The Stacks project

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

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 67.30.5) 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. $\square$

Proof. We can check on an affine étale cover of $X$ by Lemma 76.42.1. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.3. $\square$

Proof. It suffices to check this étale locally on $X$, see Lemma 76.42.1. Thus we reduce to the case of schemes which is Cohomology of Schemes, Lemma 30.23.4. $\square$

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. $\square$

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 17.22.6. By Lemma 76.42.4 the map

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 75.7). 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

the second equality by the theorem on formal functions (Cohomology of Spaces, Lemma 69.22.6). This proves the lemma. $\square$

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. $\square$

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 69.12.8. In particular we think of

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

as the full subcategory of objects annihilated by $\mathcal{K}^ e$. Moreover (1) tells us these two categories are equivalent under the completion functor (76.42.5.1).

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

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

(details omitted; see Cohomology of Schemes, Remark 30.25.1). We apply Lemma 76.42.8 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 76.42.8 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 75.7.6. $\square$

Proof. Since $f$ is a proper morphism we see that $f_*\mathcal{F}$ is a coherent $\mathcal{O}_ Y$-module (Cohomology of Spaces, Lemma 69.20.2). Thus the statement of the lemma makes sense. Consider the compositions

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

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 69.12.8 without further mention in the proof of the theorem. By Lemma 76.42.6 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 69.13.1. 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 76.40.5) 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 67.40.7). 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 67.40.4 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 10.96.6). 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 69.20.2 the $\mathcal{O}_ X$-module $\mathcal{H} = f_*\mathcal{F}'$ is coherent and by Lemma 76.42.10 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 67.40.7). 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 76.42.9 are satisfied and we conclude. $\square$