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38.28 Grothendieck's Existence Theorem, IV

This section continues the discussion in Cohomology of Schemes, Sections 30.24, 30.25, and 30.27. We will work in the following situation.

Situation 38.28.1. 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 a scheme $X$ 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 subscheme 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, Lemma 29.4.1).

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 38.28.2. In Situation 38.28.1 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 Schemes, Example 36.21.5. $\square$

Lemma 38.28.3. In Situation 38.28.1 let $K$ be as in Lemma 38.28.2. 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, Section 20.44. Hence by Derived Categories of Schemes, Lemma 36.30.7 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, Lemma 20.50.5. $\square$

Proof. Combinging Lemma 38.28.3 and Derived Categories of Schemes, Lemma 36.34.3 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, Lemma 37.69.4. $\square$

Lemma 38.28.5. In Situation 38.28.1 let $K$ be as in Lemma 38.28.2. For any quasi-compact open $U \subset X$ we have

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

in $D(A_ n)$ where $U_ n = U \cap X_ n$.

Proof. Fix $n$. By Derived Categories of Schemes, Lemma 36.33.4 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 38.28.3 and the fact that $- \otimes _ A^\mathbf {L} A_ n$ commutes with homotopy colimits we obtain the result. $\square$

Lemma 38.28.6. In Situation 38.28.1 let $K$ be as in Lemma 38.28.2. 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 $W \subset X$ 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 $U \subset X$ be an affine open neighbourhood of $x$. Say $U = \mathop{\mathrm{Spec}}(B)$. Choose a surjection $P \to B$ with $P$ smooth over $A$. By Lemma 38.28.4 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 38.28.5

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

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 $x$ is maximal. Hence in an open neighbourhood of $x$ 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 38.28.7. In Situation 38.28.1 let $K$ be as in Lemma 38.28.2. Let $W \subset X$ be as in Lemma 38.28.6. 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 support of $\mathcal{F}$. This is a closed subset as $\mathcal{F}$ is of finite presentation. By part (3) of Lemma 38.28.6 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, Lemma 37.53.9 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. $\square$

Lemma 38.28.8. Let $A = \mathop{\mathrm{lim}}\nolimits A_ n$ be a limit of a system of rings whose transition maps are surjective and with locally nilpotent kernels. Let $S = \mathop{\mathrm{Spec}}(A)$. Let $T \to S$ be a monomorphism which is locally of finite type. If $\mathop{\mathrm{Spec}}(A_ n) \to S$ factors through $T$ for all $n$, then $T = S$.

Proof. Set $S_ n = \mathop{\mathrm{Spec}}(A_ n)$. Let $T_0 \subset T$ be the common image of the factorizations $S_ n \to T$. Then $T_0$ is quasi-compact. Let $T' \subset T$ be a quasi-compact open containing $T_0$. Then $S_ n \to T$ factors through $T'$. If we can show that $T' = S$, then $T' = T = S$. Hence we may assume $T$ is quasi-compact.

Assume $T$ is quasi-compact. In this case $T \to S$ is separated and quasi-finite (Morphisms, Lemma 29.20.15). Using Zariski's Main Theorem (in the form of More on Morphisms, Lemma 37.43.3) we choose a factorization $T \to W \to S$ with $W \to S$ finite and $T \to W$ an open immersion. Write $W = \mathop{\mathrm{Spec}}(B)$. The (unique) factorizations $S_ n \to T$ may be viewed as morphisms into $W$ and we obtain

\[ A \longrightarrow B \longrightarrow \mathop{\mathrm{lim}}\nolimits A_ n = A \]

Consider the morphism $h : S = \mathop{\mathrm{Spec}}(A) \to \mathop{\mathrm{Spec}}(B) = W$ coming from the arrow on the right. Then

\[ T \times _{W, h} S \]

is an open subscheme of $S$ containing the image of $S_ n \to S$ for all $n$. To finish the proof it suffices to show that any open $U \subset S$ containing the image of $S_ n \to S$ for some $n \geq 1$ is equal to $S$. This is true because $(A, \mathop{\mathrm{Ker}}(A \to A_ n))$ is a henselian pair (More on Algebra, Lemma 15.11.3) and hence every closed point of $S$ is contained in the image of $S_ n \to S$. $\square$

Theorem 38.28.9 (Grothendieck Existence Theorem). In Situation 38.28.1 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 38.28.2, 38.28.3, 38.28.4, 38.28.5, 38.28.6, and 38.28.7 to get an open subscheme $W \subset X$ 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 38.17.1 we see that $\mathcal{F}$ is universally pure relative to $\mathop{\mathrm{Spec}}(A)$. By Theorem 38.27.5 (for explanation, see Lemma 38.27.6) 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. 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 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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