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
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 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.
\[ \xymatrix{ U \ar[r]_ j \ar[d]_ g & X \ar[d]^ f \\ V \ar[r]^{j'} & Y } \]
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
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
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
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
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 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 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 Clearly the map constructed above is given by the maps $K'_ n \to K_ n$ coming from the inclusions $(\mathcal{I}')^ n \subset \mathcal{I}^ n$.
\[ Rj'_!(K|_{U'}) \longrightarrow Rj_!K \]
\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*}
\[ K_ n = \mathcal{I}^ n\mathcal{F}^\bullet \quad \text{resp.}\quad K'_ n = (\mathcal{I}')^ n\mathcal{F}^\bullet \]
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