stacks/stacks-project

Fix error

Thanks to Bogdan Zavyalov

The error was to silently assume that X is dense in any compactification
of X but in the definition of compactifications earlier in the Stacks
project we didn't require this. This was a bit hard to fix because we
then in later lemmas used the flexibility afforded by the more general
definition of a compactification.

I fixed this by leaving the definition of a compactification alone and
by very carefully arguing that inside the category of all
compacfifications of a given scheme X the ones where X is dense is an
initial category.

I also much clarified the proof of Lemma 0B6T to boot

I also checked all other uses of compactifications in the Stacks project
and the lemmas I fixed are the only ones using compactifications in the
wrong manner...

Interestingly in the case of defining etale cohomology with proper
supports we don't make the same mistake because there we have a stronger
lemma about lower shriek functors which makes the thing work more
smoothly.
 @@ -3608,11 +3608,10 @@ \section{Upper shriek functors} \end{lemma} \begin{proof} Consider the category of compactifications of $X$ over $Y$, which is cofiltered according to More on Flatness, Theorem \ref{flat-theorem-nagata} and Lemmas \ref{flat-lemma-compactifications-cofiltered} and \ref{flat-lemma-compactifyable}. The category of compactifications of $X$ over $Y$ is defined in More on Flatness, Section \ref{flat-section-compactify}. By More on Flatness, Theorem \ref{flat-theorem-nagata} and Lemma \ref{flat-lemma-compactifyable} it is nonempty. To every choice of a compactification $$j : X \to \overline{X},\quad \overline{f} : \overline{X} \to Y @@ -3624,8 +3623,11 @@ \section{Upper shriek functors} \medskip\noindent Suppose given a morphism g : \overline{X}_1 \to \overline{X}_2 between compactifications j_i : X \to \overline{X}_i over Y. Namely, let \overline{c} be the right adjoint of between compactifications j_i : X \to \overline{X}_i over Y such that g^{-1}(j_2(X)) = j_1(X)\footnote{This may fail with our definition of compactification. See More on Flatness, Section \ref{flat-section-compactify}.}. Let \overline{c} be the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for g. Then \overline{c} \circ \overline{a}_2 = \overline{a}_1 because these functors are adjoint to @@ -3646,23 +3648,73 @@ \section{Upper shriek functors} is an isomorphism of functors which we will denote by \alpha_g. \medskip\noindent 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 functors\footnote{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.}. 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. Let \overline{d} be the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for h. Then \overline{d} \circ \overline{a}_3 = \overline{a}_2 and there is a canonical transformation Consider two compactifications j_i : X \to \overline{X}_i, i = 1, 2 of X over Y. By More on Flatness, Lemma \ref{flat-lemma-compactifications-cofiltered} part (b) we can find a compactification j : X \to \overline{X} with dense image and morphisms g_i : \overline{X} \to \overline{X}_i of compactififcatins. By More on Flatness, Lemma \ref{flat-lemma-compactifications-cofiltered} part (c) we have g_i^{-1}(j_i(X)) = j(X). Hence we get isomorpisms$$ \alpha_{g_i} : j^* \circ \overline{a} \longrightarrow j_i^* \circ \overline{a}_i $$by the previous paragraph. We obtain an isomorphism$$ \alpha_{g_2} \circ \alpha_{g_1}^{-1} : j_1^* \circ \overline{a}_1 \to j_2^* \circ \overline{a}_2 $$To finish the proof we have to show that these isomorphisms are well defined. We claim it suffices to show the composition of isomorphisms constructed in the previous paragraph is another (for a precise statement see the next paragraph). We suggest the reader check this is true on a napkin, but we will also completely spell it out in the rest of this paragraph. Namely, consider a second choice of a compactification j' : X \to \overline{X}' with dense image and morphisms of compactifications g'_i : \overline{X}' \to \overline{X}_i. By More on Flatness, Lemma \ref{flat-lemma-compactifications-cofiltered} we can find a compactification j'' : X \to \overline{X}'' with dense image and morphisms of compactifications h : \overline{X}'' \to \overline{X} and h' : \overline{X}'' \to \overline{X}'. We may even assume g_1 \circ h = g'_1 \circ h' and g_2 \circ h = g'_2 \circ h'. The result of the next paragraph gives$$ \alpha_{g_i} \circ \alpha_h = \alpha_{g_i \circ h} = \alpha_{g'_i \circ h'} = \alpha_{g'_i} \circ \alpha_{h'} $$for i = 1, 2. Since these are all isomorphisms of functors we conclude that \alpha_{g_2} \circ \alpha_{g_1}^{-1} = \alpha_{g'_2} \circ \alpha_{g'_1}^{-1} as desired. \medskip\noindent Suppose given compactifications j_i : X \to \overline{X}_i for i = 1, 2, 3. Suppose given morphisms g : \overline{X}_1 \to \overline{X}_2 and h : \overline{X}_2 \to \overline{X}_3 of compactifications such that g^{-1}(j_2(X)) = j_1(X) and h^{-1}(j_2(X)) = j_3(X). Let \overline{a}_i be as above. The claim above means that$$ \alpha_g \circ \alpha_h = \alpha_{g \circ h} : j_1^* \circ \overline{a}_1 \to j_3^* \circ \overline{a}_3 $$Let \overline{c}, resp.\ \overline{d} be the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for g, resp.\ h. Then \overline{c} \circ \overline{a}_2 = \overline{a}_1 and \overline{d} \circ \overline{a}_3 = \overline{a}_2 and there are canonical transformations$$ j_1^* \circ \overline{c} \longrightarrow j_2^* \quad\text{and}\quad j_2^* \circ \overline{d} \longrightarrow j_3^* $$of functors D^+_\QCoh(\mathcal{O}_{\overline{X}_2}) \to D^+_\QCoh(\mathcal{O}_X) and D^+_\QCoh(\mathcal{O}_{\overline{X}_3}) \to D^+_\QCoh(\mathcal{O}_X) for the same reasons as above. Denote \overline{e} the right adjoint of Lemma \ref{lemma-twisted-inverse-image} @@ -3744,7 +3796,7 @@ \section{Upper shriek functors} Then \overline{a} \circ \overline{b} is the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for the composition \overline{X} \to Z. Hence g^! = j_Y^* \circ \overline{b} and Hence g^! = i^* \circ \overline{b} and (g \circ f)^! = (X \to \overline{X})^* \circ \overline{a} \circ \overline{b}. Let U be the inverse image of Y in \overline{X} so that we get the commutative diagram @@ -3760,15 +3812,15 @@ \section{Upper shriek functors} U \to Y. Then f^! = j^* \circ \overline{a}'. We obtain$$ \gamma : (j')^* \circ \overline{a} \to \overline{a}' \circ j_Y^* \gamma : (j')^* \circ \overline{a} \to \overline{a}' \circ i^* $$by (\ref{equation-sheafy}) and we can use it to define$$ (g \circ f)^! = j_X^* \circ \overline{a} \circ \overline{b} = (j' \circ j)^* \circ \overline{a} \circ \overline{b} = j^* \circ (j')^* \circ \overline{a} \circ \overline{b} \to j^* \circ \overline{a}' \circ j_Y^* \circ \overline{b} = j^* \circ \overline{a}' \circ i^* \circ \overline{b} = f^! \circ g^! $$which is an isomorphism on objects of D_\QCoh^+(\mathcal{O}_Z) by @@ -3795,12 +3847,13 @@ \section{Upper shriek functors} }$$ We can first choose a compactification $i : Y \to \overline{Y}$ of $Y$ over $Z$ which dominates both $\overline{Y}_1$ and $\overline{Y}_2$, with dense image of $Y$ over $Z$ which dominates both $\overline{Y}_1$ and $\overline{Y}_2$, see More on Flatness, Lemma \ref{flat-lemma-compactifications-cofiltered}. By More on Flatness, Lemma \ref{flat-lemma-right-multiplicative-system} and Categories, Lemmas \ref{categories-lemma-morphisms-right-localization} and \ref{categories-lemma-equality-morphisms-right-localization} we can choose a compactification $X \to \overline{X}$ of we can choose a compactification $X \to \overline{X}$ with dense image 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 @@ -3824,6 +3877,8 @@ \section{Upper shriek functors} Z } $$and moreover the compactifications X \to \overline{X}_1 and Y \to \overline{Y}_2 have dense image. We use \overline{a}_i, \overline{a}'_i, \overline{c}, and \overline{c}' for the right adjoint of Lemma \ref{lemma-twisted-inverse-image} for @@ -3856,6 +3911,10 @@ \section{Upper shriek functors} X \ar[r] & \overline{X}_2 }$$ is cartesian (see More on Flatness, Lemma \ref{flat-lemma-compactifications-cofiltered} part (c) for A, D, E and recall that $U_i$ is the inverse image of $Y$ by $\overline{X}_i \to \overline{Y}_i$ for B, C) and hence gives rise to a base change map (\ref{equation-sheafy}) as follows $$\begin{matrix} @@ -4019,20 +4078,26 @@ \section{Upper shriek functors} the commutativity of the diagram (in greater generality). \medskip\noindent In general, choose a compactification j : X \to \overline{X} of X over Y. Since f^! is defined as j^* \circ \overline{a} we obtain \mu_f as the restriction of the map (\ref{equation-compare-with-pullback}) Let us define the map \mu_{f, K}. Choose a compactification j : X \to \overline{X} of X over Y. Since f^! is defined as j^* \circ \overline{a} we obtain \mu_{f, K} as the restriction of the map (\ref{equation-compare-with-pullback})$$ L\overline{f}^*K \otimes_{\mathcal{O}_{\overline{X}}}^\mathbf{L} \overline{a}(\mathcal{O}_Y) \longrightarrow \overline{a}(K) $$to X. To see this is independent of the choice of the compactification, we may assume given a morphism g : \overline{X}_1 \to \overline{X}_2 between compactifications j_i : X \to \overline{X}_i over Y. But now we know that the maps to X. To see this is independent of the choice of the compactification we argue as in the proof of Lemma \ref{lemma-shriek-well-defined}. We urge the reader to read the proof of that lemma first. \medskip\noindent Assume given a morphism g : \overline{X}_1 \to \overline{X}_2 between compactifications j_i : X \to \overline{X}_i over Y such that g^{-1}(j_2(X)) = j_1(X). Denote \overline{c} the right adjoint for pushforward of Lemma \ref{lemma-twisted-inverse-image} for the morphism g. The maps$$ L\overline{f}_1^*K \otimes_{\mathcal{O}_{\overline{X}}}^\mathbf{L} \overline{a}_1(\mathcal{O}_Y) @@ -4044,15 +4109,64 @@ \section{Upper shriek functors} \longrightarrow \overline{a}_2(K) $$fit into a commutative diagram by Lemma \ref{lemma-transitivity-compare-with-pullback} with two other maps given by \mu_g which restrict to an isomorphism on X by Lemma \ref{lemma-compare-on-open}. This implies the two displayed maps above restrict to the same map on the open, via the identification \overline{a}_1(K)|_X = \overline{a}_2(K)|_X used in the definition of f^!. Having said this, the commutativity of the diagram follows from the construction of the isomorphism fit into the commutative diagram$$ \xymatrix{ Lg^*(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)) \otimes^\mathbf{L} \overline{c}(\mathcal{O}_{\overline{X}_2}) \ar@{=}[d] \ar[r]_-\sigma & \overline{c}(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)) \ar[r] & \overline{c}(\overline{a}_2(K)) \ar@{=}[d] \\ L\overline{f}_1^*K \otimes^\mathbf{L} Lg^*\overline{a}_2(\mathcal{O}_Y) \otimes^\mathbf{L} \overline{c}(\mathcal{O}_{\overline{X}_2}) \ar[r]^-{1 \otimes \tau} & L\overline{f}_1^*K \otimes^\mathbf{L} \overline{a}_1(\mathcal{O}_Y) \ar[r] & \overline{a}_1(K) } $$by Lemma \ref{lemma-transitivity-compare-with-pullback}. By Lemma \ref{lemma-compare-on-open} the maps \sigma and \tau restrict to an isomorphism over X. In fact, we can say more. Recall that in the proof of Lemma \ref{lemma-shriek-well-defined} we used the map (\ref{equation-sheafy}) \gamma : j_1^* \circ \overline{c} \to j_2^* to construct our isomorphism \alpha_g : j_1^* \circ \overline{a}_1 \to j_2^* \circ \overline{a}_2. Pulling back to map \sigma by j_1 we obtain the identity map on j_2^*\left(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)\right) if we identify j_1^*\overline{c}(\mathcal{O}_{\overline{X}_2}) with \mathcal{O}_X via j_1^* \circ \overline{c} \to j_2^*, see Lemma \ref{lemma-restriction-compare-with-pullback}. Similarly, the map \tau : Lg^*\overline{a}_2(\mathcal{O}_Y) \otimes^\mathbf{L} \overline{c}(\mathcal{O}_{\overline{X}_2}) \to \overline{a}_1(\mathcal{O}_Y) = \overline{c}(\overline{a}_2(\mathcal{O}_Y)) pulls back to the identity map on j_2^*\overline{a}_2(\mathcal{O}_Y). We conclude that pulling back by j_1 and applying \gamma wherever we can we obtain a commutative diagram$$ \xymatrix{ j_2^*\left(L\overline{f}_2^*K \otimes^\mathbf{L} \overline{a}_2(\mathcal{O}_Y)\right) \ar[r] \ar[d] & j_2^*\overline{a}_2(K) \\ j_1^*L\overline{f}_1^*K \otimes^\mathbf{L} j_2^*\overline{a}_2(\mathcal{O}_Y) & j_1^*(L\overline{f}_1^*K \otimes^\mathbf{L} \overline{a}_1(\mathcal{O}_Y)) \ar[r] \ar[l]_{1 \otimes \alpha_g} & j_1^* \overline{a}_1(K) \ar[lu]_{\alpha_g} }  The commutativity of this diagram exactly tells us that the map $\mu_{f, K}$ constructed using the compactification $\overline{X}_1$ is the same as the map $\mu_{f, K}$ constructed using the compactification $\overline{X}_2$ via the identification $\alpha_g$ used in the proof of Lemma \ref{lemma-shriek-well-defined}. Some categorical arguments exactly as in the proof of Lemma \ref{lemma-shriek-well-defined} now show that $\mu_{f, K}$ is well defined (small detail omitted). \medskip\noindent Having said this, the commutativity of the diagram in the statement of our lemma follows from the construction of the isomorphism $(g \circ f)^! \to f^! \circ g^!$ (first part of the proof of Lemma \ref{lemma-upper-shriek-composition} using $\overline{X} \to \overline{Y} \to Z$) and the result