The Stacks project

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.

Remark 30.25.1. Let $X$ be a Noetherian scheme 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 open $\mathop{\mathrm{Spec}}(A) = U \subset X$ with $\mathcal{I}|_ U, \mathcal{K}|_ U$ corresponding to ideals $I, K \subset A$ denote $\alpha _ U : M \to N$ of finite $A^\wedge $-modules which corresponds to $\alpha |_ U$ via 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 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 $t \geq 1$, and

  3. there exists a finite affine open covering $X = \bigcup U_ i$ such that the conclusion of (2) holds for $\alpha _{U_ i}$.

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 use the following commutative algebra fact: suppose given an exact sequence

\[ 0 \to T \to M \to N \to Q \to 0 \]

of $A$-modules with $T$ and $Q$ annihilated by $K^ t$ for some ideal $K \subset A$. Then for every $f, g \in K^ t$ there exists a canonical map $"fg": N \to M$ such that $M \to N \to M$ is equal to multiplication by $fg$. Namely, for $y \in N$ we can pick $x \in M$ mapping to $fy$ in $N$ and then we can set $"fg"(y) = gx$. Thus it is clear that $\mathop{\mathrm{Ker}}(M/JM \to N/JN)$ and $\mathop{\mathrm{Coker}}(M/JM \to N/JN)$ are annihilated by $K^{2t}$ for any ideal $J \subset A$.

Applying the commutative algebra fact to $\alpha _{U_ i}$ and $J = I^ n$ we see that (3) implies (1). Conversely, suppose (1) holds and $M \to N$ is equal to $\alpha _ U$. Then there is a $t \geq 1$ such that $\mathop{\mathrm{Ker}}(M/I^ nM \to N/I^ nN)$ and $\mathop{\mathrm{Coker}}(M/I^ nM \to N/I^ nN)$ are annihilated by $K^ t$ for all $n$. We obtain maps $"fg" : N/I^ nN \to M/I^ nM$ which in the limit induce a map $N \to M$ as $N$ and $M$ are $I$-adically complete. Since the composition with $N \to M \to N$ is multiplication by $fg$ we conclude that $fg$ annihilates $T$ and $Q$. In other words $T$ and $Q$ are annihilated by $K^{2t}$ as desired.

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

  1. 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

  2. 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 10.96.2. 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 15.82.3. 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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