The Stacks project

Proof. We will use the equivalence of categories of Lemma 30.9.8 without further mention. For a closed subscheme $Z \subset X$ proper over $A$ in this proof we will say a coherent module on $X$ is “supported on $Z$” if it is annihilated by the ideal sheaf of $Z$ or equivalently if it is the pushforward of a coherent module on $Z$. By Proposition 30.25.4 we know that the result is true for the functor between coherent modules and systems of coherent modules supported on $Z$. Hence it suffices to show that every object of $\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_ X)$ and every object of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ is supported on a closed subscheme $Z \subset X$ proper over $A$. This holds by definition for objects of $\textit{Coh}_{\text{support proper over }A}(\mathcal{O}_ X)$. We will prove this statement for objects of $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ using the method of proof of Proposition 30.25.4. We urge the reader to read that proof first.

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 Lemma 30.10.1. 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 supported on a closed subscheme $Z \subset X$ proper over $A$, thereby completing the proof of the theorem.

Apply Chow's lemma (Lemma 30.18.1) 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 $A$. Choose an open immersion $j : Y \to Y'$ with $Y'$ projective over $A$, see Morphisms, Lemma 29.43.11. Observe that

The first equality by Morphisms, Lemma 29.5.3. By assumption and Lemma 30.26.5 part (3) we see that $f^{-1}\text{Supp}(\mathcal{F}_1)$ is proper over $A$. Hence the image of $f^{-1}\text{Supp}(\mathcal{F}_1)$ under $j$ is closed in $Y'$ by Lemma 30.26.5 part (1). Thus $\mathcal{F}'_ n = j_*f^*\mathcal{F}_ n$ is coherent on $Y'$ by Lemma 30.9.11. It follows that $(\mathcal{F}_ n')$ is an object of $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. By the projective case of Grothendieck's existence theorem (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 $\textit{Coh}(Y', I\mathcal{O}_{Y'})$. Since $\mathcal{F}'/I\mathcal{F}' = \mathcal{F}'_1$ we see that

The structure morphism $p' : Y' \to \mathop{\mathrm{Spec}}(A)$ is proper, hence $p'(\text{Supp}(\mathcal{F}') \setminus j(Y))$ is closed in $\mathop{\mathrm{Spec}}(A)$. A nonempty closed subset of $\mathop{\mathrm{Spec}}(A)$ contains a point of $V(I)$ as $I$ is contained in the Jacobson radical of $A$ by Algebra, Lemma 10.96.6. The displayed equation shows that $\text{Supp}(\mathcal{F}') \cap (p')^{-1}V(I) \subset j(Y)$ hence we conclude that $\text{Supp}(\mathcal{F}') \subset j(Y)$. Thus $\mathcal{F}'|_ Y = j^*\mathcal{F}'$ is supported on a closed subscheme $Z'$ of $Y$ proper over $A$ and $(\mathcal{F}'|_ Y)^\wedge = (f^*\mathcal{F}_ n)$.

Let $\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting out the reduced complement $X \setminus U$. By Proposition 30.19.1 the $\mathcal{O}_ X$-module $\mathcal{H} = f_*(\mathcal{F}'|_ Y)$ is coherent and by Lemma 30.25.3 there exists a morphism $\alpha : (\mathcal{F}_ n) \to \mathcal{H}^\wedge $ of $\textit{Coh}(X, \mathcal{I})$ whose kernel and cokernel are annihilated by a power $\mathcal{K}^ t$ of $\mathcal{K}$. We obtain an exact sequence

in $\textit{Coh}(X, \mathcal{I})$. If $Z_0 \subset X$ is the scheme theoretic support of $\mathcal{H}$, then it is clear that $Z_0 \subset f(Z')$ set-theoretically. Hence $Z_0$ is proper over $A$ by Lemma 30.26.3 and Lemma 30.26.5 part (2). Hence $\mathcal{H}^\wedge $ is in the subcategory defined in Lemma 30.26.11 part (2) and a fortiori in $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$. We conclude that $\mathop{\mathrm{Ker}}(\alpha )$ and $\mathop{\mathrm{Coker}}(\alpha )$ are in $\textit{Coh}_{\text{support proper over }A}(X, \mathcal{I})$ by Lemma 30.26.11 part (1). By induction hypothesis, more precisely because $\mathcal{K}^ t$ is in $\Xi $, we see that $\mathop{\mathrm{Ker}}(\alpha )$ and $\mathop{\mathrm{Coker}}(\alpha )$ are in the subcategory defined in Lemma 30.26.11 part (2). Since this is a Serre subcategory by the lemma, we conclude that the same is true for $(\mathcal{F}_ n)$ which is what we wanted to show. $\square$