The Stacks project

48.31 Preliminaries to compactly supported cohomology

In Situation 48.16.1 let $f : X \to Y$ be a morphism in the category $\textit{FTS}_ S$. Using the constructions in the previous section, we will construct a functor

\[ Rf_! : D^ b_{\textit{Coh}}(\mathcal{O}_ X) \longrightarrow \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y) \]

which reduces to the functor of Remark 48.30.5 if $f$ is an open immersion and in general is constructed using a compactification of $f$. Before we do this, we need the following lemmas to prove our construction is well defined.

Lemma 48.31.1. Let $f : X \to Y$ be a proper morphism of Noetherian schemes. Let $V \subset Y$ be an open subscheme and set $U = f^{-1}(V)$. Picture

\[ \xymatrix{ U \ar[r]_ j \ar[d]_ g & X \ar[d]^ f \\ V \ar[r]^{j'} & Y } \]

Then we have a canonical isomorphism $Rj'_! \circ Rg_* \to Rf_* \circ Rj_!$ of functors $D^ b_{\textit{Coh}}(\mathcal{O}_ U) \to \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$ where $Rj_!$ and $Rj'_!$ are as in Remark 48.30.5.

First proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Let $(K_ n)$ be a Deligne system for $U \to X$ whose restriction to $U$ is constant with value $K$. Of course this means that $(K_ n)$ represents $Rj_!K$ in $\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)$. Observe that both $Rj'_!Rg_*K$ and $Rf_*Rj_!K$ restrict to the constant pro-object with value $Rg_*K$ on $V$. This is immediate for the first one and for the second one it follows from the fact that $(Rf_*K_ n)|_ V = Rg_*(K_ n|_ U) = Rg_*K$. By the uniqueness of Deligne systems in Lemma 48.30.3 it suffices to show that $(Rf_*K_ n)$ is pro-isomorphic to a Deligne system. The lemma referenced will also show that the isomorphism we obtain is functorial.

Proof that $(Rf_*K_ n)$ is pro-isomorphic to a Deligne system. First, we observe that the question is independent of the choice of the Deligne system $(K_ n)$ corresponding to $K$ (by the aforementioned uniqueness). By Lemmas 48.30.4 and 48.30.7 if we have a distinguished triangle

\[ K \to L \to M \to K[1] \]

in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$ and the result holds for $K$ and $M$, then the result holds for $L$. Using the distinguished triangles of canonical truncations (Derived Categories, Remark 13.12.4) we reduce to the problem studied in the next paragraph.

Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Let $\mathcal{J} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals cutting out $Y \setminus V$. Denote $\mathcal{J}^ n\mathcal{F}$ the image of $f^*\mathcal{J}^ n \otimes \mathcal{F} \to \mathcal{F}$. We have to show that $(Rf_*(\mathcal{J}^ n\mathcal{F}))$ is a Deligne system. By Lemma 48.30.10 the question is local on $Y$. Thus we may assume $Y = \mathop{\mathrm{Spec}}(A)$ is affine and $\mathcal{J}$ corresponds to an ideal $I \subset A$. By Lemma 48.30.9 it suffices to show that the inverse system of cohomology modules $(H^ p(X, I^ n\mathcal{F}))$ is pro-isomorphic to the inverse system $(I^ n M)$ for some finite $A$-module $M$. This is shown in Cohomology of Schemes, Lemma 30.20.3. $\square$

Second proof. Let $K$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Let $L$ be an object of $D^ b_{\textit{Coh}}(\mathcal{O}_ Y)$. We will construct a bijection

\[ \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rj'_!Rg_*K, L) \longrightarrow \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_*Rj_!K, L) \]

functorial in $K$ and $L$. Fixing $K$ this will determine an isomorphism of pro-objects $Rf_*Rj_!K \to Rj'_!Rg_*K$ by Categories, Remark 4.22.7 and varying $K$ we obtain that this determines an isomorphism of functors. To actually produce the isomorphism we use the sequence of functorial equalities

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rj'_!Rg_*K, L) & = \mathop{\mathrm{Hom}}\nolimits _ V(Rg_*K, L|_ V) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(K, g^!(L|_ V)) \\ & = \mathop{\mathrm{Hom}}\nolimits _ U(K, f^!L|_ U)) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)}(Rj_!K, f^!L) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ Y)}(Rf_*Rj_!K, L) \end{align*}

The first equality is true by Lemma 48.30.1. The second equality is true because $g$ is proper (as the base change of $f$ to $V$) and hence $g^!$ is the right adjoint of pushforward by construction, see Section 48.16. The third equality holds as $g^!(L|_ V) = f^!L|_ U$ by Lemma 48.17.2. Since $f^!L$ is in $D^+_{\textit{Coh}}(\mathcal{O}_ X)$ by Lemma 48.17.6 the fourth equality follows from Lemma 48.30.2. The fifth equality holds again because $f^!$ is the right adjoint to $Rf_*$ as $f$ is proper. $\square$

Lemma 48.31.2. Let $j : U \to X$ be an open immersion of Noetherian schemes. Let $j' : U \to X'$ be a compactification of $U$ over $X$ (see proof) and denote $f : X' \to X$ the structure morphism. Then we have a canonical isomorphism $Rj_! \to Rf_* \circ R(j')_!$ of functors $D^ b_{\textit{Coh}}(\mathcal{O}_ U) \to \text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ where $Rj_!$ and $Rj'_!$ are as in Remark 48.30.5.

Proof. The fact that $X'$ is a compactification of $U$ over $X$ means precisely that $f : X' \to X$ is proper, that $j'$ is an open immersion, and $j = f \circ j'$. See More on Flatness, Section 38.32. If $j'(U) = f^{-1}(j(U))$, then the lemma follows immediately from Lemma 48.31.1. If $j'(U) \not= f^{-1}(j(U))$, then denote $X'' \subset X'$ the scheme theoretic closure of $j' : U \to X'$ and denote $j'' : U \to X''$ the corresponding open immersion. Picture

\[ \xymatrix{ & & X'' \ar[d]^{f'} \\ & & X' \ar[d]^ f \\ U \ar[rr]^ j \ar[rru]^{j'} \ar[rruu]^{j''} & & X } \]

By More on Flatness, Lemma 38.32.1 part (c) and the discussion above we have isomorphisms $Rf'_* \circ Rj''_! = Rj'_!$ and $R(f \circ f')_* \circ Rj''_! = Rj_!$. Since $R(f \circ f')_* = Rf_* \circ Rf'_*$ we conclude. $\square$

Remark 48.31.3. Let $X \supset U \supset U'$ be open subschemes of a Noetherian scheme $X$. Denote $j : U \to X$ and $j' : U' \to X$ the inclusion morphisms. We claim there is a canonical map

\[ Rj'_!(K|_{U'}) \longrightarrow Rj_!K \]

functorial for $K$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ U)$. Namely, by Lemma 48.30.1 we have for any $L$ in $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ the map

\begin{align*} \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)}(Rj_!K, L) & = \mathop{\mathrm{Hom}}\nolimits _ U(K, L|_ U) \\ & \to \mathop{\mathrm{Hom}}\nolimits _{U'}(K|_{U'}, L|_{U'}) \\ & = \mathop{\mathrm{Hom}}\nolimits _{\text{Pro-}D^ b_{\textit{Coh}}(\mathcal{O}_ X)}(Rj'_!(K|_{U'}), L) \end{align*}

functorial in $L$ and $K'$. The functoriality in $L$ shows by Categories, Remark 4.22.7 that we obtain a canonical map $Rj'_!(K|_{U'}) \to Rj_!K$ which is functorial in $K$ by the functoriality of the arrow above in $K$.

Here is an explicit construction of this arrow. Namely, suppose that $\mathcal{F}^\bullet $ is a bounded complex of coherent $\mathcal{O}_ X$-modules whose restriction to $U$ represents $K$ in the derived category. We have seen in the proof of Lemma 48.30.3 that such a complex always exists. Let $\mathcal{I}$, resp. $\mathcal{I}'$ be a quasi-coherent sheaf of ideals on $X$ with $V(\mathcal{I}) = X \setminus U$, resp. $V(\mathcal{I}') = X \setminus U'$. After replacing $\mathcal{I}$ by $\mathcal{I} + \mathcal{I}'$ we may assume $\mathcal{I}' \subset \mathcal{I}$. By construction $Rj_!K$, resp. $Rj'_!(K|_{U'})$ is represented by the inverse system $(K_ n)$, resp. $(K'_ n)$ of $D^ b_{\textit{Coh}}(\mathcal{O}_ X)$ with

\[ K_ n = \mathcal{I}^ n\mathcal{F}^\bullet \quad \text{resp.}\quad K'_ n = (\mathcal{I}')^ n\mathcal{F}^\bullet \]

Clearly the map constructed above is given by the maps $K'_ n \to K_ n$ coming from the inclusions $(\mathcal{I}')^ n \subset \mathcal{I}^ n$.

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