Proof. It suffices to consider the case $A=\mathbf {Z}$ . It is immediately clear that $\varpi _{\mathbf {Z}}$ has finite fibers and that it is surjective (by considering its behavior on geometric points). Furthermore, $\varpi _{\mathbf {Z}}$ is a morphism of projective $\mathbf {Z}$ -schemes, so it is proper. Thus, $\varpi _{\mathbf {Z}}$ is a finite morphism. Finally, we can see that $\varpi _{\mathbf {Z}}$ is flat by comparing dimensions of local rings in a straightforward use of ‘Miracle Flatness’ applied between geometric fibers over $\mathrm {Spec}(\mathbf {Z})$ [Reference Matsumura and Reid14, Theorem 23.1].

Proof. We will show that the base change map is an isomorphism on sections over a basis of affine opens.

Since $\pi $ is finite, it is affine; therefore, for each affine open $V \subset P$ with $V=\mathrm {Spec}(R)$ , the inverse image $\pi ^{-1}(V)=U\subset P'$ is also affine, with $U=\mathrm {Spec}(B)$ for a finite-type A-algebra B. Furthermore, the map $R \to B$ given by $\pi $ is a module-finite map between the finite-type A-algebras $R,B$ .

Similarly, the preimage $V^h$ of V in $P^h$ and $U^h$ of U in $(P')^h$ are isomorphic to the Henselian affine schemes $\mathrm {Sph}(R^h) \subset P^h, \mathrm {Sph}(B^h) \subset (P')^h$ , respectively (where the Henselizations are taken with respect to the ideals $IR,IB$ ).

We have $\mathcal {G}(U)=M$ for some B-module M (since $\mathcal {G}$ is quasi-coherent), and $\pi _*\mathcal {G}|_V$ is the sheaf associated to M viewed as an R-module via the map $R \to B$ .

We can then easily check that the base change map $(\pi _*\mathcal {G})^h \to \pi ^h_*(\mathcal {G}^h)$ of Lemma 2.9 on $V^h$ is given by the map of $R^h$ -modules

which arises from the map ; this map is an isomorphism since $R \to B$ is finite [19, Lemma 0DYE], so $((\pi _*\mathcal {G})^h)(V^h) \stackrel {\textstyle \sim }{\smash {\longrightarrow }\rule {0pt}{0.4ex}} \pi ^h_*(\mathcal {G}^h)(V^h)$ via the base change map.

Therefore, the base change map is an isomorphism on sections over a basis of affine opens. Hence, $(\pi _*\mathcal {G})^h \stackrel {\textstyle \sim }{\smash {\longrightarrow }\rule {0pt}{0.4ex}} \pi ^h_*(\mathcal {G}^h)$ , as we desired to show.

Proof. Let $P_0,P_0^{\prime }$ be the closed subschemes of $P,P'$ , respectively, defined by I. Then by Proposition 2.3, we have isomorphisms of sites $(P_0)_{\mathrm {\acute {e}t}} \to (P^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}, (P_0^{\prime })_{\mathrm {\acute {e}t}} \to ((P')^h)_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}$ .

Let $\overline {\pi }: P_0^{\prime } \to P_0$ be the base change of $\pi $ . Then by [19, Lemma 04C2], the functor $\overline {\pi }_{\mathrm {\acute {e}t},*}$ from the category of abelian sheaves on $(P_0^{\prime })_{\mathrm {\acute {e}t}}$ to the category of abelian sheaves on $(P_0)_{\mathrm {\acute {e}t}}$ is exact. Hence, for $j>0$ , we have $R^j\overline {\pi }_{\mathrm {\acute {e}t},*}\mathcal {G}=0$ for all abelian sheaves $\mathcal {G}$ on $(P_0^{\prime })_{\mathrm {\acute {e}t}}$ .

The isomorphisms of sites above tell us that similarly, for $j> 0$ , we have $R^j\pi ^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F})=0$ for all abelian sheaves $\mathcal {F}$ on $((P')^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ . This gives us the desired isomorphism of cohomology.

Proof. Fix some $i \ge 0$ and a quasi-coherent sheaf $\mathcal {G}$ on $P'$ . Then we have a commutative diagram

where the lower diagonal arrow is an isomorphism by Lemma 3.2.4. The right vertical arrow is an isomorphism because $\pi $ is finite (ensuring that $R^j\pi _{\mathrm {\acute {e}t},*}$ vanishes on all abelian sheaves for $j> 0$ ). The upper diagonal arrow is an isomorphism by Lemma 3.2.5.

It is then clear that the map $\mathrm {H}^i_{\mathrm {\acute {e}t}}(P',\mathcal {G}) \to \mathrm {H}^i_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}((P')^h,\mathcal {G}^h)$ is an isomorphism if and only if the map $\mathrm { H}^i_{\mathrm {\acute {e}t}}(P,\pi _*\mathcal {G}) \to \mathrm {H}^i_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,(\pi _*\mathcal {G})^h)$ is an isomorphism, as we desired to show.

Proof of Proposition 3.2.2.

Assume we have $(A,I)$ a Noetherian Henselian pair and $\pi : P' \to P$ a finite flat surjection of projective A-schemes. Fix a coherent sheaf $\mathcal {F}$ on P; we will show $\mathcal {F}$ satisfies GHGA comparison on ${P}$ .

Since $\pi $ is flat and surjective – hence faithfully flat – the map $\mathcal {F} \to \pi _*\pi ^*\mathcal {F}$ is injective for any coherent sheaf $\mathcal {F}$ on P.

Setting $\mathcal {G}_0:=\pi ^*\mathcal {F}$ , which is a coherent sheaf on $P'$ , we can replace $\mathcal {F}$ with $(\pi _*\mathcal {G}_0)/\mathcal {F}$ to obtain an injective map $(\pi _*\mathcal {G}_0)/\mathcal {F} \to \pi _* \mathcal {G}_1$ for some coherent sheaf $\mathcal {G}_1$ on $P'$ . We can then iteratively construct a resolution $0 \to \mathcal {F} \to \pi _*\mathcal {G}_0 \to \pi _*\mathcal {G}_1 \to \dots $ where the $\mathcal {G}_i$ are coherent sheaves on $P'$ .

In other words, for any coherent sheaf $\mathcal {F}$ on P, there exists a resolution of $\mathcal {F}$ by pushforwards of coherent sheaves on $P'$ . These sheaves $\pi _*\mathcal {G}_i$ satisfy GHGA comparison on ${P}$ by Lemma 3.2.6.

Now we consider two complexes of coherent $\mathscr {O}_P$ -modules: $\mathcal {F}[0]^\bullet $ , the complex with the only nonzero term being $\mathcal {F}$ in degree $0$ , and the complex $\mathcal {G}^\bullet $ with degree i term $\mathcal {G}^i=\pi _*\mathcal {G}_i$ . The injection $\mathcal {F} \hookrightarrow \pi _*\mathcal {G}_0$ in degree $0$ and zero maps $0 \to \pi _*\mathcal {G}_i$ in degree i for $i>0$ give a quasi-isomorphism of complexes $\mathcal {F}[0]^\bullet \to \mathcal {G}^\bullet $ . The corresponding complexes of $\mathscr {O}_{P_{\mathrm {\acute {e}t}}}$ -modules $(\mathcal {F}[0]^\bullet )_{\mathrm {\acute {e}t}}$ and $(\mathcal {G}^\bullet )_{\mathrm {\acute {e}t}}$ are also quasi-isomorphic, so we may use the hypercohomology of $(\mathcal {G}^\bullet )_{\mathrm {\acute {e}t}}$ to compute the étale cohomology of $\mathcal {F}$ . By a similar argument, for $\mathcal {H}^\bullet $ the complex with degree i term $\mathcal {H}^i=(\pi _*\mathcal {G}_i)^h$ , the hypercohomology of $(\mathcal {H}^\bullet )_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ may be used to compute the h-étale cohomology of $\mathcal {F}^h$ .

The remainder of the proof is a standard application of the hypercohomology spectral sequence.

Proof. We will show the base change map is an isomorphism by checking on stalks, using the results of [Reference Fu5, Section 5.9] on passage to limits for étale cohomology.

Since f is proper and X is locally Noetherian, the higher direct image sheaves $R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}}$ are coherent on X. Therefore, the pullback $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h$ to $(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ is a coherent sheaf of $\mathscr {O}_{(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules.

We fix a point $x \in X^h$ , and also write x for its image in X. Choose a geometric point $\overline {x}$ of X lying over x, and write $\overline {x}^h$ for the corresponding geometric point of $X^h$ lying over x. By Lemma 2.5, the stalks of $\mathscr {O}_{(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ at $\overline {x}^h$ and of $\mathscr {O}_{X_{\mathrm {\acute {e}t}}}$ at $\overline {x}$ are both naturally isomorphic to the strict Henselization $(\mathscr {O}_{X,x})^{\mathrm {sh}}$ of the local ring $\mathscr {O}_{X,x}$ along its maximal ideal.

By the construction of higher direct images as a sheafification, we can compute the stalks of both sheaves at $\overline {x}^h$ as:

(*) $$ \begin{align}\kern-9pt((R^jf_{\mathrm{\acute{e}t},*}\mathcal{F}_{\mathrm{\acute{e}t}})^h)_{\overline{x}^h} &= \varinjlim_{(U,\overline{u})} \mathrm{H}^j_{\mathrm{\acute{e}t}}(U \times_X Y,\mathcal{F}), \end{align} $$

(**) $$ \begin{align} (R^jf^h_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}},*}(\mathcal{F}^h)_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}})_{\overline{x}^h} &= \varinjlim_{(U,\overline{u})} \mathrm{H}^j_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}(U^h \times_{X^h} Y^h,\mathcal{F}^h) \end{align} $$

with $(U,\overline {u}) \to (X,\overline {x})$ the affine étale neighborhoods. This holds because the system of h-étale neighborhoods $(U^h,\overline {u}^h: U^h \to X^h)$ of $\overline {x}$ (Henselizations of affine étale neighborhoods of $\overline {x}$ as a geometric point of X) is cofinal, since by the isomorphism of sites $(X_0)_{\mathrm {\acute {e}t}} \to (X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ for $X_0$ the closed subscheme of X defined by the ideal I (Proposition 2.3), we can check cofinality modulo I. The limit of the affine étale neighborhoods $(U,\overline {u})$ of $(X,\overline {x})$ is the map $\mathrm {Spec}((\mathscr {O}_{X,x})^{\mathrm {sh}}) \to X$ , which we denote $X_{(\overline {x})} \to X$ .

Our map of sheaves $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h \to R^jf^h_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}_{\mathrm {\acute {e}t}}^h)$ is given on each stalk as the limit of the h-étale comparison maps $\mathrm { H}^j_{\mathrm {\acute {e}t}}(U \times _X Y,\mathcal {F}) \to \mathrm {H}^j_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}((U \times _X Y)^h,\mathcal {F}^h)=\mathrm {H}^j_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(U^h \times _{X^h} Y^h,\mathcal {F}^h)$ for each étale neighborhood U of $\overline {x}$ as described above.

We begin by computing the first stalk $((R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h)_{\overline {x}^h}$ . Again, by the construction of higher direct images as a sheafification, we note that that the colimit in (*) is also equal to $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})_{\overline {x}}$ , which is isomorphic to $\mathrm {H}^j_{\mathrm {\acute {e}t}}(Y \times X_{(\overline {x})},\mathcal {F}|_{Y \times X_{(\overline {x})}})$ [Reference Fu5, Corollary 5.9.5].

We now consider the other stalk, $(R^jf^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}_{\mathrm {\acute {e}t}}^h))_{\overline {x}^h}$ . For each affine étale neighborhood $(U,\overline {u})$ of $(X,\overline {x})$ , its Henselization is the Henselian affine h-étale neighborhood $(U^h,\overline {u}^h: U^h \to X^h)$ of $(X^h,\overline {x}^h)$ . If $U=\mathrm {Spec}(B)$ , then $U^h$ is isomorphic to $\mathrm {Sph}(B^h)$ for $B^h$ the I-adic Henselization of B. The limit $\varinjlim B^h$ is equal to $\varinjlim B$ , which is $(\mathscr {O}_{X,x})^{\mathrm {sh}}$ (see Lemma 2.5), so the limit of the Henselian affine h-étale neighborhoods $(U^h,\overline {u}^h: U^h \to X^h)$ is $X_{(\overline {x})} \to X$ . By the definition of f, we see that $(U \times _X Y)_{B^h} \simeq S_{B^h}$ as $B^h$ -schemes. Furthermore, we see that $U^h \times _{X^h} Y^h \simeq (S_{B^h})^h$ , the I-adic Henselization of the scheme $S_{B^h}$ . The A-algebra B is finite-type since X is finite-type over A. Therefore, by our assumption, $S_{B^h}$ satisfies coherent GHGA comparison, so by Corollary 2.14, the comparison map $\mathrm { H}^j_{\mathrm {\acute {e}t}}((U \times _X Y)_{B^h},\mathcal {F}_{B^h}) \to \mathrm {H}^j_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(U^h \times _{X^h} Y^h,\mathcal {F}^h)$ is an isomorphism.

Since $\varinjlim B^h = \varinjlim B = (\mathscr {O}_{X,x})^{\mathrm {sh}}$ , we note that both the limit of the maps $U \times _X Y \to Y$ and the limit of the maps $(U \times _X Y)_{B^h} \to Y$ are equal to the map $Y \times X_{(\overline {x})} \to Y$ . Then from (**), we can compute

$$ \begin{align*} (R^jf^h_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}},*}(\mathcal{F}^h)_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}})_{\overline{x}^h} &= \varinjlim_{(U,\overline{u})} \mathrm{H}^j_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}(U^h \times_{X^h} Y^h,\mathcal{F}^h)\\ &=\varinjlim_{(U=\mathrm{Spec}(B),\overline{u})}\mathrm{H}^j_{\mathrm{\acute{e}t}}((U \times_X Y)_{B^h},\mathcal{F}_{B^h}) \\ &= \mathrm{H}^j_{\mathrm{\acute{e}t}}(Y \times X_{(\overline{x})},\mathcal{F}|_{Y \times X_{(\overline{x})}}), \end{align*} $$

with the final equality as a consequence of [Reference Fu5, Corollary 5.9.4].

This identification arises from the natural maps, so the maps on stalks $((R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h)_{\overline {x}^h} \to (R^jf^h_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}_{\mathrm {\acute {e}t}}^h))_{\overline {x}^h}$ are isomorphisms; hence, the base change map is an isomorphism, as we desired to show.

Proof. This is a straightforward application of Proposition 3.3.3 with X projective and $S=\mathbf {P}_A^1$ .

Proof of Proposition 3.3.1.

Let $f: Y = \mathbf {P}_X^1 \to X$ be the natural map. Fix a coherent sheaf $\mathcal {F}$ on Y. We will compare the Leray spectral sequence $\mathrm {H}^i_{\mathrm {\acute {e}t}}(X,R^jf_*\mathcal {F}) \implies \mathrm {H}^{i+j}_{\mathrm {\acute {e}t}}(Y,\mathcal {F})$ to the analogous spectral sequence for $f^h$ and $\mathcal {F}^h$ .

We choose an injective resolution $\mathcal {I}^\bullet $ of $\mathcal {F}_{\mathrm {\acute {e}t}}$ by $\mathscr {O}_{Y_{\mathrm {\acute {e}t}}}$ -modules, with which we can compute the étale cohomology of $\mathcal {F}$ . Similarly, we choose an injective resolution $\mathcal {J}^\bullet $ of $\mathcal {F}^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ by $\mathscr {O}_{(Y^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules to compute the h-étale cohomology of $\mathcal {F}^h$ .

We may pull back $\mathcal {I}^\bullet $ to an exact sequence of $\mathscr {O}_{(Y^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules (with degree i term $(\mathcal {I}^i)^h$ ), which maps to $\mathcal {J}^\bullet $ since $\mathcal {J}^\bullet $ is an injective resolution. This gives us a map from the pullback of $\mathcal {I}^\bullet $ to $(Y^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ to $\mathcal {J}^\bullet $ , which can be lifted through the steps of constructing the Leray spectral sequences (applying $f_{\mathrm {\acute {e}t},*}$ or $f^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}$ , taking a Cartan-Eilenberg resolution, taking global sections, and finally filtering the total complex by rows) to give us a map of spectral sequences that must respect the filtration on limit terms.

Since f is projective (hence proper) proper and X is locally Noetherian, the higher direct image sheaves $R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}}$ are coherent on X. Therefore, the pullback $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h$ to $(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ is a coherent sheaf of $\mathscr {O}_{(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules. Then by Proposition 3.3.3, we see that pullback to the Henselization gives an isomorphism on the second sheet, since

$$ \begin{align*}\mathrm{H}^i_{\mathrm{\acute{e}t}}(X,R^jf_*\mathcal{F}) \to \mathrm{H}^i_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}(X^h,(R^jf_*\mathcal{F})^h) \simeq \mathrm{H}^i_{\mathrm{ h\mbox{-}\mathrm{\acute{e}t}}}(X^h,R^jf^h_*\mathcal{F}^h)\end{align*} $$

is an isomorphism by coherent GHGA comparison for X.

Since the map of spectral sequences is an isomorphism at the second sheet and respects the filtration on limit terms, the comparison map $\mathrm { H}^n_{\mathrm {\acute {e}t}}(Y,\mathcal {F}) \to \mathrm {H}^n_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(Y^h,\mathcal {F}^h)$ is an isomorphism for all n.

Proof. By Proposition 3.3.1 iterated and our assumption, we see that for any $d \ge 1$ , the projective A-scheme $(\mathbf {P}_A^1)^{\times d}$ satisfies coherent GHGA comparison. Then using the map $\varpi _A: (\mathbf {P}_A^1)^{\times d} \to \mathbf {P}_A^d$ described in Section 3.2, by Proposition 3.2.2, we see that $\mathbf {P}_A^d$ satisfies coherent GHGA comparison for all $d \ge 1$ .

Now consider an arbitrary projective A-scheme X. Then there exists N and a closed immersion $\iota : X \hookrightarrow \mathbf {P}_A^N$ . We apply Lemma 3.2.6 to $\iota : X \hookrightarrow \mathbf {P}_A^N$ to see that a coherent sheaf $\mathcal {F}$ on X satisfies GHGA comparison on ${X}$ if and only if $\iota _*\mathcal {F}$ satisfies GHGA comparison on ${\mathbf {P}_A^N}$ .

Therefore, since $\mathbf {P}_A^N$ satisfies coherent GHGA comparison, X does as well.

Proof. This can be checked affine-locally on X, so we can assume X is affine and then that $Y=\mathbf {P}_X^n$ . Then we can apply Proposition 3.3.3 with $S=\mathbf {P}_A^n$ to get the desired result by Lemma 3.4.1.

Proof. We will use [Reference Grothendieck6, Theorem 3.1.2], sometimes called ‘Grothendieck’s Unscrewing Lemma’ or ‘Grothendieck’s dévissage theorem’.

If we consider the full subcategory $\mathcal {C}$ of coherent $\mathscr {O}_X$ -modules consisting of $\mathcal {F}$ satisfying GHGA comparison on ${X}$ , then it is obvious that $0 \in \mathcal {C}$ and that for any short exact sequence of coherent $\mathscr {O}_X$ -modules $\mathcal {F}' \hookrightarrow \mathcal {F} \twoheadrightarrow \mathcal {F}"$ , if two of $\mathcal {F},\mathcal {F}',\mathcal {F}"$ are in $\mathcal {C}$ , all three are. (The latter statement follows by the $\delta $ -functoriality of the h-étale comparison maps and the five lemma.)

By the Unscrewing Lemma, in order to show that every coherent $\mathscr {O}_X$ -module $\mathcal {F}$ satisfies GHGA comparison on ${X}$ (or equivalently, that $\mathcal {C}$ contains all coherent $\mathscr {O}_X$ -modules), it is enough to show that for any irreducible closed subset $Z \subset X$ , there exists some $\mathcal {G}$ supported on Z and satisfying GHGA comparison on ${X}$ whose fiber at the generic point of Z has rank $1$ .

We now use Noetherian induction on X. For our inductive assumption, we assume that for any strict closed subscheme Y of X (strict meaning $Y \subsetneq X$ ) that every coherent $\mathscr {O}_Y$ -module satisfies GHGA comparison on ${Y}$ . Note that Y is necessarily proper over A. If $\mathcal {G}$ is a coherent $\mathscr {O}_X$ -module which is supported on a strict closed subscheme Y of X, then considering $\mathcal {G}$ as a coherent $\mathscr {O}_Y$ -module, it follows from the inductive assumption that $\mathcal {G}$ satisfies GHGA comparison on ${X}$ .

If X is reducible or not reduced, then any irreducible closed subset $Z \subset X$ must be contained in a strict closed subscheme Y of X, so any $\mathcal {G}$ supported on Z whose fiber at the generic point of Z has rank $1$ must satisfy GHGA comparison on ${X}$ by the inductive assumption. Thus, by the Unscrewing Lemma, every coherent $\mathscr {O}_X$ -module $\mathcal {F}$ satisfies GHGA comparison on ${X}$ if X is reducible or not reduced.

It is therefore enough to consider the case where X is integral and, given the inductive assumption, exhibit a coherent $\mathscr {O}_X$ -module $\mathcal {F}$ with generic stalk of rank $1$ satisfying GHGA comparison on ${X}$ . Then by the Unscrewing Lemma, we are done.

Since X is integral, we can use Chow’s Lemma [19, Section 02O2] to find an integral projective A-scheme $X'$ and a morphism $\pi : X' \to X$ over A such that $\pi $ is projective and surjective, as well as a dense open $U \subset X$ such that $\pi |_{\pi ^{-1}(U)}: \pi ^{-1}(U) \to U$ is an isomorphism.

Now let $\mathcal {G}=\mathscr {O}_{X'}(n)$ with $n>0$ . Since $\pi |_{\pi ^{-1}(U)}$ is an isomorphism, we see that the generic stalk of the coherent $\mathscr {O}_X$ -module $\pi _*\mathcal {G}$ is invertible. Therefore, it will suffice to show that $\pi _*\mathcal {G}$ satisfies GHGA comparison on ${X}$ for some n.

Consider the Leray spectral sequences for $\mathcal {G}_{\mathrm {\acute {e}t}}$ and $(\mathcal {G}_{\mathrm {\acute {e}t}})^h$ . Thus, we get a spectral sequence of A-modules $E_\bullet ^{\bullet ,\bullet }$ with second sheet $E_2^{i,j} = \mathrm {H}^i_{\mathrm {\acute {e}t}}(X,R^j\pi _{*}\mathcal {G})$ which converges to $\mathrm {H}^{i+j}_{\mathrm {\acute {e}t}}(X',\mathcal {G})$ (inducing finite filtrations on the limit terms), and another spectral sequence of A-modules $Q_\bullet ^{\bullet ,\bullet }$ with second sheet $Q_2^{i,j} = \mathrm {H}^i_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(X^h,R^j\pi ^h_*(\mathcal {G}^h))$ which converges to $\mathrm {H}^{i+j}_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}((X')^h,\mathcal {G}^h)$ (inducing finite filtrations on the limit terms).

Working through the construction of these Leray spectral sequences, we have a morphism $\phi :E_\bullet ^{\bullet ,\bullet } \to Q_\bullet ^{\bullet ,\bullet }$ coming from the pullback of sheaves on $X'$ to sheaves on $(X')^h$ (see also discussion in the proof of Proposition 3.3.1). Since for all j we have $((R^j\pi _*\mathcal {G})_{\mathrm {\acute {e}t}})^h=((R^j\pi _*\mathcal {G})^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ by Remark 2.4 and the map $((R^j\pi _*\mathcal {G})^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}} \to (R^j\pi ^h_*(\mathcal {G}^h))_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ is an isomorphism by Lemma 3.4.2, we see that $\phi :E_\bullet ^{\bullet ,\bullet } \to Q_\bullet ^{\bullet ,\bullet }$ is compatible with pullback of sheaves on X to sheaves on $X^h$ .

Furthermore, since we have coherent GHGA comparison for $X'$ by Lemma 3.4.1, we know that this morphism is an isomorphism on the $\infty $ -sheet (because the natural map between the limit terms is an isomorphism).

The higher direct image sheaf $(R^j\pi _*\mathcal {G})_{\mathrm {\acute {e}t}}=R^j\pi _{\mathrm {\acute {e}t},*}\mathcal {G}_{\mathrm {\acute {e}t}}$ is the sheafification of the presheaf $V \mapsto \mathrm {H}^j_{\mathrm {\acute {e}t}}(\pi ^{-1}(V),\mathcal {G})$ . For n sufficiently large, independent of the affine open V, and $j> 0$ , each of the cohomology groups $\mathrm {H}^j_{\mathrm {\acute {e}t}}(\pi ^{-1}(V),\mathcal {G})=\mathrm { H}^j_{\mathrm {\acute {e}t}}(\pi ^{-1}(V),\mathscr {O}_{X'}(n))$ vanishes. Therefore, for $j> 0$ , the sheaf $(R^j\pi _*\mathcal {G})_{\mathrm {\acute {e}t}}$ is $0$ . Hence, we also have $0=((R^j\pi _*\mathcal {G})_{\mathrm {\acute {e}t}})^h = ((R^j\pi _*\mathcal {G})^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}} \stackrel {\textstyle \sim }{\smash {\longrightarrow }\rule {0pt}{0.4ex}} (R^j\pi ^h_*(\mathcal {G}^h))_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ for $j> 0$ (see Remark 2.4, Lemma 3.4.2).

Therefore, both spectral sequences $E_\bullet ^{\bullet ,\bullet }$ and $Q_\bullet ^{\bullet ,\bullet }$ degenerate at the second sheet. Then since our morphism of spectral sequences induces an isomorphism on the $\infty $ sheet, the same is true for the second sheet, meaning the comparison map $\mathrm { H}^i_{\mathrm {\acute {e}t}}(X,\pi _*\mathcal {G}) \to \mathrm {H}^i_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(X^h,\pi ^h_*(\mathcal {G}^h)) \simeq \mathrm {H}^i_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(X^h,(\pi _*\mathcal {G})^h)$ is an isomorphism for all i. Thus, $\pi _*\mathcal {G}$ satisfies GHGA comparison on ${X}$ for large n.

Proof. We start by considering twists of the structure sheaf $\mathscr {O}_P(n)$ for all integers n.

Write $P = \mathrm {Proj}(A[x_0,x_1])$ . Then for each integer n, we have a short exact sequence of sheaves on P:

$$ \begin{align*}0 \to \mathscr{O}_P(n-1) \xrightarrow{x_1} \mathscr{O}_P(n) \to \mathcal{G}_n \to 0,\end{align*} $$

where the first map is given by multiplication by $x_1$ . Then $\mathcal {G}_n$ is the structure sheaf of the affine section $[1,0] \in \mathbf {P}_A^1=P$ , so $\mathcal {G}_n$ satisfies GHGA comparison on ${P}$ by Lemma 2.8.

Since pullback to the Henselization is exact (as the maps of local rings are flat), we have a short exact sequence of sheaves on $P^h$

$$ \begin{align*}0 \to (\mathscr{O}_P(n-1))^h \xrightarrow{x_1} (\mathscr{O}_P(n))^h \to \mathcal{G}_n^h \to 0.\end{align*} $$

These short exact sequences give us maps of the corresponding long exact sequences of étale cohomology and of h-étale cohomology. If $\mathcal {G}_n$ and $\mathscr {O}_P(n)$ satisfy GHGA comparison on ${P}$ , then the same is true for $\mathscr {O}_P(n-1)$ . Thus, using downwards inductions starting with $n=0$ , we see that for all $n \le 0$ , the twist $\mathscr {O}_P(n)$ satisfies GHGA comparison on ${P}$ .

Similarly, we see that if $\mathcal {G}_n$ and $\mathscr {O}_P(n-1)$ satisfy GHGA comparison on ${P}$ , then the same is true for $\mathscr {O}_P(n)$ . Now using induction upwards from $n=1$ , we see that for all $n> 0$ – hence for any integer n – the twist $\mathscr {O}_P(n)$ satisfies GHGA comparison on ${P}$ .

For a general coherent sheaf $\mathcal {F}$ on P, we have a short exact sequence of sheaves

$$ \begin{align*}0 \to \mathcal{K} \to \bigoplus_{\ell=0}^r \mathscr{O}_P(n_\ell) \to \mathcal{F} \to 0\end{align*} $$

for some finite collection of integers $n_0,\dots ,n_r$ . Note that $\mathcal {K}$ is also coherent as the kernel of a map between coherent sheaves.

For $j \ge 2$ , we know that $\mathrm {H}^j_{\mathrm {\acute {e}t}}(P,\mathcal {G})=0$ for all quasi-coherent $\mathcal {G}$ . Similarly, by Lemma 2.8, $\mathrm {H}^j(P^h,\mathcal {G}^h)=0$ for $j \ge 2$ because we can compute the h-étale cohomology of $\mathcal {G}^h$ on $P^h$ with the Čech complex associated to the Henselization of the standard affine open cover of P by [19, Lemma 03F7]. (Note that the Henselization of the standard affine open cover of P is necessarily also an h-étale cover.)

Therefore, the comparison map $\mathrm {H}^j_{\mathrm {\acute {e}t}}(P,\mathcal {G}) \to \mathrm {H}^j_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathcal {G}^h)$ is an isomorphism for $j \ge 2$ and all coherent $\mathcal {G}$ . Now we use downwards induction on j, assuming that for all coherent sheaves $\mathcal {G}$ we have $\mathrm {H}^{j+1}_{\mathrm {\acute {e}t}}(P,\mathcal {G}) \simeq \mathrm { H}^{j+1}_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathcal {G}^h)$ .

Writing $\mathcal {N}$ for $\bigoplus _{\ell =0}^r \mathscr {O}_P(n_\ell )$ , the long exact sequence of cohomology gives us a commutative diagram with exact rows

where we note that $\mathcal {N}=\bigoplus _{\ell =0}^r \mathscr {O}_P(n_\ell )$ satisfies GHGA comparison on ${P}$ and $\mathrm {H}^{j+1}_{\mathrm {\acute {e}t}}(P,\mathcal {K}) \simeq \mathrm {H}^{j+1}_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathcal {K}^h)$ by the inductive hypothesis on $j+1$ . By the first Four Lemma, the middle arrow $\mathrm {H}^j_{\mathrm {\acute {e}t}}(P,\mathcal {F}) \to \mathrm {H}^j_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathcal {F}^h)$ is surjective for arbitrary coherent $\mathcal {F}$ . Hence, by considering a similar commutative diagram corresponding to a presentation of $\mathcal {K}$ , the map $\mathrm {H}^j_{\mathrm {\acute {e}t}}(P,\mathcal {K}) \to \mathrm {H}^j_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathcal {K}^h)$ is also surjective. Therefore, we can use the second Four Lemma to get injectivity of the middle arrow; hence, the middle arrow is an isomorphism.

Therefore, all coherent sheaves $\mathcal {F}$ on P satisfy GHGA comparison on ${P}$ if $\mathscr {O}_P$ does.

Proof. In order to compute $\mathrm {H}^0_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h})$ , we can describe P as the union of two affine opens $U = \mathrm {Spec}(A[t])$ and $V=\mathrm {Spec}(A[1/t])$ , with $U \cap V = \mathrm {Spec}(A[t,1/t])$ . We will write $A\{t\}$ for the I-adic Henselization $(A[t])^h$ of the polynomial ring $A[t]$ , and similarly $A\{1/t\}=(A[1/t])^h, A\{t,1/t\}=(A[t,1/t])^h$ for the I-adic Henselizations of $A[1/t],A[t,1/t]$ .

We note that the map $A\{t\} \to (A[t])^\wedge $ is faithfully flat, hence injective, by [19, Lemma 0AGV]. Similarly, we have injections $A\{1/t\} \hookrightarrow (A[1/t])^\wedge $ and $A\{t,1/t\} \hookrightarrow (A[t,1/t])^\wedge .$ Elements of the completion $(A[t])^\wedge $ have the form $\sum _{n=0}^\infty a_nt^n$ such that for any $N \ge 1$ , there exists M so that $a_n \in I^N$ for $n \ge M$ . Elements of the other completions are described similarly as series in $t^{-1}$ or two-sided power series in t with coefficients tending I-adically to $0$ as the exponents go to $\pm \infty $ .

Consider the map $A\{t\} \times A\{1/t\} \to A\{t,1/t\}$ given by $(f,g) \mapsto f-g$ (we write $f,g$ also for their respective images in $A\{t,1/t\}$ ), and the similarly defined map $(A[t])^\wedge \times (A[1/t])^\wedge \to (A[t,1/t])^\wedge $ . The kernel of the first map is $\mathrm {H}^0_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h})$ by the sheaf condition; it is clear that $A \subset \mathrm {H}^0_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h})$ (via the diagonal map $A \to A\{t\} \times A\{1/t\}$ ).

Also, we see that $\mathrm {H}^0_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h})$ is contained in the kernel of $(A[t])^\wedge \times (A[1/t])^\wedge \to (A[t,1/t])^\wedge $ , which we can see is A using the explicit description of elements of these completions given above. Then $\mathrm { H}^0_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h}) = A = \mathrm {H}^0_{\mathrm {\acute {e}t}}(P,\mathscr {O}_P)$ , as we desired to show.

Proof. See [19, Theorem 07GC]. Swan gives an exposition of Popescu’s proof of this theorem in [Reference Swan20].

Proof. The map $B^h \to B^\wedge $ is faithfully flat; therefore, it is injective, so $B^h \subseteq B^\wedge $ . Elements of $B^h$ are clearly algebraic over B.

Now we assume that we have an element $f \in B^\wedge $ and a nonzero polynomial $q(x) \in B[x]$ such that $q(f)=0$ . Then since B is a G-ring, by Theorem 3.5.4 for any integer $N \ge 1$ , we can find $g_N \in B^h$ with $f-g_N \in J^NB^\wedge $ and $q(g_N)=0$ . This gives us a sequence $\{g_N\}_{N=1}^\infty $ of elements of $B^h$ converging I-adically to f in $B^\wedge $ .

Since B is a Noetherian domain, it injects into its fraction field L. The tensor product is also a finite product of fields $\prod _i L_i$ by [19, Lemma 0AH1], and is injective because $B \to B^h$ is flat.

There are only finitely many roots of q in $\prod _i L_i$ , so there are only finitely many roots of q in $B^h \hookrightarrow \prod _i L_i$ . Therefore, we have a subsequence $\{g_{N_r}\}_{r=1}^\infty $ of $\{g_N\}_{N=1}^\infty $ which are all equal to a single element $g_0 \in B^h$ with $q(g_0)=0$ , so $f = g_0 \in B^h$ , as we desired to show.

Proof. This is shown by de Jong in the Stacks Project Blog post at [Reference de Jong9]; we provide the proof here for the reader’s convenience.

The element f is a two-sided power series in t with coefficients going I-adically to $0$ as the exponent goes to $\pm \infty $ . We separate f into $f_++f_-$ with $f_+$ a power series in t and $f_-$ a power series in $1/t=t^{-1}$ . It suffices to show that $f_+$ is algebraic over $A[t]$ , by the symmetry of t and $1/t$ .

We can assume the constant term of f is part of $f_+$ . If f is algebraic over $A[t,1/t]$ , we have a relation $\sum _{i=0}^m P_if^i=0$ with $P_i \in A[t,1/t], m> 0, P_m \ne 0$ . By multiplying by an appropriate power of t, we can assume that the $P_i$ are in $A[t]$ .

Let $A(t)$ denote the fraction field of $A[t]$ . We see by the above that the $A(t)$ -span of the powers of f is finite-dimensional. Therefore, the $A(t)$ -span of $\{f^{p^i}\}_{i=0}^\infty $ is also finite-dimensional, meaning we have a relation $\sum _{i=0}^r Q_if^{p^i}$ with $Q_i \in A(t), r>0, Q_r \ne 0$ . By clearing denominators, we can assume that the $Q_i$ are polynomials in t.

Now because $p=0 \in A$ , we know that $\left (\sum _{i=0}^r Q_if_+^{p^i}\right )+\left (\sum _{i=0}^r Q_if_-^{p^i}\right ) = 0.$ We can consider the coefficients of large positive powers of t on the left side of this equation; in particular, for very large m, we know that the coefficient of $t^m$ in $\sum _{i=0}^r Q_if_-^{p^i}$ is $0$ by our choice of $f_-$ . Hence, in order for this equation to hold, for m sufficiently large, the coefficient of $t^m$ in $\sum _{i=0}^r Q_if_+^{p^i}$ must also be $0$ .

Therefore, $\sum _{i=0}^r Q_if_+^{p^i}$ is actually a polynomial in t. Let $\sum _{i=0}^r Q_if_+^{p^i}=:Q \in A[t]$ . As we assumed $Q_r \ne 0$ , we see that $Q-\sum _{i=0}^r Q_if_+^{p^i}=0$ is a nonzero polynomial relation, so $f_+$ is algebraic over $A[t]$ , as desired.

Proof. By Proposition 3.5.2, we have an isomorphism $\mathrm {H}^0_{\mathrm {\acute {e}t}}(P,\mathscr {O}_P) \simeq \mathrm {H}^0_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h})$ . We also have isomorphisms $\mathrm {H}^j_{\mathrm {\acute {e}t}}(P,\mathscr {O}_P) \simeq \mathrm { H}^j_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h}) = 0 $ for $j \ge 2$ by Lemma 2.8 and [19, Lemma 03F7] (see the discussion in the proof of Proposition 3.5.1).

Therefore, in order to show that $\mathscr {O}_P$ satisfies GHGA comparison on ${P}$ , it suffices to show that $\mathrm {H}^1_{\mathrm {\acute {e}t}}(P,\mathscr {O}_P) \simeq \mathrm {H}^1_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(P^h,\mathscr {O}_{P^h})$ via the natural map. By Corollary 2.14, in fact it is enough to show that the Zariski comparison map $\mathrm {H}^1(P,\mathscr {O}_P) \to \mathrm {H}^1(P^h,\mathscr {O}_{P^h})$ is an isomorphism, or equivalently that $\mathrm {H}^1(P^h,\mathscr {O}_{P^h})=0$ , in order to prove that $\mathscr {O}_P$ satisfies GHGA comparison on ${P}$ .

Using [19, Lemma 01ET] and applying Theorem 2.7, we know we can compute $\mathrm {H}^1(P^h,\mathscr {O}_{P^h})$ with Čech cohomology associated to the Henselization of the affine cover of P given by the two affine opens $U = \mathrm {Spec}(A[t])$ and $V=\mathrm {Spec}(A[1/t])$ , with intersection $U\, \cap \,V = \mathrm {Spec}(A[t,1/t])$ . Thus, $\mathrm { H}^1(P,\mathscr {O}_P) \to \mathrm {H}^1(P^h,\mathscr {O}_{P^h})$ is an isomorphism if and only if the map $A\{t\} \times A\{1/t\} \to A\{t,1/t\}$ given by $(f,g) \mapsto f-g$ is surjective.Footnote ³

1. (a) We assume that A is a Noetherian $\mathbf {F}_p$ -algebra domain and G-ring.

   Since A is a Noetherian domain and G-ring, the same is true for $A[t]$ and $A[t,1/t]$ by [19, Proposition 07PV]. Therefore, we can apply Lemma 3.5.5 to $B=A[t],A[1/t]$ and $A[t,1/t]$ . We then see that showing that the map $A\{t\} \times A\{1/t\} \to A\{t,1/t\}$ is surjective amounts to showing that for $f \in (A[t,1/t])^\wedge $ (the I-adic completion) which is algebraic over $A[t,1/t]$ , we can write f as a sum $f=f_++f_-$ such that $f_+ \in (A[t])^\wedge , f_- \in (A[1/t])^\wedge $ with $f_+$ algebraic over $A[t]$ and $f_-$ algebraic over $A[1/t]$ . This can be done by Lemma 3.5.6, so $\mathrm {H}^1(P^h,\mathscr {O}_{P^h})=0$ . It follows that $\mathscr {O}_P$ satisfies GHGA comparison on ${P}$ for A which is a domain and G-ring, as discussed above.

2. (b) Now assume that A is a Noetherian normal reduced G-ring. It follows that A is a finite product of normal domains $\prod _i A_i$ by [19, Lemma 030C]. Therefore, $\mathrm {Spec}(A)$ is the disjoint union of the $\mathrm {Spec}(A_i)$ , and P is the disjoint union of $\mathbf {P}_{A_i}^1=:P_i$ . These decompositions into components carry over to the Henselizations and $P^h$ is the disjoint union of $P_i^h$ . Thus, $\mathrm {H}^1(P^h,\mathscr {O}_{P^h})=\bigoplus _i \mathrm { H}^1(P_i^h,\mathscr {O}_{P_i^h})$ .

   The $A_i$ are clearly Noetherian domains and G-rings, so by the previous case (a), each $\mathrm {H}^1(P_i^h,\mathscr {O}_{P_i^h})$ is $0$ , so $\mathrm {H}^1(P^h,\mathscr {O}_{P^h})=0$ and $\mathscr {O}_P$ satisfies GHGA comparison on ${P}$ in this case as well.

3. (c) Finally, we treat the general Noetherian case. By hypothesis, A is an $\mathbf {F}_p$ -algebra. We can write A as the filtered direct limit of its finitely generated subalgebras $A = \varinjlim A_\lambda $ . Let $I_\lambda = I \cap A_\lambda $ , and $A_\lambda ^h$ be the Henselization of $A_\lambda $ along $I_\lambda $ . We see that $A=\varinjlim A_\lambda ^h, I = \varinjlim I_\lambda ^h$ by [19, Lemma 0A04].

   Since $A_\lambda $ for each $\lambda $ is a finitely generated $\mathbf {F}_p$ -algebra, we see that $A_\lambda ^h$ is the quotient of the Henselization (along some ideal) of a polynomial ring $\mathbf {F}_p[t_1,\dots ,t_r]$ . Note that the Henselization of $\mathbf {F}_p[t_1,\dots ,t_r]$ is a Noetherian normal reduced G-ring. (See [19, Lemma 033B, Lemma 0AH3, Lemma 033C, Lemma 037D, Lemma 0AGV].)

   Therefore, each pair $(A_\lambda ^h,I_\lambda ^h)$ is the quotient of some Henselian pair $(A_\lambda ',I_\lambda ')$ with $A_\lambda '$ a Noetherian normal reduced G-ring. Let $\smash {P_\lambda :=\mathbf {P}_{A_\lambda ^h}^1, P_\lambda ' := \mathbf {P}_{A_\lambda '}^1}$ .

   By the previous case (b), for all $\lambda $ , the structure sheaf $\mathscr {O}_{P_\lambda '}$ satisfies GHGA comparison on ${P_\lambda '}$ ; then by Proposition 3.5.1, in fact, $P_\lambda '$ satisfies coherent GHGA comparison.

   Because $\mathrm {Spec}(A_\lambda ^h) \hookrightarrow \mathrm {Spec}(A_\lambda ')$ is a closed immersion, so is $\smash {P_\lambda \hookrightarrow P_\lambda '.}$ Closed immersions are finite, so we can apply Lemma 3.2.6 – the pushforward of $\mathscr {O}_{P_\lambda }$ is coherent, so $\mathscr {O}_{P_\lambda }$ satisfies GHGA comparison on ${P_\lambda }$ .

   Therefore, for all $\lambda $ , the map $A_\lambda ^h\{t\} \times A_\lambda ^h\{1/t\} \to A_\lambda ^h\{t,1/t\} $ is surjective. Since Henselization commutes with direct limits [19, Lemma 0A04], we see that $\varinjlim A_\lambda ^h\{t\} = A\{t\}$ . Similarly, we see that $\varinjlim A_\lambda ^h\{1/t\} = A\{1/t\}$ and $\varinjlim A_\lambda ^h\{t,1/t\} = A\{t,1/t\}$ .

   Then $A\{t\} \times A\{1/t\} \to A\{t,1/t\}$ is the colimit of surjective maps, so it is surjective. Therefore $\mathrm { H}^1(P^h,\mathscr {O}_{P^h})=0$ .

Thus, for $(A,I)$ a general Noetherian Henselian pair in characteristic $p>0$ , we have shown that $\mathrm {H}^1(P^h,\mathscr {O}_{P^h})=0$ , so $\mathscr {O}_P$ satisfies GHGA comparison on ${P}$ , as we desired to show.

Proof. This follows from Theorem 3.4.3, Proposition 3.5.1 and Lemma 3.5.7 because if B is an A-algebra with $A \to B$ essentially finite-type, then B and $B^h$ have positive characteristic if A does.

Proof of Theorem 3.6.1.

Fix $X \to \mathrm {Spec}(A)$ proper and $\mathcal {F}$ a quasi-coherent sheaf on X. We will show that $\mathcal {F}$ satisfies GHGA comparison on ${X}$ .

For $X \to \mathrm {Spec}(A)$ proper, we can find a closed immersion $\iota : X \hookrightarrow X'$ with $X'$ proper and finitely presented over A. (See, for example, [19, Lemma 09ZR]).

Now by Lemma 3.2.6, we may reduce to the situation of $X \to \mathrm {Spec}(A)$ both proper and finitely presented. By [19, Lemma 01PJ], $\mathcal {F}$ can be written as the filtered colimit of finitely presented, quasi-coherent $\mathscr {O}_X$ -modules, so we can reduce to the case of $\mathcal {F}$ finitely presented using [19, Lemma 01FF].

Write A as the filtered direct limit of its subalgebras $A_i$ which are finitely generated over the finite field $\mathbf {F}_p$ . Set $I_i=I \cap A_i$ . Then clearly, $\varinjlim I_i=I$ . Hence, by the universal property of Henselization, $A=\varinjlim A_i^h,I=\varinjlim I_i^h$ for $(A_i^h,I_i^h)$ the Henselization of the pair $(A_i,I_i)$ as Henselization commutes with filtered colimits. Furthermore, the $A_i^h$ are Noetherian by [19, Lemma 0AGV].

Since we are now assuming that X is finitely presented over A, there exists $i_0$ and a finitely presented $A_{i_0}^h$ -scheme $X_{i_0}$ such that by [19, Lemma 01ZM]. We may assume that $X_{i_0}$ is proper over $A_{i_0}^h$ by [19, Lemma 081F].

Furthermore, as $\mathcal {F}$ is finitely presented, we can increase $i_0$ as necessary so that there exists a coherent sheaf $\mathcal {F}_{i_0}$ on $X_{i_0}$ so that $\mathcal {F}=\mathcal {F}_{i_0} \times _{X_{i_0}} X$ by [19, Lemma 01ZR].

Setting for $A_{i_0} \subset A_i$ and defining $\mathcal {F}_i$ similarly, if we let $Y_i$ be the closed subscheme of $X_i$ corresponding to $I_i^h$ , we see that we are in the situations of Lemmas A.1 and A.2. Therefore, we see that $\varinjlim _i \mathrm {H}^j(X_i,\mathcal {F}_i) \simeq \mathrm {H}^j(X,\mathcal {F})$ and $\varinjlim _i \mathrm {H}^j(X_i^h,\mathcal {F}_i^h) \simeq \mathrm { H}^j(X^h,\mathcal {F}^h)$ for all $j \ge 0$ . Because each $A_i^h$ is a Noetherian $\mathbf {F}_p$ -algebra, we know that the natural map $\mathrm { H}^j(X_i,\mathcal {F}_i) \to \mathrm {H}^j(X_i^h,\mathcal {F}_i^h)$ is an isomorphism for all i and all $j \ge 0$ by Theorem 3.5.8, so $\mathcal {F}$ also satisfies GHGA comparison on ${X}$ , as we desired to show.

Proof. Since f is proper and X is locally Noetherian, the higher direct image sheaves $R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}}$ are coherent on X. Therefore, the pullback $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h$ to $(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ is a coherent sheaf of $\mathscr {O}_{(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules.

We proceed as in the proof of Proposition 3.3.3 to show the base change map is an isomorphism by checking on stalks; fix a point $x \in X^h$ , and choose a geometric point $\overline {x}$ of X lying over x, with $\overline {x}^h$ the corresponding geometric point of $X^h$ . Recall the equations computing the stalks:

(*) $$ \begin{align}\kern-11pt ((R^jf_{\mathrm{\acute{e}t},*}\mathcal{F}_{\mathrm{\acute{e}t}})^h)_{\overline{x}^h} &= \varinjlim_{(U,\overline{u})} \mathrm{H}^j_{\mathrm{\acute{e}t}}(U \times_X Y,\mathcal{F}), \end{align} $$

(**) $$ \begin{align} (R^jf^h_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}},*}(\mathcal{F}^h)_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}})_{\overline{x}^h} &= \varinjlim_{(U,\overline{u})} \mathrm{H}^j_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}(U^h \times_{X^h} Y^h,\mathcal{F}^h), \end{align} $$

where $(U,\overline {u}) \to (X,\overline {x})$ are affine étale neighborhoods. The remainder of the argument to show that both stalks can be identified with $\mathrm {H}^j_{\mathrm {\acute {e}t}}(Y \times X_{(\overline {x})},\mathcal {F}|_{Y \times X_{(\overline {x})}})$ via the natural maps is almost exactly as in the proof of Proposition 3.3.3. The only difference in the proof is how we show that for $U=\mathrm {Spec}(B), U^h=\mathrm {Sph}(B^h)$ , the comparison map $\mathrm {H}^j_{\mathrm {\acute {e}t}}((U \times _X Y)_{B^h},\mathcal {F}_{B^h}) \to \mathrm {H}^j_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(U^h \times _{X^h} Y^h,\mathcal {F}^h)$ is an isomorphism, which we now describe.

Note that the $B^h$ -scheme $U':=(U \times _X Y)_{B^h}$ is proper over $B^h$ since $f: Y \to X$ is proper. Furthermore, we see that $U^h \times _{X^h} Y^h$ is isomorphic to the I-adic Henselization $(U')^h$ .

Because A is an $\mathbf {F}_p$ -algebra and B is a finite-type A-algebra, both $B,B^h$ are Noetherian $\mathbf {F}_p$ -algebras [19, Lemma 0AGV]. Therefore, the proper $B^h$ -scheme $U'$ satisfies coherent GHGA comparison by Theorem 3.5.8; it follows from Corollary 2.14 that the comparison map

$$ \begin{align*}\mathrm{H}^j_{\mathrm{\acute{e}t}}((U \times_X Y)_{B^h},\mathcal{F}_{B^h}) =\mathrm{H}^j_{\mathrm{\acute{e}t}}(U',\mathcal{F}_{U'})\to \mathrm{H}^j_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}((U')^h,\mathcal{F}^h)=\mathrm{H}^j_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}(U^h \times_{X^h} Y^h,\mathcal{F}^h)\end{align*} $$

is an isomorphism.

The rest is precisely as the proof of Proposition 3.3.3.

Proof of Theorem 3.6.2.

Fix a quasi-coherent sheaf $\mathcal {F}$ on Y and let $Y^h$ be the I-adic Henselization of Y (which is a Henselian scheme over $\mathrm {Sph}(A)$ ). We will show that the canonical h-étale cohomology comparison map $\mathrm {H}^n_{\mathrm {\acute {e}t}}(Y,\mathcal {F}) \to \mathrm {H}^n_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(Y^h,\mathcal {F}^h)$ is an isomorphism for all n, so $\mathcal {F}$ satisfies GHGA comparison on ${Y}$ .

Since $f: Y \to X$ is proper, we can find a closed immersion $\iota : Y \hookrightarrow Y'$ with $Y'$ proper and finitely presented over X. (See, for example, [19, Lemma 09ZR]).

Now by Lemma 3.2.6, $\mathcal {F}$ satisfies GHGA comparison on ${Y}$ if and only if $\iota _*\mathcal {F}$ satisfies GHGA comparison on ${Y'}$ . Thus, we may reduce to the situation where f is both proper and finitely presented. By [19, Lemma 01PJ], $\mathcal {F}$ can be written as the filtered colimit of finitely presented, quasi-coherent $\mathscr {O}_Y$ -modules, so we can reduce to the case of $\mathcal {F}$ finitely presented using [19, Lemma 01FF].

As in the proof of Theorem 3.6.1, we can write the pair $(A,I)$ as the filtered direct limit of $(A_i, I_i:=I \cap A_i)$ for the finitely generated $\mathbf {F}_p$ -subalgebras $A_i$ of A. In fact, $A=\varinjlim A_i^h,I =\varinjlim I_i^h$ for $(A_i^h,I_i^h)$ the Henselization of the pair $(A_i,I_i)$ , and the $A_i^h$ are Noetherian by [19, Lemma 0AGV].

Since we are now assuming that $f: Y \to X$ is a morphism of finitely presented A-schemes, there exists $i_0$ and a morphism of finitely presented $A_{i_0}^h$ -schemes $f_{i_0}: Y_{i_0} \to X_{i_0}$ such that and f is the base change of $f_{i_0}$ by [19, Lemma 01ZM]. We may assume that $f_{i_0}$ is proper by [19, Lemma 081F].

Furthermore, as $\mathcal {F}$ is finitely presented, we can increase $i_0$ as necessary so that there exists a coherent sheaf $\mathcal {F}_{i_0}$ on $Y_{i_0}$ so that $\mathcal {F}=\mathcal {F}_{i_0} \times _{Y_{i_0}} Y$ by [19, Lemma 01ZR].

Set for $A_{i_0} \subset A_i$ and define $Y_i, f_i, \mathcal {F}_i$ similarly. Then for all $i \ge i_0$ and all $j \ge 0$ , the map $(R^j(f_i)_{\mathrm {\acute {e}t},*}(\mathcal {F}_i)_{\mathrm {\acute {e}t}})^h \to R^j(f_i^h)_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}_i^h)$ of sheaves on $(X_i^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ arising from the base change map of Lemma 2.9 is an isomorphism by Lemma 3.6.3.

Fix j and consider for all $i \ge i_0$ the pullbacks of the sheaves $R^j(f_i)_{\mathrm {\acute {e}t},*}(\mathcal {F}_i)_{\mathrm {\acute {e}t}}$ from $(X_i)_{\mathrm {\acute {e}t}}$ to $X_{\mathrm {\acute {e}t}}$ . The limit of these pullbacks is the sheaf $R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}}$ by [Reference Fu5, Corollary 5.9.6]. It follows that the limit of the pullbacks of $(R^j(f_i)_{\mathrm {\acute {e}t},*}(\mathcal {F}_i)_{\mathrm {\acute {e}t}})^h$ to $(X^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ is $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h$ (where $X_i^h$ is the Henselization of $X_i$ along the closed subscheme $X_i'$ corresponding to $I_i^h$ ; consider Remark 2.4).

A similar argument (again implicitly using the equivalence of $(Y_i^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ and $(Y_i')_{\mathrm {\acute {e}t}}$ for $Y_i'$ the closed subscheme of $Y_i$ corresponding to $I_i^h$ and Remark 2.4) finds that the limit of the pullbacks of $R^j(f_i^h)_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}_i^h)$ to $X_{\mathrm {\acute {e}t}}$ is $R^jf^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}^h)$ . The maps $(R^j(f_i)_{\mathrm {\acute {e}t},*}(\mathcal {F}_i)_{\mathrm {\acute {e}t}})^h \to R^j(f_i^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}_i^h)$ from Lemma 2.9 are compatible with these limits, so we have an isomorphism $(R^jf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h \to R^jf^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}^h)_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}$ for each j.

To complete the proof, we will compare the Leray spectral sequence $\mathrm {H}^m_{\mathrm {\acute {e}t}}(X,R^nf_*\mathcal {F}) \implies \mathrm { H}^{m+n}_{\mathrm {\acute {e}t}}(Y,\mathcal {F})$ , and the analogous spectral sequence for $f^h$ and $\mathcal {F}^h$ . (We use $m,n$ to avoid confusion with the indices i above.)

As in the proof of Proposition 3.3.1, we choose an injective resolution $\mathcal {I}^\bullet $ of $\mathcal {F}_{\mathrm {\acute {e}t}}$ by $\mathscr {O}_{Y_{\mathrm {\acute {e}t}}}$ -modules, with which we can compute the étale cohomology of $\mathcal {F}$ . Similarly, we choose an injective resolution $\mathcal {J}^\bullet $ of $\mathcal {F}^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ by $\mathscr {O}_{(Y^h)_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules to compute the h-étale cohomology of $\mathcal {F}^h$ .

We may pull back $\mathcal {I}^\bullet $ to an exact sequence of $\mathscr {O}_{(Y^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}}$ -modules (with degree m term $(\mathcal {I}^m)^h$ ), which maps to $\mathcal {J}^\bullet $ since $\mathcal {J}^\bullet $ is an injective resolution. This gives us a map from the pullback of $\mathcal {I}^\bullet $ to $(Y^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ to $\mathcal {J}^\bullet $ , which can be lifted through the steps of constructing the Leray spectral sequences to give us a map of spectral sequences that must respect the filtration on limit terms.

By hypothesis, the map $\mathrm {H}^m_{\mathrm {\acute {e}t}}(X,R^nf_*\mathcal {F}) \to \mathrm {H}^m_{\mathrm { h\mbox {-}\mathrm {\acute {e}t}}}(X^h,(R^nf_*\mathcal {F})^h)$ is an isomorphism, and we showed above that $(R^nf_{\mathrm {\acute {e}t},*}\mathcal {F}_{\mathrm {\acute {e}t}})^h \to R^nf^h_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}},*}(\mathcal {F}^h)_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}$ for all n. Thus, pullback to the Henselization gives an isomorphism on the second sheet

$$ \begin{align*}\mathrm{H}^m_{\mathrm{\acute{e}t}}(X,R^nf_*\mathcal{F}) \to \mathrm{H}^m_{\mathrm{h\mbox{-}\mathrm{\acute{e}t}}}(X^h,(R^nf_*\mathcal{F})^h) \simeq \mathrm{H}^m_{\mathrm{ h\mbox{-}\mathrm{\acute{e}t}}}(X^h,R^nf^h_*\mathcal{F}^h)\end{align*} $$

for our map of spectral sequences Since this map respects the filtration on limit terms, the comparison map $\mathrm {H}^n_{\mathrm {\acute {e}t}}(Y,\mathcal {F}) \to \mathrm {H}^n_{\mathrm {h\mbox {-}\mathrm {\acute {e}t}}}(Y^h,\mathcal {F}^h)$ is an isomorphism for all n. It follows that $\mathcal {F}$ satisfies GHGA comparison on ${Y}$ by Corollary 2.14.

Proof. As in the proof of Theorem 3.6.1, we can write A as the filtered direct limit of its subalgebras $A_i$ which are finitely generated over $\mathbf {Z}$ . Let $I_i=I \cap A_i$ . If $(A_i^h,I_i^h)$ is the Henselization of $(A_i,I_i)$ for each i, we have $A= \varinjlim A_i^h$ . Because X is finitely presented, for some $i_0$ , we have $X=X_{i_0} \times _{A_{i_0}^h} A$ for some proper morphism $X_{i_0} \to \mathrm {Spec}(A_{i_0}^h)$ , and similarly for $X',Y$ .

By [19, Lemma 01ZM], we can assume that the map $X \to X'$ arises from some map $X_{i_0 } \to X_{i_0}^{\prime }$ for some index $i_0$ . Similarly, using Lemma A.3, we can assume (possibly after increasing $i_0$ ) that the map $X^h \to Y^h$ arises from some map $X_{i_0}^h \to Y_{i_0}^h$ . Note that $X_{i_0},X_{i_0}^{\prime },Y_{i_0}$ are proper and finitely presented $A_{i_0}^h$ -schemes.

In the same way, we can (again, possibly after further increasing $i_0$ ) use Lemma A.5 to find finitely presented $\mathscr {O}_{X_{i_0}^h}$ -modules $\mathcal {G}_{i_0}, \mathcal {G}_{i_0}^{\prime }$ with a map $\mathcal {G}_{i_0} \to \mathcal {G}_{i_0}^{\prime }$ which pulls back to $\mathcal {G} \to \mathcal {G}'$ via the map $X^h \to X_{i_0}^h$ .

Since $A_{i_0}^h$ has been chosen to be the Henselization of a finitely generated $\mathbf {Z}$ -algebra, we see that $A_{i_0}^h$ is a Noetherian G-ring by [19, Lemma 0AH3]. Then letting $A_0=A_{i_0}^h,I_0=I_{i_0}^h$ , we get the desired result.

Proof. Since A is a Noetherian G-ring, the map $A \to A^\wedge $ from A to its I-adic completion is regular by [19, Lemma 0AH2]. We can now leverage Popescu’s theorem (Theorem 3.5.3) to see that $A \to A^\wedge $ is a filtered colimit of smooth ring maps. Furthermore, the map $A \to A^\wedge $ is faithfully flat by [19, Lemma 0AGV]. Then $X' \to X$ must be flat and surjective since $\mathrm {Spec}(A^\wedge ) \to \mathrm {Spec}(A)$ is.

We then have a commutative cube (†) below, with left and right faces Cartesian. The zigzag arrows represent pullback of sheaves, such as with the leftmost zigzag arrow showing that $\mathcal {G}'$ is the pullback of $\mathcal {G}$ .

We would like to find a coherent sheaf $\mathcal {F}$ on X such that $\mathcal {F}^h \simeq \mathcal {G}$ . In order to do so, we will use Popescu’s theorem. Since $A \to A^\wedge $ is a filtered colimit of smooth ring maps, there exists a smooth A-algebra $C_0$ such that $\mathcal {F}'$ is the pullback of a coherent sheaf $\mathcal {F}_0$ on the proper $C_0$ -scheme $X_{C_0}=X \times _A C_0$ . Let $\mathcal {G}_0 := \mathcal {G} \times _{X^h} X_{C_0}^h$ ; we then (possibly replacing $C_0$ by some further smooth A-algebra in the colimit yielding $A^\wedge $ ) can use Lemma A.5 to arrange that $\mathcal {F}_0^h \simeq \mathcal {G}_0$ descending the isomorphism $(\mathcal {F}')^h \simeq \mathcal {G}'$ .

(†)

This gives us a larger commutative diagram (††) below, again with all left and right faces Cartesian. The arrow from $\mathcal {F}_0$ to $\mathcal {F}$ represents that we will construct $\mathcal {F}$ as a pullback of $\mathcal {F}_0$ along the dashed arrow from X to $X_{C_0}$ , which is a section of the morphism $X_{C_0} \to X$ .

(††)

The map $C_0 \to A^\wedge $ gives us a map $C_0 \to A/I$ . We can find an étale A-algebra $A'$ with $A'/IA' \simeq A/I$ and a map $C_0 \to A'$ lifting the map $C_0 \to A/I$ by [19, Lemma 07M7]. Because A is Henselian, the map $A \to A'$ has a section lifting the isomorphism $A/I \simeq A'/IA'$ , so we have a section $C_0 \to A$ over A lifting the map $C_0 \to A/I$ .

This map $C_0 \to A$ gives us the desired dashed arrow $X \to X_{C_0}$ over X. As in (††), let $\mathcal {F}$ be the pullback of $\mathcal {F}_0$ along this map. Now we wish to show that $\mathcal {F}^h$ is isomorphic to $\mathcal {G}$ , which is represented by the zigzag arrow with a question mark in (††).

We recall that we have an isomorphism $\mathcal {F}_0^h \stackrel {\textstyle \sim }{\smash {\longrightarrow }\rule {0pt}{0.4ex}} \mathcal {G}_0$ , represented by the second horizontal zigzag arrow of (††). We can pull back this map along the map $X_{C_0}^h \to X^h$ , giving us an isomorphism $\mathcal {F}^h \stackrel {\textstyle \sim }{\smash {\longrightarrow }\rule {0pt}{0.4ex}} \mathcal {G}$ (here, we use that $X^h \to X_{C_0}^h \to X^h$ is the identity map). Hence, $\mathcal {G}$ is algebraizable, as we desired to show.

Proof. Applying Lemma 4.1.1 to the the injective morphism $\mathcal {G} \hookrightarrow \mathcal {F}^h$ of finitely presented $\mathscr {O}_{X^h}$ -submodules, we can obtain a map of Henselian pairs $(A_0,I_0) \to (A,I)$ with $A_0$ a Noetherian G-ring, as well as

• • a proper and finitely presented $A_{0}$ -scheme $X_{0}$ for which ;

• • a finitely presented sheaf of $\mathscr {O}_{X_{0}}$ -modules $\mathcal {F}_{0}$ for which $\mathcal {F} = \mathcal {F}_{0} \times _{X_{0}} X$ ;

• • and a morphism of finitely presented sheaves of $\mathscr {O}_{{X_0}^h}$ -modules $\mathcal {G}_{0} \to \mathcal {F}_0^h$ which pulls back to the injective morphism $\mathcal {G} \hookrightarrow \mathcal {F}^h$ by the map $X^h \to X_0^h$ .

Since pullback is right exact, we can replace $\mathcal {G}_{0}$ with its sheafified image in $\mathcal {F}_{0}^h$ , as that will also pull back to $\mathcal {G}$ . Thus, we can identify $\mathcal {G}_{0}$ with a coherent subsheaf of $\mathcal {F}_{0}^h$ .

As in Remark 4.1.2, if $\mathcal {G}_0$ is algebraizable as the Henselization of some coherent subsheaf of $\mathcal {F}_0$ , then $\mathcal {G}$ is also algebraizable (as the Henselization of a coherent subsheaf of $\mathcal {F}$ ). Thus, replacing A with $A_0$ , we have reduced to the case where A is a Noetherian G-ring. By Lemma 4.2.1 and Remark 4.2.2, we can furthermore assume that A is I-adically complete (and we will no longer need to assume that A is a G-ring).

Consider the formal scheme $X^\wedge \to \mathrm {Spf}(A)$ which is the I-adic completion of X, and is proper over $\mathrm {Spf}(A)$ . The map $X^\wedge \to X$ factors through $X^h$ .

We write $\mathcal {F}^\wedge ,\mathcal {G}^\wedge $ for the pullbacks of $\mathcal {F},\mathcal {G}$ along the maps $X^\wedge \to X, X^\wedge \to X^h$ , respectively. Then clearly $\mathcal {G}^\wedge \subset \mathcal {F}^\wedge $ , so by formal GAGA [Reference Grothendieck6, Theorem 5.1.4], we have a coherent sheaf $\mathcal {G}_1 \subset \mathcal {F}$ on X such that $\mathcal {G}_1^\wedge =(\mathcal {G}_1^h)^\wedge \simeq \mathcal {G}^\wedge $ as subsheaves of $\mathcal {F}^\wedge $ .

To show that the finitely presented sheaves $\mathcal {G}_1^h$ and $\mathcal {G}$ are equal as subsheaves of $\mathcal {F}^h$ , it suffices to work over Henselian-affine opens of $X^h$ arising from affine opens of X. Since $\mathcal {G}_1^h,\mathcal {G}$ are finitely presented, we can consider a new affine setting: $\mathrm {Spec}(B^h)$ for B a finite-type A-algebra (with I-adic Henselization and completion $B^h, B^\wedge $ , respectively) and $\mathcal {F}^h=\widetilde {M}$ , $\mathcal {G}=\widetilde {N}$ , $\mathcal {G}_1^h=\widetilde {N'}$ for M a finitely generated $B^h$ -module with finitely generated submodules $N,N'$ : we wish to show that if and are equal as $B^\wedge $ -submodules of , then N and $N'$ are equal as $B^h$ -submodules of M.