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77.12 Grothendieck's Existence Theorem

This section is the analogue of More on Flatness, Section 38.28 and continues the discussion in More on Morphisms of Spaces, Section 76.42. We will work in the following situation.

Situation 77.12.1. Here we have an inverse system of rings $(A_ n)$ with surjective transition maps whose kernels are locally nilpotent. Set $A = \mathop{\mathrm{lim}}\nolimits A_ n$. We have an algebraic space $X$ separated and of finite presentation over $A$. We set $X_ n = X \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_ n)$ and we view it as a closed subspace of $X$. We assume further given a system $(\mathcal{F}_ n, \varphi _ n)$ where $\mathcal{F}_ n$ is a finitely presented $\mathcal{O}_{X_ n}$-module, flat over $A_ n$, with support proper over $A_ n$, and

\[ \varphi _ n : \mathcal{F}_ n \otimes _{\mathcal{O}_{X_ n}} \mathcal{O}_{X_{n - 1}} \longrightarrow \mathcal{F}_{n - 1} \]

is an isomorphism (notation using the equivalence of Morphisms of Spaces, Lemma 67.14.1).

Our goal is to see if we can find a quasi-coherent sheaf $\mathcal{F}$ on $X$ such that $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_{X_ n}$ for all $n$.

Lemma 77.12.2. In Situation 77.12.1 consider

\[ K = R\mathop{\mathrm{lim}}\nolimits _{D_\mathit{QCoh}(\mathcal{O}_ X)}(\mathcal{F}_ n) = DQ_ X(R\mathop{\mathrm{lim}}\nolimits _{D(\mathcal{O}_ X)}\mathcal{F}_ n) \]

Then $K$ is in $D^ b_{\mathit{QCoh}}(\mathcal{O}_ X)$ and in fact $K$ has nonzero cohomology sheaves only in degrees $\geq 0$.

Proof. Special case of Derived Categories of Spaces, Example 75.19.5. $\square$

Lemma 77.12.3. In Situation 77.12.1 let $K$ be as in Lemma 77.12.2. For any perfect object $E$ of $D(\mathcal{O}_ X)$ we have

  1. $M = R\Gamma (X, K \otimes ^\mathbf {L} E)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\Gamma (X_ n, \mathcal{F}_ n \otimes ^\mathbf {L} E|_{X_ n}) = M \otimes _ A^\mathbf {L} A_ n$ in $D(A_ n)$,

  2. $N = R\mathop{\mathrm{Hom}}\nolimits _ X(E, K)$ is a perfect object of $D(A)$ and there is a canonical isomorphism $R\mathop{\mathrm{Hom}}\nolimits _{X_ n}(E|_{X_ n}, \mathcal{F}_ n) = N \otimes _ A^\mathbf {L} A_ n$ in $D(A_ n)$.

In both statements $E|_{X_ n}$ denotes the derived pullback of $E$ to $X_ n$.

Proof. Proof of (2). Write $E_ n = E|_{X_ n}$ and $N_ n = R\mathop{\mathrm{Hom}}\nolimits _{X_ n}(E_ n, \mathcal{F}_ n)$. Recall that $R\mathop{\mathrm{Hom}}\nolimits _{X_ n}(-, -)$ is equal to $R\Gamma (X_ n, R\mathop{\mathcal{H}\! \mathit{om}}\nolimits (-, -))$, see Cohomology on Sites, Section 21.36. Hence by Derived Categories of Spaces, Lemma 75.25.8 we see that $N_ n$ is a perfect object of $D(A_ n)$ whose formation commutes with base change. Thus the maps $N_ n \otimes _{A_ n}^\mathbf {L} A_{n - 1} \to N_{n - 1}$ coming from $\varphi _ n$ are isomorphisms. By More on Algebra, Lemma 15.97.3 we find that $R\mathop{\mathrm{lim}}\nolimits N_ n$ is perfect and that its base change back to $A_ n$ recovers $N_ n$. On the other hand, the exact functor $R\mathop{\mathrm{Hom}}\nolimits _ X(E, -) : D_\mathit{QCoh}(\mathcal{O}_ X) \to D(A)$ of triangulated categories commutes with products and hence with derived limits, whence

\[ R\mathop{\mathrm{Hom}}\nolimits _ X(E, K) = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ X(E, \mathcal{F}_ n) = R\mathop{\mathrm{lim}}\nolimits R\mathop{\mathrm{Hom}}\nolimits _ X(E_ n, \mathcal{F}_ n) = R\mathop{\mathrm{lim}}\nolimits N_ n \]

This proves (2). To see that (1) holds, translate it into (2) using Cohomology on Sites, Lemma 21.48.4. $\square$

Proof. Combinging Lemma 77.12.3 and Derived Categories of Spaces, Lemma 75.25.7 we see that $R\Gamma (X, K \otimes ^\mathbf {L} E)$ is pseudo-coherent in $D(A)$ for all pseudo-coherent $E$ in $D(\mathcal{O}_ X)$. Thus the lemma follows from More on Morphisms of Spaces, Lemma 76.51.4. $\square$

Lemma 77.12.5. In Situation 77.12.1 let $K$ be as in Lemma 77.12.2. For any ├ętale morphism $U \to X$ with $U$ quasi-compact and quasi-separated we have

\[ R\Gamma (U, K) \otimes _ A^\mathbf {L} A_ n = R\Gamma (U_ n, \mathcal{F}_ n) \]

in $D(A_ n)$ where $U_ n = U \times _ X X_ n$.

Proof. Fix $n$. By Derived Categories of Spaces, Lemma 75.27.3 there exists a system of perfect complexes $E_ m$ on $X$ such that $R\Gamma (U, K) = \text{hocolim} R\Gamma (X, K \otimes ^\mathbf {L} E_ m)$. In fact, this formula holds not just for $K$ but for every object of $D_\mathit{QCoh}(\mathcal{O}_ X)$. Applying this to $\mathcal{F}_ n$ we obtain

\begin{align*} R\Gamma (U_ n, \mathcal{F}_ n) & = R\Gamma (U, \mathcal{F}_ n) \\ & = \text{hocolim}_ m R\Gamma (X, \mathcal{F}_ n \otimes ^\mathbf {L} E_ m) \\ & = \text{hocolim}_ m R\Gamma (X_ n, \mathcal{F}_ n \otimes ^\mathbf {L} E_ m|_{X_ n}) \end{align*}

Using Lemma 77.12.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

Lemma 77.12.6. In Situation 77.12.1 let $K$ be as in Lemma 77.12.2. Denote $X_0 \subset |X|$ the closed subset consisting of points lying over the closed subset $\mathop{\mathrm{Spec}}(A_1) = \mathop{\mathrm{Spec}}(A_2) = \ldots $ of $\mathop{\mathrm{Spec}}(A)$. There exists an open subspace $W \subset X$ containing $X_0$ such that

  1. $H^ i(K)|_ W$ is zero unless $i = 0$,

  2. $\mathcal{F} = H^0(K)|_ W$ is of finite presentation, and

  3. $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_{X_ n}$.

Proof. Fix $n \geq 1$. By construction there is a canonical map $K \to \mathcal{F}_ n$ in $D_\mathit{QCoh}(\mathcal{O}_ X)$ and hence a canonical map $H^0(K) \to \mathcal{F}_ n$ of quasi-coherent sheaves. This explains the meaning of part (3).

Let $x \in X_0$ be a point. We will find an open neighbourhood $W$ of $x$ such that (1), (2), and (3) are true. Since $X_0$ is quasi-compact this will prove the lemma. Let $U \to X$ be an ├ętale morphism with $U$ affine and $u \in U$ a point mapping to $x$. Since $|U| \to |X|$ is open it suffices to find an open neighbourhood of $u$ in $U$ where (1), (2), and (3) are true. Say $U = \mathop{\mathrm{Spec}}(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma 77.12.4 and the definition of relative pseudo-coherence there exists a bounded above complex $F^\bullet $ of finite free $P$-modules representing $Ri_*K$ where $i : U \to \mathop{\mathrm{Spec}}(P)$ is the closed immersion induced by the presentation. Let $M_ n$ be the $B$-module corresponding to $\mathcal{F}_ n|_ U$. By Lemma 77.12.5

\[ H^ i(F^\bullet \otimes _ A A_ n) = \left\{ \begin{matrix} 0 & \text{if} & i \not= 0 \\ M_ n & \text{if} & i = 0 \end{matrix} \right. \]

Let $i$ be the maximal index such that $F^ i$ is nonzero. If $i \leq 0$, then (1), (2), and (3) are true. If not, then $i > 0$ and we see that the rank of the map

\[ F^{i - 1} \to F^ i \]

in the point $u$ is maximal. Hence in an open neighbourhood of $u$ inside $\mathop{\mathrm{Spec}}(P)$ the rank is maximal. Thus after replacing $P$ by a principal localization we may assume that the displayed map is surjective. Since $F^ i$ is finite free we may choose a splitting $F^{i - 1} = F' \oplus F^ i$. Then we may replace $F^\bullet $ by the complex

\[ \ldots \to F^{i - 2} \to F' \to 0 \to \ldots \]

and we win by induction on $i$. $\square$

Lemma 77.12.7. In Situation 77.12.1 let $K$ be as in Lemma 77.12.2. Let $W \subset X$ be as in Lemma 77.12.6. Set $\mathcal{F} = H^0(K)|_ W$. Then, after possibly shrinking the open $W$, the support of $\mathcal{F}$ is proper over $A$.

Proof. Fix $n \geq 1$. Let $I_ n = \mathop{\mathrm{Ker}}(A \to A_ n)$. By More on Algebra, Lemma 15.11.3 the pair $(A, I_ n)$ is henselian. Let $Z \subset W$ be the scheme theoretic support of $\mathcal{F}$. This is a closed subspace as $\mathcal{F}$ is of finite presentation. By part (3) of Lemma 77.12.6 we see that $Z \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A_ n)$ is equal to the support of $\mathcal{F}_ n$ and hence proper over $\mathop{\mathrm{Spec}}(A/I)$. By More on Morphisms of Spaces, Lemma 76.36.10 we can write $Z = Z_1 \amalg Z_2$ with $Z_1, Z_2$ open and closed in $Z$, with $Z_1$ proper over $A$, and with $Z_1 \times _{\mathop{\mathrm{Spec}}(A)} \mathop{\mathrm{Spec}}(A/I_ n)$ equal to the support of $\mathcal{F}_ n$. In other words, $|Z_2|$ does not meet $X_0$. Hence after replacing $W$ by $W \setminus Z_2$ we obtain the lemma. $\square$

Theorem 77.12.8 (Grothendieck Existence Theorem). In Situation 77.12.1 there exists a finitely presented $\mathcal{O}_ X$-module $\mathcal{F}$, flat over $A$, with support proper over $A$, such that $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ X} \mathcal{O}_{X_ n}$ for all $n$ compatibly with the maps $\varphi _ n$.

Proof. Apply Lemmas 77.12.2, 77.12.3, 77.12.4, 77.12.5, 77.12.6, and 77.12.7 to get an open subspace $W \subset X$ containing all points lying over $\mathop{\mathrm{Spec}}(A_ n)$ and a finitely presented $\mathcal{O}_ W$-module $\mathcal{F}$ whose support is proper over $A$ with $\mathcal{F}_ n = \mathcal{F} \otimes _{\mathcal{O}_ W} \mathcal{O}_{X_ n}$ for all $n \geq 1$. (This makes sense as $X_ n \subset W$.) By Lemma 77.3.6 we see that $\mathcal{F}$ is universally pure relative to $\mathop{\mathrm{Spec}}(A)$. By Theorem 77.11.7 (for explanation, see Lemma 77.11.8) there exists a universal flattening $S' \to \mathop{\mathrm{Spec}}(A)$ of $\mathcal{F}$ and moreover the morphism $S' \to \mathop{\mathrm{Spec}}(A)$ is a monomorphism of finite presentation. In particular $S'$ is a scheme (this follows from the proof of the theorem but it also follows a postoriori by Morphisms of Spaces, Proposition 67.50.2). Since the base change of $\mathcal{F}$ to $\mathop{\mathrm{Spec}}(A_ n)$ is $\mathcal{F}_ n$ we find that $\mathop{\mathrm{Spec}}(A_ n) \to \mathop{\mathrm{Spec}}(A)$ factors (uniquely) through $S'$ for each $n$. By More on Flatness, Lemma 38.28.8 we see that $S' = \mathop{\mathrm{Spec}}(A)$. This means that $\mathcal{F}$ is flat over $A$. Finally, since the scheme theoretic support $Z$ of $\mathcal{F}$ is proper over $\mathop{\mathrm{Spec}}(A)$, the morphism $Z \to X$ is closed. Hence the pushforward $(W \to X)_*\mathcal{F}$ is supported on $W$ and has all the desired properties. $\square$

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