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Characterization essential image of R\epsilon_*

Thanks to Bhargav Bhatt

Presumably this doesn't work with unbounded dudes...?
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aisejohan committed Oct 27, 2018
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  1. +19 −167 etale-cohomology.tex
  2. +257 −8 sites-cohomology.tex
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@@ -19740,7 +19740,7 @@ \section{Almost blow up squares and the h topology}
$X' \setminus E \to X \setminus Z$.
\medskip\noindent
We are going to give a criterion for ``h sheafifiness'' for
We are going to give a criterion for ``h sheafiness'' for
objects in the derived category of the big fppf site
$(\Sch/S)_{fppf}$ of a scheme $S$. On the same underlying category
we have a second topology, namely the h topology
@@ -19758,90 +19758,23 @@ \section{Almost blow up squares and the h topology}
$$
and we can ask: what is the essential image of this functor?
\medskip\noindent
Let $K$ be an object of $D((\Sch/S)_{fppf})$ and consider an almost
blow up square (\ref{equation-almost-blow-up-square}) in $(\Sch/S)_{fppf}$.
Choose a K-injective complex $\mathcal{I}^\bullet$ of abelian sheaves
representing $K$. Then we get a complex of complexes
(use $-1$ times the canonical map for one of the four arrows)
$$
\mathcal{I}^\bullet(X) \xrightarrow{\alpha}
\mathcal{I}^\bullet(Z) \oplus
\mathcal{I}^\bullet(X') \xrightarrow{\beta}
\mathcal{I}^\bullet(E)
$$
Since $\beta \circ \alpha = 0$ we get a canonical map
$$
c^K_{X, X', Z, E} : \mathcal{I}^\bullet(X) \longrightarrow C(\beta)^\bullet[-1]
$$
This map is canonical in the sense that a different choice
of K-injective complex representing $K$ determines an isomorphic
arrow in the derived category of abelian groups.
We are interested in the question of whether
$c^K_{X, X', Z, E}$ is a quasi-isomorphism\footnote{If so, then
we obtain a canonical distinguished diagram
$$
R\Gamma_{fppf}(X, K) \to
R\Gamma_{fppf}(Z, K) \oplus
R\Gamma_{fppf}(X', K) \to
R\Gamma_{fppf}(E, K) \to
R\Gamma_{fppf}(X, K)[1]
$$
However, simply asking for the existence of a distinguished diagram
of this form isn't precise enough, hence the slightly cumbersome
formulation.}.
\begin{lemma}
\label{lemma-blow-up-square-h}
With notation as above, if $K$ is in the essential image
of $R\epsilon_*$, then the maps $c^K_{X, X', Z, E}$
of $R\epsilon_*$, then the maps $c^K_{X, Z, X', E}$ of
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-c-square}
are quasi-isomorphisms.
\end{lemma}
\begin{proof}
Say $K = R\epsilon_*L$. In this case we can choose
$\mathcal{I}^\bullet = \epsilon_*\mathcal{J}^\bullet$
were $\mathcal{J}^\bullet$ is a K-injective complex of abelian
sheaves on $(\Sch/S)_h$ representing $L$. See
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-K-injective-flat}.
Then applying More on Flatness, Lemma \ref{flat-lemma-blow-up-square-h}
we conclude that
$$
0 \to
\mathcal{J}^\bullet(X) \xrightarrow{\alpha}
\mathcal{J}^\bullet(Z) \oplus
\mathcal{J}^\bullet(X') \xrightarrow{\beta}
\mathcal{J}^\bullet(E)
\to 0
$$
is a short exact sequence of complexes of abelian groups
(for exactness on the right see proof of
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-square-triangle})
which forces $c^K_{X, X', Z, E}$ to be a quasi-isomorphism.
\end{proof}
\begin{lemma}
\label{lemma-two-out-of-three-blow-up-square}
Let $K_1 \to K_2 \to K_3 \to K_1[1]$ be a distinguished
triangle in $D((\Sch/S)_{fppf})$.
If $c^{K_i}_{X, X', Z, E}$ is a quasi-isomorphism for
two $i$ out of $\{1, 2, 3\}$, then it is a quasi-isomorphism
for the third $i$.
\end{lemma}
\begin{proof}
By rotating the triangle we may assume $c^{K_1}_{X, X', Z, E}$ and
$c^{K_2}_{X, X', Z, E}$ are quasi-isomorphisms. Choose a map
$f : \mathcal{I}^\bullet_1 \to \mathcal{I}^\bullet_2$ of
K-injective complexes representing $K_1 \to K_2$.
Then $K_3$ is represented by the K-injective complex
$C(f)^\bullet$, see
Derived Categories, Lemma \ref{derived-lemma-triangle-K-injective}.
Then the morphism $c^{K_3}_{X, X', Z, E}$ is an isomorphism
as it is the third leg in a map of distinguished triangles
in $K(\textit{Ab})$ whose other two legs are quasi-isomorphisms.
Some details omitted; use
Derived Categories, Lemma \ref{derived-lemma-third-isomorphism-triangle}.
Denote ${}^\#$ sheafification in the h topology.
We have seen in More on Flatness, Lemma \ref{flat-lemma-blow-up-square-h}
that $h_X^\# = h_Z^\# \amalg_{h_E^\#} h_{X'}^\#$. On the other hand,
the map $h_E^\# \to h_{X'}^\#$ is injective as $E \to X'$ is a
monomorphism. Thus this lemma is a special case of
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-descent-squares-helper}
(which itself is a formal consequence of
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-square-triangle}).
\end{proof}
\begin{proposition}
@@ -19855,92 +19788,11 @@ \section{Almost blow up squares and the h topology}
\end{proposition}
\begin{proof}
One direction we have seen in Lemma \ref{lemma-blow-up-square-h}.
On the other hand, if $K$ is a sheaf, then the proposition follows from
More on Flatness, Proposition \ref{flat-proposition-check-h}.
In general we will use an induction argument to finish the proof.
\medskip\noindent
Namely, let $K$ be an object of $D^+((\Sch/S)_{fppf})$ satisfying
the condition on almost blow up squares.
Choose a bounded below complex $\mathcal{K}^\bullet$ of
abelian sheaves on $(\Sch/S)_{fppf}$ representing $K$.
We will show by induction on $n$ that we may assume for $p \leq n$ we
have $\mathcal{K}^p = \epsilon_*\mathcal{J}^p$ for some
injective abelian sheaf $\mathcal{J}^p$ on $(\Sch/S)_h$.
The assertion is true for $n \ll 0$ because $\mathcal{K}^\bullet$
is bounded below.
\medskip\noindent
Induction step. Assume we have $\mathcal{K}^p = \epsilon_*\mathcal{J}^p$
for some injective abelian sheaf $\mathcal{J}^p$ on $(\Sch/S)_h$
for $p \leq n$. Denote $\mathcal{J}^\bullet$ the bounded complex
of injective abelian sheaves on $(\Sch/S)_h$ made from these sheaves
and the maps between them. Consider the short exact sequence of complexes
$$
0 \to \sigma_{\geq n + 1}\mathcal{K}^\bullet \to
\mathcal{K}^\bullet \to \epsilon_*\mathcal{J}^\bullet \to 0
$$
where $\sigma_{\geq n + 1}$ denotes the ``stupid'' truncation.
By Lemma \ref{lemma-blow-up-square-h} the object
$\epsilon_*\mathcal{J}^\bullet$ satisfies the condition on
almost blow up squares.
By Lemma \ref{lemma-two-out-of-three-blow-up-square}
we conclude that $\sigma_{\geq n + 1}\mathcal{K}^\bullet$
satisfies the condition on almost blow up squares.
We conclude that for every almost blow up square
(\ref{equation-almost-blow-up-square})
the sequence
$$
0 \to H^{n + 1}(X, \sigma_{\geq n + 1}\mathcal{K}^\bullet) \to
H^{n + 1}(X', \sigma_{\geq n + 1}\mathcal{K}^\bullet) \oplus
H^{n + 1}(Z, \sigma_{\geq n + 1}\mathcal{K}^\bullet) \to
H^{n + 1}(E, \sigma_{\geq n + 1}\mathcal{K}^\bullet)
$$
is exact (use the footnote above and use
there is no nonzero cohomology in degrees $< n + 1$).
We conclude that
$\Ker(\mathcal{K}^{n + 1} \to \mathcal{K}^{n + 2})$
is an h sheaf by
We prove this by applying
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-descent-squares}
whose hypotheses hold by
Lemma \ref{lemma-blow-up-square-h} and
More on Flatness, Proposition \ref{flat-proposition-check-h}.
Thus we may choose an injective abelian sheaf
$\mathcal{J}^{n + 1}$ on $(\Sch/S)_h$ and an injection
$$
\Ker(\mathcal{K}^{n + 1} \to \mathcal{K}^{n + 2})
\to \epsilon_*\mathcal{J}^{n + 1}
$$
Since $\epsilon_*\mathcal{J}$ is also an injective abelian
sheaf on $(\Sch/S)_{fppf}$
(see Cohomology on Sites, Lemma
\ref{sites-cohomology-lemma-pushforward-injective-flat})
we can extend the map above to a map
$\mathcal{K}^{n + 1} \to \epsilon_*\mathcal{J}^{n + 1}$.
Then the complex $\mathcal{K}^\bullet$ is quasi-isomorphic to the complex
$$
\ldots \to
\epsilon_*\mathcal{J}^n \to
\epsilon_*\mathcal{J}^{n + 1} \to
\frac{\epsilon_*\mathcal{J}^{n + 1} \oplus \mathcal{K}^{n + 2}}{\mathcal{K}^{n + 1}}
\to
\mathcal{K}^{n + 3} \to \ldots
$$
This finishes the induction step.
\medskip\noindent
The induction procedure described above actually produces a sequence of
quasi-isomorphisms of complexes
$$
\mathcal{K}^\bullet \to
\mathcal{K}_{n_0}^\bullet \to
\mathcal{K}_{n_0 + 1}^\bullet \to
\mathcal{K}_{n_0 + 2}^\bullet \to \ldots
$$
where $\mathcal{K}_n^\bullet \to \mathcal{K}_{n + 1}^\bullet$
is an isomorphism in degrees $\leq n$ and such that
$\mathcal{K}_n^p = \epsilon_*\mathcal{I}^p$ for $p \leq n$.
Taking the ``limit'' of these maps therefore gives
a quasi-isomorphism $\mathcal{K}^\bullet \to \epsilon_*\mathcal{I}^\bullet$
which proves the proposition.
\end{proof}
\begin{lemma}
@@ -19954,11 +19806,11 @@ \section{Almost blow up squares and the h topology}
\end{lemma}
\begin{proof}
The proof of this lemma is exactly the same as the proof of
Proposition \ref{proposition-check-h} using
We prove this by applying
Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-descent-squares}
whose hypotheses hold by
Lemma \ref{lemma-blow-up-square-h} and
More on Flatness, Lemma \ref{flat-lemma-refine-check-h}
instead of
More on Flatness, Proposition \ref{flat-proposition-check-h}.
\end{proof}
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