hcju/stacksqa huggingface dataset:
By Lemma \ref{lemma-compare-h-etale} the lemmas in Cohomology on Sites, Section \ref{sites-cohomology-section-compare-general} all apply to our current setting. To translate the results observe that the category $\mathcal{A}_X$ of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-A} is the full subcategory of $\textit{Ab}((\Sch/X)_h)$ consisting of sheaves of the form $a_X^{-1}\mathcal{F}$ where $\mathcal{F}$ is an abelian torsion sheaf on $X_\etale$. \medskip\noindent Part (1) is equivalent to $(V_n)$ for all $n$ which holds by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general}. \medskip\noindent Part (2) follows by applying $\epsilon_Y^{-1}$ to the conclusion of Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-implies-C-general}. \medskip\noindent Part (3) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general} part (1) because $\pi_X^{-1}K$ is in $D^+_{\mathcal{A}'_X}((\Sch/X)_\etale)$ and $a_X^{-1} = \epsilon_X^{-1} \circ a_X^{-1}$. \medskip\noindent Part (4) follows from Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-V-C-all-n-general} part (2) for the same reason.
We first reduce the statement to the case where $K$ is given by a single abelian sheaf. Namely, represent $K$ by a bounded below complex $\mathcal{F}^\bullet$ of torsion abelian sheaves. This is possible by Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-torsion}. By the case of a sheaf we see that $\mathcal{F}^n = a_{X, *} a_X^{-1} \mathcal{F}^n$ and that the sheaves $R^qa_{X, *}a_X^{-1}\mathcal{F}^n$ are zero for $q > 0$. By Leray's acyclicity lemma (Derived Categories, Lemma \ref{derived-lemma-leray-acyclicity}) applied to $a_X^{-1}\mathcal{F}^\bullet$ and the functor $a_{X, *}$ we conclude. From now on assume $K = \mathcal{F}$ where $\mathcal{F}$ is a torsion abelian sheaf. \medskip\noindent By Lemma \ref{lemma-describe-pullback-pi-h} we have $a_{X, *}a_X^{-1}\mathcal{F} = \mathcal{F}$. Thus it suffices to show that $R^qa_{X, *}a_X^{-1}\mathcal{F} = 0$ for $q > 0$. For this we can use $a_X = \epsilon_X \circ \pi_X$ and the Leray spectral sequence (Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-relative-Leray}). By Lemma \ref{lemma-V-C-all-n-etale-h} we have $R^i\epsilon_{X, *}(a_X^{-1}\mathcal{F}) = 0$ for $i > 0$ and $\epsilon_{X, *}a_X^{-1}\mathcal{F} = \pi_X^{-1}\mathcal{F}$. By Lemma \ref{lemma-cohomological-descent-etale} we have $R^j\pi_{X, *}(\pi_X^{-1}\mathcal{F}) = 0$ for $j > 0$. This concludes the proof.
In this proof we drop the subscript ${}_{small}$ from our pullback and pushforward functors. Denote $X_1 = X \times_Y X$ and denote $f_1 : X_1 \to Y$ the morphism $f \circ \text{pr}_0 = f \circ \text{pr}_1$. Let $(\mathcal{F}, \varphi)$ be a descent datum for $\{X \to Y\}$. Let us set $\mathcal{F}_1 = \text{pr}_0^{-1}\mathcal{F}$. We may think of $\varphi$ as defining an isomorphism $\mathcal{F}_1 \to \text{pr}_1^{-1}\mathcal{F}$. We claim that the rule which sends a descent datum $(\mathcal{F}, \varphi)$ to the sheaf $$ \mathcal{G} = \text{Equalizer}\left( \xymatrix{ f_*\mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & f_{1, *}\mathcal{F}_1 } \right) $$ is a quasi-inverse to the functor in the statement of the lemma. The first of the two arrows comes from the map $$ f_*\mathcal{F} \to f_*\text{pr}_{0, *}\text{pr}_0^{-1}\mathcal{F} = f_{1, *}\mathcal{F}_1 $$ and the second arrow comes from the map $$ f_*\mathcal{F} \to f_* \text{pr}_{1, *}\text{pr}_1^{-1}\mathcal{F} \xleftarrow{\varphi} f_* \text{pr}_{0, *} \text{pr}_0^{-1}\mathcal{F} = f_{1, *}\mathcal{F}_1 $$ where the arrow pointing left is invertible. To prove this works we have to show that the canonical map $f^{-1}\mathcal{G} \to \mathcal{F}$ is an isomorphism; details omitted. In order to prove this it suffices to check after pulling back by any collection of morphisms $\Spec(k) \to Y$ where $k$ is an algebraically closed field. Namely, the corresponing base changes $X_k \to X$ are jointly surjective and we can check whether a map of sheaves on $X_\etale$ is an isomorphism by looking at stalks on geometric points, see Theorem \ref{theorem-exactness-stalks}. By Lemma \ref{lemma-integral-pushforward-commutes-with-base-change} the construction of $\mathcal{G}$ from the descent datum $(\mathcal{F}, \varphi)$ commutes with any base change. Thus we may assume $Y$ is the spectrum of an algebraically closed point (note that base change preserves the properties of the morphism $f$, see Morphisms, Lemma \ref{morphisms-lemma-base-change-surjective} and \ref{morphisms-lemma-base-change-finite}). In this case the morphism $X \to Y$ has a section, so we know that the functor is an equivalence by Lemma \ref{lemma-glue-etale-sheaf-section}. However, the reader may show that the functor is an equivalence if and only if the construction above is a quasi-inverse; details omitted. This finishes the proof.
Exactly as in the proof of Lemma \ref{lemma-glue-etale-sheaf-integral-surjective} we claim a quasi-inverse is given by the functor sending $(\mathcal{F}, \varphi)$ to $$ \mathcal{G} = \text{Equalizer}\left( \xymatrix{ f_*\mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & f_{1, *}\mathcal{F}_1 } \right) $$ and in order to prove this it suffices to show that $f^{-1}\mathcal{G} \to \mathcal{F}$ is an isomorphism. This we may check locally, hence we may and do assume $Y$ is affine. Then we can find a finite surjective morphism $Z \to Y$ such that there exists an open covering $Z = \bigcup W_i$ such that $W_i \to Y$ factors through $X$. See More on Morphisms, Lemma \ref{more-morphisms-lemma-dominate-fppf-finite}. Applying Lemma \ref{lemma-glue-etale-sheaf-check-after-base-change} we see that it suffices to prove the lemma after replacing $Y$ by $Z$ and $Z \times_Y Z$ and $f$ by its base change. Thus we may assume $f$ has sections Zariski locally. Of course, using that the problem is local on $Y$ we reduce to the case where we have a section which is Lemma \ref{lemma-glue-etale-sheaf-section}.
We will work with the $2$-fibre product category as constructed in Categories, Example \ref{categories-example-2-fibre-product-categories}. The functor sends $\mathcal{F}$ to the triple $(f^{-1}\mathcal{F}, i^{-1}\mathcal{F}, c)$ where $c : j^{-1}f^{-1}\mathcal{F} \to g^{-1}i^{-1}\mathcal{F}$ is the canonical isomorphism. We will construct a quasi-inverse functor. Let $(\mathcal{F}', \mathcal{G}, \alpha)$ be an object of the right hand side of the arrow. We obtain an isomorphism $$ i^{-1}f_*\mathcal{F}' = g_*j^{-1}\mathcal{F}' \xrightarrow{g_*\alpha} g_*g^{-1}\mathcal{G} $$ The first equality is Lemma \ref{lemma-proper-base-change-f-star}. Using this we obtain maps $i_*\mathcal{G} \to i_*g_*g^{-1}\mathcal{G}$ and $f'_*\mathcal{F}' \to i_*g_*g^{-1}\mathcal{G}$. We set $$ \mathcal{F} = f_*\mathcal{F}' \times_{i_*g_*g^{-1}\mathcal{G}} i_*\mathcal{G} $$ and we claim that $\mathcal{F}$ is an object of the left hand side of the arrow whose image in the right hand side is isomorphic to the triple we started out with. Let us compute the stalk of $\mathcal{F}$ at a geometric point $\overline{x}$ of $X$. \medskip\noindent If $\overline{x}$ is not in $Z$, then on the one hand $\overline{x}$ comes from a unique geometric point $\overline{x}'$ of $X'$ and $\mathcal{F}'_{\overline{x}'} = (f_*\mathcal{F}')_{\overline{x}}$ and on the other hand we have $(i_*\mathcal{G})_{\overline{x}}$ and $(i_*g_*g^{-1}\mathcal{G})_{\overline{x}}$ are singletons. Hence we see that $\mathcal{F}_{\overline{x}}$ equals $\mathcal{F}'_{\overline{x}'}$. \medskip\noindent If $\overline{x}$ is in $Z$, i.e., $\overline{x}$ is the image of a geometric point $\overline{z}$ of $Z$, then we obtain $(i_*\mathcal{G})_{\overline{x}} = \mathcal{G}_{\overline{z}}$ and $$ (i_*g_*g^{-1}\mathcal{G})_{\overline{x}} = (g_*g^{-1}\mathcal{G})_{\overline{z}} = \Gamma(E_{\overline{z}}, g^{-1}\mathcal{G}|_{E_{\overline{z}}}) $$ (by the proper base change for pushforward used above) and similarly $$ (f_*\mathcal{F}')_{\overline{x}} = \Gamma(X'_{\overline{x}}, \mathcal{F}'|_{X'_{\overline{x}}}) $$ Since we have the identification $E_{\overline{z}} = X'_{\overline{x}}$ and since $\alpha$ defines an isomorphism between the sheaves $\mathcal{F}'|_{X'_{\overline{x}}}$ and $g^{-1}\mathcal{G}|_{E_{\overline{z}}}$ we conclude that we get $$ \mathcal{F}_{\overline{x}} = \mathcal{G}_{\overline{z}} $$ in this case. \medskip\noindent To finish the proof, we observe that there are canonical maps $i^{-1}\mathcal{F} \to \mathcal{G}$ and $f^{-1}\mathcal{F} \to \mathcal{F}'$ compatible with $\alpha$ which on stalks produce the isomorphisms we saw above. We omit the careful construction of these maps.
To prove the lemma we will check conditions (1), (2), and (3) of More on Flatness, Lemma \ref{flat-lemma-refine-check-h-stack}. \medskip\noindent Condition (1) holds because we have glueing for sheaves (and Zariski coverings are \'etale coverings). See Sites, Lemma \ref{sites-lemma-glue-sheaves}. \medskip\noindent To see condition (2), suppose that $f : X \to Y$ is a surjective, flat, proper morphism of finite presentation over $S$ with $Y$ affine. Then we have descent for $\{X \to Y\}$ by either Lemma \ref{lemma-glue-etale-sheaf-fppf-cover} or Lemma \ref{lemma-glue-etale-sheaf-proper-surjective}. \medskip\noindent Condition (3) follows immediately from the more general Lemma \ref{lemma-glue-etale-sheaf-modification}.
The notation $K|_{X'}$ stands for $b_{small}^{-1}K$. Choose a bounded below complex $\mathcal{F}^\bullet$ of abelian sheaves representing $K$. Observe that $i_*(\mathcal{F}^\bullet|_Z)$ represents $Ri_*(K|_Z)$ because $i_*$ is exact (Proposition \ref{proposition-finite-higher-direct-image-zero}). Choose a quasi-isomorphism $b_{small}^{-1}\mathcal{F}^\bullet \to \mathcal{I}^\bullet$ where $\mathcal{I}^\bullet$ is a bounded below complex of injective abelian sheaves on $X'_\etale$. This map is adjoint to a map $\mathcal{F}^\bullet \to b_*(\mathcal{I}^\bullet)$ and $b_*(\mathcal{I}^\bullet)$ represents $Rb_*(K|_{X'})$. We have $\pi_*(\mathcal{I}^\bullet|_E) = (b_*\mathcal{I}^\bullet)|_Z$ by Lemma \ref{lemma-proper-base-change-f-star} and by Lemma \ref{lemma-proper-base-change} this complex represents $R\pi_*(K|_E)$. Hence the map $$ Ri_*(K|_Z) \oplus Rb_*(K|_{X'}) \to Rc_*(K|_E) $$ is represented by the surjective map of bounded below complexes $$ i_*(\mathcal{F}^\bullet|_Z) \oplus b_*(\mathcal{I}^\bullet) \to i_*\left(b_*(\mathcal{I}^\bullet)|_Z\right) $$ To get our distinguished triangle it suffices to show that the canonical map $\mathcal{F}^\bullet \to i_*(\mathcal{F}^\bullet|_Z) \oplus b_*(\mathcal{I}^\bullet)$ maps quasi-isomorphically onto the kernel of the map of complexes displayed above (namely a short exact sequence of complexes determines a distinguished triangle in the derived category, see Derived Categories, Section \ref{derived-section-canonical-delta-functor}). We may check this on stalks at a geometric point $\overline{x}$ of $X$. If $\overline{x}$ is not in $Z$, then $X' \to X$ is an isomorphism over an open neighbourhood of $\overline{x}$. Thus, if $\overline{x}'$ denotes the corresponding geometric point of $X'$ in this case, then we have to show that $$ \mathcal{F}^\bullet_{\overline{x}} \to \mathcal{I}^\bullet_{\overline{x}'} $$ is a quasi-isomorphism. This is true by our choice of $\mathcal{I}^\bullet$. If $\overline{x}$ is in $Z$, then $b_(\mathcal{I}^\bullet)_{\overline{x}} \to  i_*\left(b_*(\mathcal{I}^\bullet)|_Z\right)_{\overline{x}}$ is an isomorphism of complexes of abelian groups. Hence the kernel is equal to $i_*(\mathcal{F}^\bullet|_Z)_{\overline{x}} = \mathcal{F}^\bullet_{\overline{x}}$ as desired.
If $K$ exists, then Lemma \ref{lemma-blow-up-square-cohomological-descent} tells us a distinguished triangle that it fits in. Thus we simply choose a distinguished triangle $$ K \to Ri_*(L) \oplus Rb_*(K') \to Rc_*(L|_E) \to K[1] $$ where $c = i \circ \pi = b \circ j$. Here the map $Ri_*(L) \to Rc_*(L|_E)$ is $Ri_*$ applied to the adjunction mapping $E \to R\pi_*(L|_E)$. The map $Rb_*(K') \to Rc_*(L|_E)$ is the composition of the canonical map $Rb_*(K') \to Rc_*(K'|_E)) = R$ and $Rc_*(\gamma)$. The maps $g$ and $f$ of the statement of the lemma are the adjoints of these maps. If we restrict this distinguished triangle to $Z$ then the map $Rb_*(K) \to Rc_*(L|_E)$ becomes an isomorphism by the base change theorem (Lemma \ref{lemma-proper-base-change}) and hence the map $g : K|_Z \to L$ is an isomorphism. Looking at the distinguished triangle we see that $f : K|_{X'} \to K'$ is an isomorphism over $X' \setminus E = X \setminus Z$. Moreover, we have $\gamma \circ f|_E = g|_E$ by construction. Then since $\gamma$ and $g$ are isomorphisms we conclude that $f$ induces isomorphisms on stalks at geometric points of $E$ as well. Thus $f$ is an isomorphism. \medskip\noindent For the final statement, we may replace $K'$ by $K|_{X'}$, $L$ by $K|_Z$, and $\gamma$ by the canonical identification. Observe that $\alpha$ and $\beta$ induce a commutative square $$ \xymatrix{ K \ar[r] \ar@{..>}[d] & Ri_*(K|_Z) \oplus Rb_*(K|_{X'}) \ar[r] \ar[d]_{\beta \oplus \alpha} & Rc_*(K|_E) \ar[r] \ar[d]_{\alpha|_E} & K[1] \ar@{..>}[d] \\ M \ar[r] & Ri_*(M|_Z) \oplus Rb_*(M|_{X'}) \ar[r] & Rc_*(M|_E) \ar[r] & M[1] } $$ Thus by the axioms of a derived category we get a dotted arrow producing a morphism of distinguished triangles.
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}).
We prove this using the criterion of Lemma \ref{lemma-refine-check-h}. Before check the conditions, we note that for a quasi-compact and quasi-separated object $X$ of $(\Sch/S)_{fppf}$ we have $$ H^i_{fppf}(X, \mathcal{O}^{perf}) = \colim_F H^i_{fppf}(X, \mathcal{O}) $$ See Cohomology on Sites, Lemma \ref{sites-cohomology-lemma-colim-works-over-collection}. We will also use that $H^i_{fppf}(X, \mathcal{O}) = H^i(X, \mathcal{O})$, see Descent, Proposition \ref{descent-proposition-same-cohomology-quasi-coherent}. \medskip\noindent Let $A, f, J$ be as in More on Flatness, Example \ref{flat-example-one-generator} and consider the associated almost blow up square. Since $X$, $X'$, $Z$, $E$ are affine, we have no higher cohomology of $\mathcal{O}$. Hence we only have to check that $$ 0 \to \mathcal{O}^{perf}(X) \to \mathcal{O}^{perf}(X') \oplus \mathcal{O}^{perf}(Z) \to \mathcal{O}^{perf}(E) \to 0 $$ is a short exact sequence. This was shown in (the proof of) More on Flatness, Lemma \ref{flat-lemma-h-sheaf-colim-F}. \medskip\noindent Let $X, X', Z, E$ be as in More on Flatness, Example \ref{flat-example-two-generators}. Since $X$ and $Z$ are affine we have $H^p(X, \mathcal{O}_X) = H^p(Z, \mathcal{O}_X) = 0$ for $p > 0$. By More on Flatness, Lemma \ref{flat-lemma-funny-blow-up} we have $H^p(X', \mathcal{O}_{X'}) = 0$ for $p > 0$. Since $E = \mathbf{P}^1_Z$ and $Z$ is affine we also have $H^p(E, \mathcal{O}_E) = 0$ for $p > 0$. As in the previous paragraph we reduce to checking that $$ 0 \to \mathcal{O}^{perf}(X) \to \mathcal{O}^{perf}(X') \oplus \mathcal{O}^{perf}(Z) \to \mathcal{O}^{perf}(E) \to 0 $$ is a short exact sequence. This was shown in (the proof of) More on Flatness, Lemma \ref{flat-lemma-h-sheaf-colim-F}.
This is proved in Descent, Lemma \ref{descent-lemma-sheaf-condition-holds}. We indicate the proof here. As established in Lemma \ref{lemma-fpqc-sheaves}, it is enough to check the sheaf property on Zariski coverings and faithfully flat morphisms of affine schemes. The sheaf property for Zariski coverings is standard scheme theory, since $\Gamma(U, i^\ast \mathcal{F}) = \mathcal{F}(U)$ when $i : U \hookrightarrow S$ is an open immersion. \medskip\noindent For $\left\{\Spec(B)\to \Spec(A)\right\}$ with $A\to B$ faithfully flat and $\mathcal{F}|_{\Spec(A)} = \widetilde{M}$ this corresponds to the fact that $M = H^0\left((B/A)_\bullet \otimes_A M \right)$, i.e., that \begin{align*} 0 \to M \to B \otimes_A M \to B \otimes_A B \otimes_A M \end{align*} is exact by Lemma \ref{lemma-descent-modules}.
We have proved these facts (and more) in the preceding chapters. Here is a list of references: (1) Morphisms, Lemma \ref{morphisms-lemma-etale-over-field}. (2) Morphisms, Lemmas \ref{morphisms-lemma-etale-flat-etale-fibres} and \ref{morphisms-lemma-flat-unramified-etale}. (3) Algebra, Lemma \ref{algebra-lemma-etale-standard-smooth}. (4) Morphisms, Lemma \ref{morphisms-lemma-base-change-etale}. (5) Morphisms, Lemma \ref{morphisms-lemma-composition-etale}. (6) Follows formally from (4) and (5). (7) Morphisms, Lemmas \ref{morphisms-lemma-etale-locally-quasi-finite} and \ref{morphisms-lemma-locally-quasi-finite-rel-dimension-0}. (8) See Algebra, Lemmas \ref{algebra-lemma-reduced-goes-up} and \ref{algebra-lemma-Rk-goes-up}, see also more results of this kind in \'Etale Morphisms, Section \ref{etale-section-properties-permanence}. (9) See Morphisms, Lemma \ref{morphisms-lemma-fppf-open} and \ref{morphisms-lemma-etale-flat}. (10) See Morphisms, Lemma \ref{morphisms-lemma-etale-permanence}.
Let's discuss the case of sheaves of sets. For any sheaf $\mathcal{G}$ on $Z$ the morphism $i_{small}^{-1}i_{small, *}\mathcal{G} \to \mathcal{G}$ is an isomorphism by Lemma \ref{lemma-stalk-pushforward-closed-immersion} (and Theorem \ref{theorem-exactness-stalks}). This implies formally that $i_{small, *}$ is fully faithful, see Sites, Lemma \ref{sites-lemma-exactness-properties}. It is clear that $i_{small, *}\mathcal{G}|_{U_\etale} \cong *$ where $U = X \setminus Z$. Conversely, suppose that $\mathcal{F}$ is a sheaf of sets on $X$ such that $\mathcal{F}|_{U_\etale} \cong *$. Consider the adjunction mapping $$ \mathcal{F} \longrightarrow i_{small, *}i_{small}^{-1}\mathcal{F} $$ Combining Lemmas \ref{lemma-stalk-pushforward-closed-immersion} and \ref{lemma-stalk-pullback} we see that it is an isomorphism. This finishes the proof of (1). The proof of (2) is identical.
We may factor $f$ as $$ \xymatrix{ X \ar[r]^h & Z \ar[r]^i & Y } $$ where $h$ is integral, universally injective and surjective and $i : Z \to Y$ is a closed immersion. Apply Proposition \ref{proposition-closed-immersion-pushforward} to $i$ and apply Theorem \ref{theorem-topological-invariance} to $h$.
Let $X_{\overline{y}}^{sh}$ denote the fiber product $X \times_Y \Spec(\mathcal{O}_{Y, \overline{y}}^{sh})$. By Theorem \ref{theorem-higher-direct-images} the stalk of $R^qf_*\mathcal{F}$ at $\overline{y}$ is computed by $H_\etale^q(X_{\overline{y}}^{sh}, \mathcal{F})$. Since $f$ is finite, $X_{\bar y}^{sh}$ is finite over $\Spec(\mathcal{O}_{Y, \overline{y}}^{sh})$, thus $X_{\bar y}^{sh} = \Spec(A)$ for some ring $A$ finite over $\mathcal{O}_{Y, \bar y}^{sh}$. Since the latter is strictly henselian, Lemma \ref{lemma-finite-over-henselian} implies that $A$ is a finite product of henselian local rings $A = A_1 \times \ldots \times A_r$. Since the residue field of $\mathcal{O}_{Y, \overline{y}}^{sh}$ is separably closed the same is true for each $A_i$. Hence $A_i$ is strictly henselian. This implies that $X_{\overline{y}}^{sh} = \coprod_{i = 1}^r \Spec(A_i)$. The vanishing of Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian} implies that $(R^qf_*\mathcal{F})_{\overline{y}} = 0$ for $q > 0$ which implies (2) by Theorem \ref{theorem-exactness-stalks}. Part (1) follows from the corresponding statement of Lemma \ref{lemma-vanishing-etale-cohomology-strictly-henselian}.
Exactness of $j_!$ is very general, see Modules on Sites,  Lemma \ref{sites-modules-lemma-extension-by-zero-exact}. Of course it does also follow from the description of stalks. The formula for the stalk follows from Modules on Sites, Lemma \ref{sites-modules-lemma-stalk-j-shriek} and the description of points of the small \'etale site in terms of geometric points, see Lemma \ref{lemma-points-small-etale-site}. \medskip\noindent For later use we note that the isomorphism \begin{align*} (j_!\mathcal{F})_{\overline{x}} & = (j_{p!}\mathcal{F})_{\overline{x}} \\ & = \colim_{(V, \overline{v})} j_{p!}\mathcal{F}(V) \\ & = \colim_{(V, \overline{v})} \bigoplus\nolimits_{\varphi : V \to U} \mathcal{F}(V \xrightarrow{\varphi} U) \\ & \to \bigoplus\nolimits_{\overline{u} : \Spec(k) \to U,\ j(\overline{u}) = \overline{x}} \mathcal{F}_{\bar{u}}. \end{align*} constructed in Modules on Sites, Lemma \ref{sites-modules-lemma-stalk-j-shriek} sends $(V, \overline{v}, \varphi, s)$ to the class of $s$ in the stalk of $\mathcal{F}$ at $\overline{u} = \varphi(\overline{v})$.
This is just a reformulation of Lemma \ref{lemma-h-sheaf-colim-F}. Recall that $\mathcal{O}^h = \mathcal{O}^{perf} = \colim_F \mathcal{O}$, see More on Flatness, Lemma \ref{flat-lemma-char-p}. By Lemma \ref{lemma-h-sheaf-colim-F} we see that $\mathcal{O}^{perf}$ viewed as an object of $D((\Sch/S)_{fppf})$ is of the form $R\epsilon_*K$ for some $K \in D((\Sch/S)_h)$. Then $K = \epsilon^{-1}\mathcal{O}^{perf}$ which is actually equal to $\mathcal{O}^{perf}$ because $\mathcal{O}^{perf}$ is an h sheaf. See Cohomology on Sites, Section \ref{sites-cohomology-section-compare}. Hence $R\epsilon_*\mathcal{O}^{perf} = \mathcal{O}^{perf}$ (with apologies for the confusing notation). Thus the lemma now follows from Leray $$ R\Gamma_h(S, \mathcal{O}^{perf}) = R\Gamma_{fppf}(S, R\epsilon_*\mathcal{O}^{perf}) = R\Gamma_{fppf}(S, \mathcal{O}^{perf}) $$ and the fact that $$ H^i_{fppf}(S, \mathcal{O}^{perf}) = H^i_{fppf}(S, \colim_F \mathcal{O}) = \colim_F H^i_{fppf}(S, \mathcal{O}) $$ as $S$ is quasi-compact and quasi-separated (see proof of Lemma \ref{lemma-h-sheaf-colim-F}).
The derived category $D_\QCoh(\mathcal{O}_X)$ has direct sums, is compactly generated, and $D_{perf}(\mathcal{O}_X)$ is the full subcategory of compact objects, see Derived Categories of Schemes, Lemma \ref{perfect-lemma-quasi-coherence-direct-sums}, Theorem \ref{perfect-theorem-bondal-van-den-Bergh}, and Proposition \ref{perfect-proposition-compact-is-perfect}. By Lemma \ref{lemma-van-den-bergh} we may assume $F(E) = \Hom_X(E, K)$ for some $K \in \Ob(D_\QCoh(\mathcal{O}_X))$. Then it follows that $K$ is in $D^b_{\textit{Coh}}(\mathcal{O}_X)$ by Lemma \ref{lemma-characterize-dbcoh-projective}.