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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.
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aisejohan committed Nov 16, 2019
1 parent da4e7fb commit 93cea2a4c6dd66215c5e986ad9cca8aaa2c6fb0e
Showing with 214 additions and 60 deletions.
  1. +1 −0 CONTRIBUTORS
  2. +157 −43 duality.tex
  3. +56 −17 flat.tex
@@ -354,6 +354,7 @@ Alex Youcis
Jize Yu
John Yu
Felipe Zaldivar
Bogdan Zavyalov
Maciek Zdanowicz
Dingxin Zhang
Keke Zhang
@@ -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

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