## 30.25 Grothendieck's existence theorem, II

In this section we discuss Grothendieck's existence theorem in the proper case. Before we give the statement and proof, we need to develop a bit more theory regarding the categories $\textit{Coh}(X, \mathcal{I})$ of coherent formal modules introduced in Section 30.23.

Lemma 30.25.2. Let $X$ be a Noetherian scheme. Let $\mathcal{I}, \mathcal{K} \subset \mathcal{O}_ X$ be quasi-coherent sheaves of ideals. Let $X_ e \subset X$ be the closed subscheme cut out by $\mathcal{K}^ e$. Let $\mathcal{I}_ e = \mathcal{I}\mathcal{O}_{X_ e}$. Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. Assume

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

there exists a coherent sheaf $\mathcal{H}$ on $X$ 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 (30.23.3.1).

**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 Lemma 30.9.8. In particular we may identify $\textit{Coh}(\mathcal{O}_{X_ e})$ with the category of coherent $\mathcal{O}_ X$-modules annihilated by $\mathcal{K}^ e$ and $\textit{Coh}(X_ e, \mathcal{I}_ e)$ as the full subcategory of $\textit{Coh}(X, \mathcal{I})$ of objects annihilated by $\mathcal{K}^ e$. Moreover (1) tells us these two categories are equivalent under the completion functor (30.23.3.1).

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 $\textit{Coh}(X, \mathcal{I})$. 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 $\mathcal{G}_{e + 1}/\mathcal{K}^ e\mathcal{G}_{e + 1} \to \mathcal{G}_ e$. Hence $(\mathcal{G}_ e)$ is an object of $\textit{Coh}(X, \mathcal{K})$. 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 $\mathcal{F}_ n/\mathcal{K}^ e\mathcal{F}_ n \to \mathcal{H}/(\mathcal{I}^ n + \mathcal{K}^ e)\mathcal{H}$, see Remark 30.25.1. Whereupon we conclude that $\mathcal{K}^{4t}$ annihilates the kernel and the cokernel of the maps

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

see Remark 30.25.1. We apply Lemma 30.23.6 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 30.23.6 gives a canonical isomorphism $\mathcal{F}/\mathcal{I}^ n\mathcal{F} \to \mathcal{F}_ n$ compatible with all the morphisms in sight. This finishes the proof of the lemma.
$\square$

Lemma 30.25.3. Let $Y$ be a Noetherian scheme. Let $\mathcal{J}, \mathcal{K} \subset \mathcal{O}_ Y$ be quasi-coherent sheaves of ideals. Let $f : X \to Y$ be a proper morphism which 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 (Proposition 30.19.1). 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 we may assume that $Y = \mathop{\mathrm{Spec}}(B)$ is affine. Let $J \subset B$ corresponds to the ideal $\mathcal{J}$. Set

\[ M_ n = H^0(X, \mathcal{F}/\mathcal{I}^ n\mathcal{F}) \quad \text{and}\quad M = H^0(X, \mathcal{F}) \]

By Lemma 30.20.4 and Theorem 30.20.5 the inverse limit of the modules $M_ n$ equals the completion $M^\wedge = \mathop{\mathrm{lim}}\nolimits M/J^ nM$. Set $N_ n = H^0(Y, \mathcal{G}_ n)$ and $N = \mathop{\mathrm{lim}}\nolimits N_ n$. Via the equivalence of categories of Lemma 30.23.1 the finite $B^\wedge $ modules $N$ and $M^\wedge $ correspond to $(\mathcal{G}_ n)$ and $f_*\mathcal{F}^\wedge $. It follows from this that $\alpha $ has to be the morphism of $\textit{Coh}(Y, \mathcal{J})$ corresponding to the homomorphism

\[ \mathop{\mathrm{lim}}\nolimits \gamma _ n : N = \mathop{\mathrm{lim}}\nolimits _ n N_ n \longrightarrow \mathop{\mathrm{lim}}\nolimits M_ n = M^\wedge \]

of finite $B^\wedge $-modules.

We still have to show that the kernel and cokernel of $\alpha $ are annihilated by a power of $\mathcal{K}$. Set $Y' = \mathop{\mathrm{Spec}}(B^\wedge )$ and $X' = Y' \times _ Y X$. Let $\mathcal{K}'$, $\mathcal{J}'$, $\mathcal{G}'_ n$ and $\mathcal{I}'$, $\mathcal{F}'$ be the pullback of $\mathcal{K}$, $\mathcal{J}$, $\mathcal{G}_ n$ and $\mathcal{I}$, $\mathcal{F}$, to $Y'$ and $X'$. The projection morphism $f' : X' \to Y'$ is the base change of $f$ by $Y' \to Y$. Note that $Y' \to Y$ is a flat morphism of schemes as $B \to B^\wedge $ is flat by Algebra, Lemma 10.97.2. Hence $f'_*\mathcal{F}'$, resp. $f'_*(f')^*\mathcal{G}_ n'$ is the pullback of $f_*\mathcal{F}$, resp. $f_*f^*\mathcal{G}_ n$ to $Y'$ by Lemma 30.5.2. The uniqueness of our construction shows the pullback of $\alpha $ to $Y'$ is the corresponding map $\alpha '$ constructed for the situation on $Y'$. Moreover, to check that the kernel and cokernel of $\alpha $ are annihilated by $\mathcal{K}^ t$ it suffices to check that the kernel and cokernel of $\alpha '$ are annihilated by $(\mathcal{K}')^ t$. Namely, to see this we need to check this for kernels and cokernels of the maps $\alpha _ n$ and $\alpha '_ n$ (see Remark 30.25.1) and the ring map $B \to B^\wedge $ induces an equivalence of categories between modules annihilated by $J^ n$ and $(J')^ n$, see More on Algebra, Lemma 15.88.3. Thus we may assume $B$ is complete with respect to $J$.

Assume $Y = \mathop{\mathrm{Spec}}(B)$ is affine, $\mathcal{J}$ corresponds to the ideal $J \subset B$, and $B$ is complete with respect to $J$. In this case $(\mathcal{G}_ n)$ is in the essential image of the functor $\textit{Coh}(\mathcal{O}_ Y) \to \textit{Coh}(Y, \mathcal{J})$. Say $\mathcal{G}$ is a coherent $\mathcal{O}_ Y$-module such that $(\mathcal{G}_ n) = \mathcal{G}^\wedge $. Note that $f^*(\mathcal{G}^\wedge ) = (f^*\mathcal{G})^\wedge $. Hence Lemma 30.24.1 tells us that $\beta $ comes from an isomorphism $b : f^*\mathcal{G} \to \mathcal{F}$ and $\alpha $ is the completion functor applied to

\[ \mathcal{G} \to f_*f^*\mathcal{G} \cong f_*\mathcal{F} \]

Hence we are trying to verify that the kernel and cokernel of the adjunction map $c : \mathcal{G} \to f_*f^*\mathcal{G}$ are annihilated by a power of $\mathcal{K}$. However, since the restriction $f|_{f^{-1}(V)} : f^{-1}(V) \to V$ is an isomorphism we see that $c|_ V$ is an isomorphism. Thus the coherent sheaves $\mathop{\mathrm{Ker}}(c)$ and $\mathop{\mathrm{Coker}}(c)$ are supported on $V(\mathcal{K})$ hence are annihilated by a power of $\mathcal{K}$ (Lemma 30.10.2) as desired.
$\square$

The following proposition is the form of Grothendieck's existence theorem which is most often used in practice.

Proposition 30.25.4. Let $A$ be a Noetherian ring complete with respect to an ideal $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of schemes. Set $\mathcal{I} = I\mathcal{O}_ X$. Then the functor (30.23.3.1) is an equivalence.

**Proof.**
We have already seen that (30.23.3.1) is fully faithful in Lemma 30.24.1. Thus it suffices to show that the functor is essentially surjective.

Consider the collection $\Xi $ of quasi-coherent sheaves of ideals $\mathcal{K} \subset \mathcal{O}_ X$ such that every object $(\mathcal{F}_ n)$ annihilated by $\mathcal{K}$ is in the essential image. 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 $. (This uses the correspondence by coherent modules annihilated by $\mathcal{K}$ and coherent modules on the closed subscheme corresponding to $\mathcal{K}$, see Lemma 30.9.8.) Let $(\mathcal{F}_ n)$ be an object of $\textit{Coh}(X, \mathcal{I})$. We will show that this object is in the essential image of the functor (30.23.3.1), thereby completion the proof of the proposition.

Apply Chow's lemma (Lemma 30.18.1) to find a proper surjective morphism $f : X' \to X$ which is an isomorphism over a dense open $U \subset X$ such that $X'$ is projective over $A$. Let $\mathcal{K}$ be the quasi-coherent sheaf of ideals cutting out the reduced complement $X \setminus U$. By the projective case of Grothendieck's existence theorem (Lemma 30.24.3) there exists a coherent module $\mathcal{F}'$ on $X'$ such that $(\mathcal{F}')^\wedge \cong (f^*\mathcal{F}_ n)$. By Proposition 30.19.1 the $\mathcal{O}_ X$-module $\mathcal{H} = f_*\mathcal{F}'$ is coherent and by Lemma 30.25.3 there exists a morphism $(\mathcal{F}_ n) \to \mathcal{H}^\wedge $ of $\textit{Coh}(X, \mathcal{I})$ whose kernel and cokernel are annihilated by a power of $\mathcal{K}$. The powers $\mathcal{K}^ e$ are all in $\Xi $ so that (30.23.3.1) is an equivalence for the closed subschemes $X_ e = V(\mathcal{K}^ e)$. We conclude by Lemma 30.25.2.
$\square$

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