The Stacks project

74.42 Grothendieck's existence theorem

In this section we discuss Grothendieck's existence theorem for algebraic spaces. Instead of developing a theory of “formal algebraic spaces” we temporarily develop a bit of language that replaces the notion of a “coherent module on a Noetherian adic formal space”.

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Below we will consider inverse systems $(\mathcal{F}_ n)$ of coherent $\mathcal{O}_ X$-modules such that

  1. $\mathcal{F}_ n$ is annihilated by $\mathcal{I}^ n$, and

  2. the transition maps induce isomorphisms $\mathcal{F}_{n + 1}/\mathcal{I}^ n\mathcal{F}_{n + 1} \to \mathcal{F}_ n$.

A morphism $\alpha : (\mathcal{F}_ n) \to (\mathcal{G}_ n)$ of such inverse systems is simply a compatible system of morphisms $\alpha _ n : \mathcal{F}_ n \to \mathcal{G}_ n$. Let us denote the category of these inverse systems with $\textit{Coh}(X, \mathcal{I})$. We will develop some theory regarding these systems that will parallel to the corresponding results in the case of schemes, see Cohomology of Schemes, Sections 30.24, 30.25, 30.27, and 30.28.

Functoriality. Let $f : X \to Y$ be a morphism of Noetherian algebraic spaces over a scheme $S$, and let $\mathcal{J} \subset \mathcal{O}_ Y$ be a quasi-coherent sheaf of ideals. Set $\mathcal{I} = f^{-1}\mathcal{J}\mathcal{O}_ X$. In this situation there is a functor

\[ f^* : \textit{Coh}(Y, \mathcal{J}) \longrightarrow \textit{Coh}(X, \mathcal{I}) \]

which sends $(\mathcal{G}_ n)$ to $(f^*\mathcal{G}_ n)$. Compare with Cohomology of Schemes, Lemma 30.23.9. If $f$ is étale, then we may think of this as simply the restriction of the system to $X$, see Properties of Spaces, Equation 64.26.1.1.

Étale descent. Let $S$ be a scheme. Let $U_0 \to X$ be a surjective étale morphism of Noetherian algebraic spaces. Set $U_1 = U_0 \times _ X U_0$ and $U_2 = U_0 \times _ X U_0 \times _ X U_0$. Let $\mathcal{I} \subset \mathcal{O}_{X}$ be a quasi-coherent sheaf of ideals. Set $\mathcal{I}_ i = \mathcal{I}|_{U_ i}$. In this situation we obtain a diagram of categories

\[ \xymatrix{ \textit{Coh}(X, \mathcal{I}) \ar[r] & \textit{Coh}(U_0, \mathcal{I}_0) \ar@<0.5ex>[r] \ar@<-0.5ex>[r] & \textit{Coh}(U_1, \mathcal{I}_1) \ar@<1ex>[r] \ar[r] \ar@<-1ex>[r] & \textit{Coh}(U_2, \mathcal{I}_2) } \]

an the first arrow presents $\textit{Coh}(X, \mathcal{I})$ as the homotopy limit of the right part of the diagram. More precisely, given a descent datum, i.e., a pair $((\mathcal{G}_ n), \varphi )$ where $(\mathcal{G}_ n)$ is an object of $\textit{Coh}(U_0, \mathcal{I}_0)$ and $\varphi : \text{pr}_0^*(\mathcal{G}_ n) \to \text{pr}_1^*(\mathcal{G}_ n)$ is an isomorphism in $\textit{Coh}(U_1, \mathcal{I}_1)$ satisfying the cocycle condition in $\textit{Coh}(U_2, \mathcal{I}_2)$, then there exists a unique object $(\mathcal{F}_ n)$ of $\textit{Coh}(X, \mathcal{I})$ whose associated canonical descent datum is isomorphic to $((\mathcal{G}_ n), \varphi )$. Compare with Descent on Spaces, Definition 72.3.3. The proof of this statement follows immediately by applying Descent on Spaces, Proposition 72.4.1 to the descent data $(\mathcal{G}_ n, \varphi _ n)$ for varying $n$.

Lemma 74.42.1. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals.

  1. The category $\textit{Coh}(X, \mathcal{I})$ is abelian.

  2. Exactness in $\textit{Coh}(X, \mathcal{I})$ can be checked étale locally.

  3. For any flat morphism $f : X' \to X$ of Noetherian algebraic spaces the functor $f^* : \textit{Coh}(X, \mathcal{I}) \to \textit{Coh}(X', f^{-1}\mathcal{I}\mathcal{O}_{X'})$ is exact.

Proof. Proof of (1). Choose an affine scheme $U_0$ and a surjective étale morphism $U_0 \to X$. Set $U_1 = U_0 \times _ X U_0$ and $U_2 = U_0 \times _ X U_0 \times _ X U_0$ as in our discussion of étale descent above. The categories $\textit{Coh}(U_ i, \mathcal{I}_ i)$ are abelian (Cohomology of Schemes, Lemma 30.23.2) and the pullback functors are exact functors $\textit{Coh}(U_0, \mathcal{I}_0) \to \textit{Coh}(U_1, \mathcal{I}_1)$ and $\textit{Coh}(U_1, \mathcal{I}_1) \to \textit{Coh}(U_2, \mathcal{I}_2)$ (Cohomology of Schemes, Lemma 30.23.9). The lemma then follows formally from the description of $\textit{Coh}(X, \mathcal{I})$ as a category of descent data. Some details omitted; compare with the proof of Groupoids, Lemma 39.14.6.

Part (2) follows immediately from the discussion in the previous paragraph. In the situation of (3) choose a commutative diagram

\[ \xymatrix{ U' \ar[d] \ar[r] & U \ar[d] \\ X' \ar[r] & X } \]

where $U'$ and $U$ are affine schemes and the vertical morphisms are surjective étale. Then $U' \to U$ is a flat morphism of Noetherian schemes (Morphisms of Spaces, Lemma 65.30.5) whence the pullback functor $\textit{Coh}(U, \mathcal{I}\mathcal{O}_ U) \to \textit{Coh}(U', \mathcal{I}\mathcal{O}_{U'})$ is exact by Cohomology of Schemes, Lemma 30.23.9. Since we can check exactness in $\textit{Coh}(X, \mathcal{O}_ X)$ on $U$ and similarly for $X', U'$ the assertion follows. $\square$

Lemma 74.42.2. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. A map $(\mathcal{F}_ n) \to (\mathcal{G}_ n)$ is surjective in $\textit{Coh}(X, \mathcal{I})$ if and only if $\mathcal{F}_1 \to \mathcal{G}_1$ is surjective.

Proof. We can check on an affine étale cover of $X$ by Lemma 74.42.1. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.3. $\square$

Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. There is a functor

74.42.2.1
\begin{equation} \label{spaces-more-morphisms-equation-completion-functor} \textit{Coh}(\mathcal{O}_ X) \longrightarrow \textit{Coh}(X, \mathcal{I}), \quad \mathcal{F} \longmapsto \mathcal{F}^\wedge \end{equation}

which associates to the coherent $\mathcal{O}_ X$-module $\mathcal{F}$ the object $\mathcal{F}^\wedge = (\mathcal{F}/\mathcal{I}^ n\mathcal{F})$ of $\textit{Coh}(X, \mathcal{I})$.

Proof. It suffices to check this étale locally on $X$, see Lemma 74.42.1. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.4. $\square$

Lemma 74.42.4. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Set $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{F}, \mathcal{G})$. Then

\[ \mathop{\mathrm{lim}}\nolimits H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) = \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge , \mathcal{G}^\wedge ). \]

Proof. Since $\mathcal{H}$ is a sheaf on $X_{\acute{e}tale}$ and since we have étale descent for objects of $\textit{Coh}(X, \mathcal{I})$ it suffices to prove this étale locally. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.5. $\square$

We introduce the setting that we will focus on throughout the rest of this section.

Situation 74.42.5. Here $A$ is a Noetherian ring complete with respect to an ideal $I$. Also $f : X \to \mathop{\mathrm{Spec}}(A)$ is a finite type separated morphism of algebraic spaces and $\mathcal{I} = I\mathcal{O}_ X$.

In this situation we denote

\[ \textit{Coh}_{\text{support proper over } A}(\mathcal{O}_ X) \]

be the full subcategory of $\textit{Coh}(\mathcal{O}_ X)$ consisting of those coherent $\mathcal{O}_ X$-modules whose support is proper over $\mathop{\mathrm{Spec}}(A)$, or equivalently whose scheme theoretic support is proper over $\mathop{\mathrm{Spec}}(A)$, see Derived Categories of Spaces, Lemma 73.7.7. Similarly, we let

\[ \textit{Coh}_{\text{support proper over } A}(X, \mathcal{I}) \]

be the full subcategory of $\textit{Coh}(X, \mathcal{I})$ consisting of those objects $(\mathcal{F}_ n)$ such that the support of $\mathcal{F}_1$ is proper over $\mathop{\mathrm{Spec}}(A)$. Since the support of a quotient module is contained in the support of the module, it follows that (74.42.2.1) induces a functor

74.42.5.1
\begin{equation} \label{spaces-more-morphisms-equation-completion-functor-proper-over-A} \textit{Coh}_{\text{support proper over }A}(\mathcal{O}_ X) \longrightarrow \textit{Coh}_{\text{support proper over }A}(X, \mathcal{I}) \end{equation}

Our first result is that this functor is fully faithful.

Lemma 74.42.6. In Situation 74.42.5. Let $\mathcal{F}$, $\mathcal{G}$ be coherent $\mathcal{O}_ X$-modules. Assume that the intersection of the supports of $\mathcal{F}$ and $\mathcal{G}$ is proper over $\mathop{\mathrm{Spec}}(A)$. Then the map

\[ \mathop{Mor}\nolimits _{\textit{Coh}(\mathcal{O}_ X)}(\mathcal{F}, \mathcal{G}) \longrightarrow \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{F}^\wedge , \mathcal{G}^\wedge ) \]

coming from (74.42.2.1) is a bijection. In particular, (74.42.5.1) is fully faithful.

Proof. Let $\mathcal{H} = \mathop{\mathcal{H}\! \mathit{om}}\nolimits _{\mathcal{O}_ X}(\mathcal{G}, \mathcal{F})$. This is a coherent $\mathcal{O}_ X$-module because its restriction of schemes étale over $X$ is coherent by Modules, Lemma 17.21.5. By Lemma 74.42.4 the map

\[ \mathop{\mathrm{lim}}\nolimits _ n H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) \to \mathop{Mor}\nolimits _{\textit{Coh}(X, \mathcal{I})} (\mathcal{G}^\wedge , \mathcal{F}^\wedge ) \]

is bijective. Let $i : Z \to X$ be the scheme theoretic support of $\mathcal{H}$. It is clear that $Z$ is a closed subspace such that $|Z|$ is contained in the intersection of the supports of $\mathcal{F}$ and $\mathcal{G}$. Hence $Z \to \mathop{\mathrm{Spec}}(A)$ is proper by assumption (see Derived Categories of Spaces, Section 73.7). Write $\mathcal{H} = i_*\mathcal{H}'$ for some coherent $\mathcal{O}_ Z$-module $\mathcal{H}'$. We have $i_*(\mathcal{H}'/I^ n\mathcal{H}') = \mathcal{H}/I^ n\mathcal{H}$. Hence we obtain

\begin{align*} \mathop{\mathrm{lim}}\nolimits _ n H^0(X, \mathcal{H}/\mathcal{I}^ n\mathcal{H}) & = \mathop{\mathrm{lim}}\nolimits _ n H^0(Z, \mathcal{H}'/\mathcal{I}^ n\mathcal{H}') \\ & = H^0(Z, \mathcal{H}') \\ & = H^0(X, \mathcal{H}) \\ & = \mathop{Mor}\nolimits _{\textit{Coh}(\mathcal{O}_ X)}(\mathcal{F}, \mathcal{G}) \end{align*}

the second equality by the theorem on formal functions (Cohomology of Spaces, Lemma 67.21.6). This proves the lemma. $\square$

Remark 74.42.7. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_ X$ be quasi-coherent sheaves of ideals. Let $\alpha : (\mathcal{F}_ n) \to (\mathcal{G}_ n)$ be a morphism of $\textit{Coh}(X, \mathcal{I})$. Given an affine scheme $U = \mathop{\mathrm{Spec}}(A)$ and a surjective étale morphism $U \to X$ denote $I, K \subset A$ the ideals corresponding to the restrictions $\mathcal{I}|_ U, \mathcal{K}|_ U$. Denote $\alpha _ U : M \to N$ of finite $A^\wedge $-modules which corresponds to $\alpha |_ U$ via Cohomology of Schemes, Lemma 30.23.1. We claim the following are equivalent

  1. there exists an integer $t \geq 1$ such that $\mathop{\mathrm{Ker}}(\alpha _ n)$ and $\mathop{\mathrm{Coker}}(\alpha _ n)$ are annihilated by $\mathcal{K}^ t$ for all $n \geq 1$,

  2. for any (or some) affine open $\mathop{\mathrm{Spec}}(A) = U \subset X$ as above the modules $\mathop{\mathrm{Ker}}(\alpha _ U)$ and $\mathop{\mathrm{Coker}}(\alpha _ U)$ are annihilated by $K^ t$ for some integer $t \geq 1$.

If these equivalent conditions hold we will say that $\alpha $ is a map whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. To see the equivalence we refer to Cohomology of Schemes, Remark 30.25.1.

Lemma 74.42.8. Let $S$ be a scheme. Let $X$ be a Noetherian algebraic space over $S$ and let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $\mathcal{G}$ be a coherent $\mathcal{O}_ X$-module, $(\mathcal{F}_ n)$ an object of $\textit{Coh}(X, \mathcal{I})$, and $\alpha : (\mathcal{F}_ n) \to \mathcal{G}^\wedge $ a map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$. Then there exists a unique (up to unique isomorphism) triple $(\mathcal{F}, a, \beta )$ where

  1. $\mathcal{F}$ is a coherent $\mathcal{O}_ X$-module,

  2. $a : \mathcal{F} \to \mathcal{G}$ is an $\mathcal{O}_ X$-module map whose kernel and cokernel are annihilated by a power of $\mathcal{I}$,

  3. $\beta : (\mathcal{F}_ n) \to \mathcal{F}^\wedge $ is an isomorphism, and

  4. $\alpha = a^\wedge \circ \beta $.

Proof. The uniqueness and étale descent for objects of $\textit{Coh}(X, \mathcal{I})$ and $\textit{Coh}(\mathcal{O}_ X)$ implies it suffices to construct $(\mathcal{F}, a, \beta )$ étale locally on $X$. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.6. $\square$

Lemma 74.42.9. In Situation 74.42.5. Let $\mathcal{K} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $X_ e \subset X$ be the closed subspace cut out by $\mathcal{K}^ e$. Let $\mathcal{I}_ e = \mathcal{I}\mathcal{O}_{X_ e}$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})$. Assume

  1. the functor $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_ e}) \to \textit{Coh}_{\text{support proper over } A}(X_ e, \mathcal{I}_ e)$ is an equivalence for all $e \geq 1$, and

  2. there exists an object $\mathcal{H}$ of $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_ X)$ and a map $\alpha : (\mathcal{F}_ n) \to \mathcal{H}^\wedge $ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.

Then $(\mathcal{F}_ n)$ is in the essential image of (74.42.5.1).

Proof. During this proof we will use without further mention that for a closed immersion $i : Z \to X$ the functor $i_*$ gives an equivalence between the category of coherent modules on $Z$ and coherent modules on $X$ annihilated by the ideal sheaf of $Z$, see Cohomology of Spaces, Lemma 67.12.8. In particular we think of

\[ \textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_ e}) \subset \textit{Coh}_{\text{support proper over } A}(\mathcal{O}_ X) \]

as the full subcategory of consisting of modules annihilated by $\mathcal{K}^ e$ and

\[ \textit{Coh}_{\text{support proper over } A}(X_ e, \mathcal{I}_ e) \subset \textit{Coh}_{\text{support proper over } A}(X, \mathcal{I}) \]

as the full subcategory of objects annihilated by $\mathcal{K}^ e$. Moreover (1) tells us these two categories are equivalent under the completion functor (74.42.5.1).

Applying this equivalence we get a coherent $\mathcal{O}_ X$-module $\mathcal{G}_ e$ annihilated by $\mathcal{K}^ e$ corresponding to the system $(\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n)$ of $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})$. The maps $\mathcal{F}_ n/\mathcal{K}^{e + 1}\mathcal{F}_ n \to \mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n$ correspond to canonical maps $\mathcal{G}_{e + 1} \to \mathcal{G}_ e$ which induce isomorphisms $\mathcal{G}_{e + 1}/\mathcal{K}^ e\mathcal{G}_{e + 1} \to \mathcal{G}_ e$. We obtain an object $(\mathcal{G}_ e)$ of the category $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{K})$. The map $\alpha $ induces a system of maps

\[ \mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n \longrightarrow \mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H} \]

whence maps $\mathcal{G}_ e \to \mathcal{H}/\mathcal{K}^ e\mathcal{H}$ (by the equivalence of categories again). Let $t \geq 1$ be an integer, which exists by assumption (2), such that $\mathcal{K}^ t$ annihilates the kernel and cokernel of all the maps $\mathcal{F}_ n \to \mathcal{H}/\mathcal{I}^ n\mathcal{H}$. Then $\mathcal{K}^{2t}$ annihilates the kernel and cokernel of the maps $\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n \to \mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H}$ (details omitted; see Cohomology of Schemes, Remark 30.25.1). Whereupon we conclude that $\mathcal{K}^{4t}$ annihilates the kernel and the cokernel of the maps

\[ \mathcal{G}_ e \longrightarrow \mathcal{H}/\mathcal{K}^ e\mathcal{H}, \]

(details omitted; see Cohomology of Schemes, Remark 30.25.1). We apply Lemma 74.42.8 to obtain a coherent $\mathcal{O}_ X$-module $\mathcal{F}$, a map $a : \mathcal{F} \to \mathcal{H}$ and an isomorphism $\beta : (\mathcal{G}_ e) \to (\mathcal{F}/\mathcal{K}^ e\mathcal{F})$ in $\textit{Coh}(X, \mathcal{K})$. Working backwards, for a given $n$ the triple $(\mathcal{F}/\mathcal{I}^ n\mathcal{F}, a \bmod \mathcal{I}^ n, \beta \bmod \mathcal{I}^ n)$ is a triple as in the lemma for the morphism $\alpha _ n \bmod \mathcal{K}^ e : (\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n) \to (\mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H})$ of $\textit{Coh}(X, \mathcal{K})$. Thus the uniqueness in Lemma 74.42.8 gives a canonical isomorphism $\mathcal{F}/\mathcal{I}^ n\mathcal{F} \to \mathcal{F}_ n$ compatible with all the morphisms in sight.

To finish the proof of the lemma we still have to show that the support of $\mathcal{F}$ is proper over $A$. By construction the kernel of $a : \mathcal{F} \to \mathcal{H}$ is annihilated by a power of $\mathcal{K}$. Hence the support of this kernel is contained in the support of $\mathcal{G}_1$. Since $\mathcal{G}_1$ is an object of $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_{X_1})$ we see this is proper over $A$. Combined with the fact that the support of $\mathcal{H}$ is proper over $A$ we conclude that the support of $\mathcal{F}$ is proper over $A$ by Derived Categories of Spaces, Lemma 73.7.6. $\square$

Lemma 74.42.10. Let $S$ be a scheme. Let $f : X \to Y$ be a representable proper morphism of Noetherian algebraic spaces over $S$. Let $\mathcal{J}, \mathcal{K} \subset \mathcal{O}_ Y$ be quasi-coherent sheaves of ideals. Assume $f$ is an isomorphism over $V = Y \setminus V(\mathcal{K})$. Set $\mathcal{I} = f^{-1}\mathcal{J} \mathcal{O}_ X$. Let $(\mathcal{G}_ n)$ be an object of $\textit{Coh}(Y, \mathcal{J})$, let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module, and let $\beta : (f^*\mathcal{G}_ n) \to \mathcal{F}^\wedge $ be an isomorphism in $\textit{Coh}(X, \mathcal{I})$. Then there exists a map

\[ \alpha : (\mathcal{G}_ n) \longrightarrow (f_*\mathcal{F})^\wedge \]

in $\textit{Coh}(Y, \mathcal{J})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$.

Proof. Since $f$ is a proper morphism we see that $f_*\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module (Cohomology of Spaces, Lemma 67.20.2). Thus the statement of the lemma makes sense. Consider the compositions

\[ \gamma _ n : \mathcal{G}_ n \to f_*f^*\mathcal{G}_ n \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}). \]

Here the first map is the adjunction map and the second is $f_*\beta _ n$. We claim that there exists a unique $\alpha $ as in the lemma such that the compositions

\[ \mathcal{G}_ n \xrightarrow {\alpha _ n} f_*\mathcal{F}/\mathcal{J}^ nf_*\mathcal{F} \to f_*(\mathcal{F}/\mathcal{I}^ n\mathcal{F}) \]

equal $\gamma _ n$ for all $n$. Because of the uniqueness and étale descent for $\textit{Coh}(Y, \mathcal{J})$ it suffices to prove this étale locally on $Y$. Thus we may assume $Y$ is the spectrum of a Noetherian ring. As $f$ is representable we see that $X$ is a scheme as well. Thus we reduce to the case of schemes, see proof of Cohomology of Schemes, Lemma 30.25.3. $\square$

Proof. We will use the equivalence of categories of Cohomology of Spaces, Lemma 67.12.8 without further mention in the proof of the theorem. By Lemma 74.42.6 the functor is fully faithful. Thus we need to prove the functor is essentially surjective.

Consider the collection $\Xi $ of quasi-coherent sheaves of ideals $\mathcal{K} \subset \mathcal{O}_ X$ such that the statement holds for every object $(\mathcal{F}_ n)$ of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ annihilated by $\mathcal{K}$. We want to show $(0)$ is in $\Xi $. If not, then since $X$ is Noetherian there exists a maximal quasi-coherent sheaf of ideals $\mathcal{K}$ not in $\Xi $, see Cohomology of Spaces, Lemma 67.13.1. After replacing $X$ by the closed subscheme of $X$ corresponding to $\mathcal{K}$ we may assume that every nonzero $\mathcal{K}$ is in $\Xi $. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$. We will show that this object is in the essential image, thereby completing the proof of the theorem.

Apply Chow's lemma (Lemma 74.40.5) to find a proper surjective morphism $f : Y \to X$ which is an isomorphism over a dense open $U \subset X$ such that $Y$ is H-quasi-projective over $A$. Note that $Y$ is a scheme and $f$ representable. Choose an open immersion $j : Y \to Y'$ with $Y'$ projective over $A$, see Morphisms, Lemma 29.43.11. Let $T_ n$ be the scheme theoretic support of $\mathcal{F}_ n$. Note that $|T_ n| = |T_1|$, hence $T_ n$ is proper over $A$ for all $n$ (Morphisms of Spaces, Lemma 65.40.7). Then $f^*\mathcal{F}_ n$ is supported on the closed subscheme $f^{-1}T_ n$ which is proper over $A$ (by Morphisms of Spaces, Lemma 65.40.4 and properness of $f$). In particular, the composition $f^{-1}T_ n \to Y \to Y'$ is closed (Morphisms, Lemma 29.41.7). Let $T'_ n \subset Y'$ be the corresponding closed subscheme; it is contained in the open subscheme $Y$ and equal to $f^{-1}T_ n$ as a closed subscheme of $Y$. Let $\mathcal{F}_ n'$ be the coherent $\mathcal{O}_{Y'}$-module corresponding to $f^*\mathcal{F}_ n$ viewed as a coherent module on $Y'$ via the closed immersion $f^{-1}T_ n = T'_ n \subset Y'$. Then $(\mathcal{F}_ n')$ is an object of $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. By the projective case of Grothendieck's existence theorem (Cohomology of Schemes, Lemma 30.24.3) there exists a coherent $\mathcal{O}_{Y'}$-module $\mathcal{F}'$ and an isomorphism $(\mathcal{F}')^\wedge \cong (\mathcal{F}'_ n)$ in $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. Let $Z' \subset Y'$ be the scheme theoretic support of $\mathcal{F}'$. Since $\mathcal{F}'/I\mathcal{F}' = \mathcal{F}'_1$ we see that $Z' \cap V(I\mathcal{O}_{Y'}) = T'_1$ set-theoretically. The structure morphism $p' : Y' \to \mathop{\mathrm{Spec}}(A)$ is proper, hence $p'(Z' \cap (Y' \setminus Y))$ is closed in $\mathop{\mathrm{Spec}}(A)$. If nonempty, then it would contain a point of $V(I)$ as $I$ is contained in the Jacobson radical of $A$ (Algebra, Lemma 10.95.6). But we've seen above that $Z' \cap (p')^{-1}V(I) = T'_1 \subset Y$ hence we conclude that $Z' \subset Y$. Thus $\mathcal{F}'|_ Y$ is supported on a closed subscheme of $Y$ proper over $A$.

Let $\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting out the reduced complement $X \setminus U$. By Cohomology of Spaces, Lemma 67.20.2 the $\mathcal{O}_ X$-module $\mathcal{H} = f_*\mathcal{F}'$ is coherent and by Lemma 74.42.10 there exists a morphism $\alpha : (\mathcal{F}_ n) \to \mathcal{H}^\wedge $ in the category $\textit{Coh}_{\text{support proper over } A}(X, \mathcal{I})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. Let $Z_0 \subset X$ be the scheme theoretic support of $\mathcal{H}$. It is clear that $|Z_0| \subset f(|Z'|)$. Hence $Z_0 \to \mathop{\mathrm{Spec}}(A)$ is proper (Morphisms of Spaces, Lemma 65.40.7). Thus $\mathcal{H}$ is an object of $\textit{Coh}_{\text{support proper over } A}(\mathcal{O}_ X)$. Since each of the sheaves of ideals $\mathcal{K}^ e$ is an element of $\Xi $ we see that the assumptions of Lemma 74.42.9 are satisfied and we conclude. $\square$

Remark 74.42.12 (Unwinding Grothendieck's existence theorem). Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Write $S = \mathop{\mathrm{Spec}}(A)$ and $S_ n = \mathop{\mathrm{Spec}}(A/I^ n)$. Let $X \to S$ be a morphism of algebraic spaces that is separated and of finite type. For $n \geq 1$ we set $X_ n = X \times _ S S_ n$. Picture:

\[ \xymatrix{ X_1 \ar[r]_{i_1} \ar[d] & X_2 \ar[r]_{i_2} \ar[d] & X_3 \ar[r] \ar[d] & \ldots & X \ar[d] \\ S_1 \ar[r] & S_2 \ar[r] & S_3 \ar[r] & \ldots & S } \]

In this situation we consider systems $(\mathcal{F}_ n, \varphi _ n)$ where

  1. $\mathcal{F}_ n$ is a coherent $\mathcal{O}_{X_ n}$-module,

  2. $\varphi _ n : i_ n^*\mathcal{F}_{n + 1} \to \mathcal{F}_ n$ is an isomorphism, and

  3. $\text{Supp}(\mathcal{F}_1)$ is proper over $S_1$.

Theorem 74.42.11 says that the completion functor

\[ \begin{matrix} \text{coherent }\mathcal{O}_ X\text{-modules }\mathcal{F} \\ \text{with support proper over }A \end{matrix} \quad \longrightarrow \quad \begin{matrix} \text{systems }(\mathcal{F}_ n) \\ \text{as above} \end{matrix} \]

is an equivalence of categories. In the special case that $X$ is proper over $A$ we can omit the conditions on the supports.


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