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59.106 Almost blow up squares and the h topology

In this section we continue the discussion in More on Flatness, Section 38.37. For the convenience of the reader we recall that an almost blow up square is a commutative diagram

59.106.0.1
\begin{equation} \label{etale-cohomology-equation-almost-blow-up-square} \vcenter { \xymatrix{ E \ar[d] \ar[r] & X' \ar[d]^ b \\ Z \ar[r] & X } } \end{equation}

of schemes satisfying the following conditions:

  1. $Z \to X$ is a closed immersion of finite presentation,

  2. $E = b^{-1}(Z)$ is a locally principal closed subscheme of $X'$,

  3. $b$ is proper and of finite presentation,

  4. the closed subscheme $X'' \subset X'$ cut out by the quasi-coherent ideal of sections of $\mathcal{O}_{X'}$ supported on $E$ (Properties, Lemma 28.24.5) is the blow up of $X$ in $Z$.

It follows that the morphism $b$ induces an isomorphism $X' \setminus E \to X \setminus Z$.

We are going to give a criterion for “h sheafiness” for objects in the derived category of the big fppf site $(\mathit{Sch}/S)_{fppf}$ of a scheme $S$. On the same underlying category we have a second topology, namely the h topology (More on Flatness, Section 38.34). Recall that fppf coverings are h coverings (More on Flatness, Lemma 38.34.5). Hence we may consider the morphism

\[ \epsilon : (\mathit{Sch}/S)_ h \longrightarrow (\mathit{Sch}/S)_{fppf} \]

See Cohomology on Sites, Section 21.26. In particular, we have a fully faithful functor

\[ R\epsilon _* : D((\mathit{Sch}/S)_ h) \longrightarrow D((\mathit{Sch}/S)_{fppf}) \]

and we can ask: what is the essential image of this functor?

Lemma 59.106.1. With notation as above, if $K$ is in the essential image of $R\epsilon _*$, then the maps $c^ K_{X, Z, X', E}$ of Cohomology on Sites, Lemma 21.25.1 are quasi-isomorphisms.

Proof. Denote ${}^\# $ sheafification in the h topology. We have seen in More on Flatness, Lemma 38.37.7 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 21.28.3 (which itself is a formal consequence of Cohomology on Sites, Lemma 21.25.3). $\square$

Proposition 59.106.2. Let $K$ be an object of $D^+((\mathit{Sch}/S)_{fppf})$. Then $K$ is in the essential image of $R\epsilon _* : D((\mathit{Sch}/S)_ h) \to D((\mathit{Sch}/S)_{fppf})$ if and only if $c^ K_{X, X', Z, E}$ is a quasi-isomorphism for every almost blow up square (59.106.0.1) in $(\mathit{Sch}/S)_ h$ with $X$ affine.

Proof. We prove this by applying Cohomology on Sites, Lemma 21.28.2 whose hypotheses hold by Lemma 59.106.1 and More on Flatness, Proposition 38.37.9. $\square$

Lemma 59.106.3. Let $K$ be an object of $D^+((\mathit{Sch}/S)_{fppf})$. Then $K$ is in the essential image of $R\epsilon _* : D((\mathit{Sch}/S)_ h) \to D((\mathit{Sch}/S)_{fppf})$ if and only if $c^ K_{X, X', Z, E}$ is a quasi-isomorphism for every almost blow up square as in More on Flatness, Examples 38.37.10 and 38.37.11.

Proof. We prove this by applying Cohomology on Sites, Lemma 21.28.2 whose hypotheses hold by Lemma 59.106.1 and More on Flatness, Lemma 38.37.12 $\square$


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