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 21.26.1 are quasi-isomorphisms.

## 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

of schemes satisfying the following conditions:

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

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

$b$ is proper and of finite presentation,

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 38.34.6). Hence we may consider the morphism

See Cohomology on Sites, Section 21.27. In particular, we have a fully faithful functor

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

**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 21.29.3 (which itself is a formal consequence of Cohomology on Sites, Lemma 21.26.3).
$\square$

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 21.29.2 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 21.29.2 whose hypotheses hold by Lemma 59.106.1 and More on Flatness, Lemma 38.37.12
$\square$

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