textit{FTS}_ S with h étale. Then there is a canonical morphism
Rg_!(h^*K) \longrightarrow Rf_!K
functorial in K in D^ b_{\textit{Coh}}(\mathcal{O}_ X). We define this transformation using the sequence of maps
\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_!K, L) & = \mathop{\mathrm{Hom}}\nolimits _ X(K, f^!L) \\ & \to \mathop{\mathrm{Hom}}\nolimits _ U(h^*K, h^*(f^!L)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(h^*K, h^!f^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(h^*K, g^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rg_!(h^*K), L) \end{align*}
functorial in L and K. Here we have used Proposition 48.32.2 twice, we have used the equality h^* = h^! of Lemma 48.18.2, and we have used the equality h^! \circ f^! = g^! of Lemma 48.16.3. The functoriality in L shows by Categories, Remark 4.22.7 that we obtain a canonical map Rg_!(h^*K) \to Rf_!K in \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y) which is functorial in K by the functoriality of the arrow above in K.
Remark 48.32.9. In Remarks 48.32.7 and 48.32.8 we have seen that the construction of compactly supported cohomology is covariant with respect to open immersions and étale morphisms. In fact, the correct generality is that given a commutative diagram
\xymatrix{ U \ar[rr]_ h \ar[rd]_ g & & X \ar[ld]^ f \\ & Y }
of \textit{FTS}_ S with h flat and quasi-finite there exists a canonical transformation
Rg_! \circ h^* \longrightarrow Rf_!
As in Remark 48.32.8 this map can be constructed using a transformation of functors h^* \to h^! on D^+_{\textit{Coh}}(\mathcal{O}_ X). Recall that h^!K = h^*K \otimes \omega _{U/X} where \omega _{U/X} = h^!\mathcal{O}_ X is the relative dualizing sheaf of the flat quasi-finite morphism h (see Lemmas 48.17.9 and 48.21.6). Recall that \omega _{U/X} is the same as the relative dualizing module which will be constructed in Discriminants, Remark 49.2.11 by Discriminants, Lemma 49.15.1. Thus we can use the trace element \tau _{U/X} : \mathcal{O}_ U \to \omega _{U/X} which will be constructed in Discriminants, Remark 49.4.7 to define our transformation. If we ever need this, we will precisely formulate and prove the result here.
Comments (0)