Situation 69.22.1. Here $A$ is a Noetherian ring and $I \subset A$ is an ideal. Also, $f : X \to \mathop{\mathrm{Spec}}(A)$ is a proper morphism of algebraic spaces and $\mathcal{F}$ is a coherent sheaf on $X$.
69.22 The theorem on formal functions
This section is the analogue of Cohomology of Schemes, Section 30.20. We encourage the reader to read that section first.
In this situation we denote $I^ n\mathcal{F}$ the quasi-coherent submodule of $\mathcal{F}$ generated as an $\mathcal{O}_ X$-module by products of local sections of $\mathcal{F}$ and elements of $I^ n$. In other words, it is the image of the map $f^*\widetilde{I} \otimes _{\mathcal{O}_ X} \mathcal{F} \to \mathcal{F}$.
Lemma 69.22.2. In Situation 69.22.1. Set $B = \bigoplus _{n \geq 0} I^ n$. Then for every $p \geq 0$ the graded $B$-module $\bigoplus _{n \geq 0} H^ p(X, I^ n\mathcal{F})$ is a finite $B$-module.
Proof. Let $\mathcal{B} = \bigoplus I^ n\mathcal{O}_ X = f^*\widetilde{B}$. Then $\bigoplus I^ n\mathcal{F}$ is a finite type graded $\mathcal{B}$-module. Hence the result follows from Lemma 69.20.4. $\square$
Lemma 69.22.3. In Situation 69.22.1. For every $p \geq 0$ there exists an integer $c \geq 0$ such that
the multiplication map $I^{n - c} \otimes H^ p(X, I^ c\mathcal{F}) \to H^ p(X, I^ n\mathcal{F})$ is surjective for all $n \geq c$, and
the image of $H^ p(X, I^{n + m}\mathcal{F}) \to H^ p(X, I^ n\mathcal{F})$ is contained in the submodule $I^{m - c} H^ p(X, I^ n\mathcal{F})$ for all $n \geq 0$, $m \geq c$.
Proof. By Lemma 69.22.2 we can find $d_1, \ldots , d_ t \geq 0$, and $x_ i \in H^ p(X, I^{d_ i}\mathcal{F})$ such that $\bigoplus _{n \geq 0} H^ p(X, I^ n\mathcal{F})$ is generated by $x_1, \ldots , x_ t$ over $B = \bigoplus _{n \geq 0} I^ n$. Take $c = \max \{ d_ i\} $. It is clear that (1) holds. For (2) let $b = \max (0, n - c)$. Consider the commutative diagram of $A$-modules
\[ \xymatrix{ I^{n + m - c - b} \otimes I^ b \otimes H^ p(X, I^ c\mathcal{F}) \ar[r] \ar[d] & I^{n + m - c} \otimes H^ p(X, I^ c\mathcal{F}) \ar[r] & H^ p(X, I^{n + m}\mathcal{F}) \ar[d] \\ I^{n + m - c - b} \otimes H^ p(X, I^ n\mathcal{F}) \ar[rr] & & H^ p(X, I^ n\mathcal{F}) } \]
By part (1) of the lemma the composition of the horizontal arrows is surjective if $n + m \geq c$. On the other hand, it is clear that $n + m - c - b \geq m - c$. Hence part (2). $\square$
Lemma 69.22.4. In Situation 69.22.1. Fix $p \geq 0$.
There exists a $c_1 \geq 0$ such that for all $n \geq c_1$ we have
The inverse system
satisfies the Mittag-Leffler condition (see Homology, Definition 12.31.2).
In fact for any $p$ and $n$ there exists a $c_2(n) \geq n$ such that
for all $k \geq c_2(n)$.
\[ \mathop{\mathrm{Ker}}( H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}) ) \subset I^{n - c_1}H^ p(X, \mathcal{F}). \]
\[ \left(H^ p(X, \mathcal{F}/I^ n\mathcal{F})\right)_{n \in \mathbf{N}} \]
\[ \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}/I^ k\mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) = \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) \]
Proof. Let $c_1 = \max \{ c_ p, c_{p + 1}\} $, where $c_ p, c_{p +1}$ are the integers found in Lemma 69.22.3 for $H^ p$ and $H^{p + 1}$. We will use this constant in the proofs of (1), (2) and (3).
Let us prove part (1). Consider the short exact sequence
\[ 0 \to I^ n\mathcal{F} \to \mathcal{F} \to \mathcal{F}/I^ n\mathcal{F} \to 0 \]
From the long exact cohomology sequence we see that
\[ \mathop{\mathrm{Ker}}( H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}) ) = \mathop{\mathrm{Im}}( H^ p(X, I^ n\mathcal{F}) \to H^ p(X, \mathcal{F}) ) \]
Hence by our choice of $c_1$ we see that this is contained in $I^{n - c_1}H^ p(X, \mathcal{F})$ for $n \geq c_1$.
Note that part (3) implies part (2) by definition of the Mittag-Leffler condition.
Let us prove part (3). Fix an $n$ throughout the rest of the proof. Consider the commutative diagram
\[ \xymatrix{ 0 \ar[r] & I^ n\mathcal{F} \ar[r] & \mathcal{F} \ar[r] & \mathcal{F}/I^ n\mathcal{F} \ar[r] & 0 \\ 0 \ar[r] & I^{n + m}\mathcal{F} \ar[r] \ar[u] & \mathcal{F} \ar[r] \ar[u] & \mathcal{F}/I^{n + m}\mathcal{F} \ar[r] \ar[u] & 0 } \]
This gives rise to the following commutative diagram
\[ \xymatrix{ H^ p(X, I^ n\mathcal{F}) \ar[r] & H^ p(X, \mathcal{F}) \ar[r] & H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^ n\mathcal{F}) \\ H^ p(X, I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^ p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u] & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^ a } \]
If $m \geq c_1$ we see that the image of $a$ is contained in $I^{m - c_1} H^{p + 1}(X, I^ n\mathcal{F})$. By the Artin-Rees lemma (see Algebra, Lemma 10.51.3) there exists an integer $c_3(n)$ such that
\[ I^ N H^{p + 1}(X, I^ n\mathcal{F}) \cap \mathop{\mathrm{Im}}(\delta ) \subset \delta \left(I^{N - c_3(n)}H^ p(X, \mathcal{F}/I^ n\mathcal{F})\right) \]
for all $N \geq c_3(n)$. As $H^ p(X, \mathcal{F}/I^ n\mathcal{F})$ is annihilated by $I^ n$, we see that if $m \geq c_3(n) + c_1 + n$, then
\[ \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) = \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) \]
In other words, part (3) holds with $c_2(n) = c_3(n) + c_1 + n$. $\square$
Theorem 69.22.5 (Theorem on formal functions). In Situation 69.22.1. Fix $p \geq 0$. The system of maps define an isomorphism of limits where the left hand side is the completion of the $A$-module $H^ p(X, \mathcal{F})$ with respect to the ideal $I$, see Algebra, Section 10.96. Moreover, this is in fact a homeomorphism for the limit topologies.
\[ H^ p(X, \mathcal{F})/I^ nH^ p(X, \mathcal{F}) \longrightarrow H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]
\[ H^ p(X, \mathcal{F})^\wedge \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]
Proof. In fact, this follows immediately from Lemma 69.22.4. We spell out the details. Set $M = H^ p(X, \mathcal{F})$ and $M_ n = H^ p(X, \mathcal{F}/I^ n\mathcal{F})$. Denote $N_ n = \mathop{\mathrm{Im}}(M \to M_ n)$. By the description of the limit in Homology, Section 12.31 we have
\[ \mathop{\mathrm{lim}}\nolimits _ n M_ n = \{ (x_ n) \in \prod M_ n \mid \varphi _ i(x_ n) = x_{n - 1}, \ n = 2, 3, \ldots \} \]
Pick an element $x = (x_ n) \in \mathop{\mathrm{lim}}\nolimits _ n M_ n$. By Lemma 69.22.4 part (3) we have $x_ n \in N_ n$ for all $n$ since by definition $x_ n$ is the image of some $x_{n + m} \in M_{n + m}$ for all $m$. By Lemma 69.22.4 part (1) we see that there exists a factorization
\[ M \to N_ n \to M/I^{n - c_1}M \]
of the reduction map. Denote $y_ n \in M/I^{n - c_1}M$ the image of $x_ n$ for $n \geq c_1$. Since for $n' \geq n$ the composition $M \to M_{n'} \to M_ n$ is the given map $M \to M_ n$ we see that $y_{n'}$ maps to $y_ n$ under the canonical map $M/I^{n' - c_1}M \to M/I^{n - c_1}M$. Hence $y = (y_{n + c_1})$ defines an element of $\mathop{\mathrm{lim}}\nolimits _ n M/I^ nM$. We omit the verification that $y$ maps to $x$ under the map
\[ M^\wedge = \mathop{\mathrm{lim}}\nolimits _ n M/I^ nM \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n M_ n \]
of the lemma. We also omit the verification on topologies. $\square$
Lemma 69.22.6. Let $A$ be a ring. Let $I \subset A$ be an ideal. Assume $A$ is Noetherian and complete with respect to $I$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism of algebraic spaces. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then for all $p \geq 0$.
\[ H^ p(X, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \]
Proof. This is a reformulation of the theorem on formal functions (Theorem 69.22.5) in the case of a complete Noetherian base ring. Namely, in this case the $A$-module $H^ p(X, \mathcal{F})$ is finite (Lemma 69.20.3) hence $I$-adically complete (Algebra, Lemma 10.97.1) and we see that completion on the left hand side is not necessary. $\square$
Lemma 69.22.7. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$ and let $\mathcal{F}$ be a quasi-coherent sheaf on $Y$. Assume
$Y$ locally Noetherian,
$f$ proper, and
$\mathcal{F}$ coherent.
Let $\overline{y}$ be a geometric point of $Y$. Consider the “infinitesimal neighbourhoods”
\[ \xymatrix{ X_ n = \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^ n) \times _ Y X \ar[r]_-{i_ n} \ar[d]_{f_ n} & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}/\mathfrak m_{\overline{y}}^ n) \ar[r]^-{c_ n} & Y } \]
of the fibre $X_1 = X_{\overline{y}}$ and set $\mathcal{F}_ n = i_ n^*\mathcal{F}$. Then we have
\[ \left(R^ pf_*\mathcal{F}\right)_{\overline{y}}^\wedge \cong \mathop{\mathrm{lim}}\nolimits _ n H^ p(X_ n, \mathcal{F}_ n) \]
as $\mathcal{O}_{Y, \overline{y}}^\wedge $-modules.
Proof. This is just a reformulation of a special case of the theorem on formal functions, Theorem 69.22.5. Let us spell it out. Note that $\mathcal{O}_{Y, \overline{y}}$ is a Noetherian local ring, see Properties of Spaces, Lemma 66.24.4. Consider the canonical morphism $c : \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}}) \to Y$. This is a flat morphism as it identifies local rings. Denote $f' : X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, \overline{y}})$ the base change of $f$ to this local ring. We see that $c^*R^ pf_*\mathcal{F} = R^ pf'_*\mathcal{F}'$ by Lemma 69.11.2. Moreover, we have canonical identifications $X_ n = X'_ n$ for all $n \geq 1$.
Hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a strictly henselian Noetherian local ring $A$ with maximal ideal $\mathfrak m$ and that $\overline{y} \to Y$ is equal to $\mathop{\mathrm{Spec}}(A/\mathfrak m) \to Y$. It follows that
\[ \left(R^ pf_*\mathcal{F}\right)_{\overline{y}} = \Gamma (Y, R^ pf_*\mathcal{F}) = H^ p(X, \mathcal{F}) \]
because $(Y, \overline{y})$ is an initial object in the category of étale neighbourhoods of $\overline{y}$. The morphisms $c_ n$ are each closed immersions. Hence their base changes $i_ n$ are closed immersions as well. Note that $i_{n, *}\mathcal{F}_ n = i_{n, *}i_ n^*\mathcal{F} = \mathcal{F}/\mathfrak m^ n\mathcal{F}$. By the Leray spectral sequence for $i_ n$, and Lemma 69.12.9 we see that
\[ H^ p(X_ n, \mathcal{F}_ n) = H^ p(X, i_{n, *}\mathcal{F}) = H^ p(X, \mathcal{F}/\mathfrak m^ n\mathcal{F}) \]
Hence we may indeed apply the theorem on formal functions to compute the limit in the statement of the lemma and we win. $\square$
Here is a lemma which we will generalize later to fibres of dimension $ > 0$, namely the next lemma.
Lemma 69.22.8. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume
$Y$ locally Noetherian,
$f$ is proper, and
$X_{\overline{y}}$ has discrete underlying topological space.
Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$ for all $p > 0$.
Proof. Let $\kappa (\overline{y})$ be the residue field of the local ring of $\mathcal{O}_{Y, \overline{y}}$. As in Lemma 69.22.7 we set $X_{\overline{y}} = X_1 = \mathop{\mathrm{Spec}}(\kappa (\overline{y})) \times _ Y X$. By Morphisms of Spaces, Lemma 67.34.8 the morphism $f : X \to Y$ is quasi-finite at each of the points of the fibre of $X \to Y$ over $\overline{y}$. It follows that $X_{\overline{y}} \to \overline{y}$ is separated and quasi-finite. Hence $X_{\overline{y}}$ is a scheme by Morphisms of Spaces, Proposition 67.50.2. Since it is quasi-compact its underlying topological space is a finite discrete space. Then it is an affine scheme by Schemes, Lemma 26.11.8. By Lemma 69.17.3 it follows that the algebraic spaces $X_ n$ are affine schemes as well. Moreover, the underlying topological of each $X_ n$ is the same as that of $X_1$. Hence it follows that $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > 0$. Hence we see that $(R^ pf_*\mathcal{F})_{\overline{y}}^\wedge = 0$ by Lemma 69.22.7. Note that $R^ pf_*\mathcal{F}$ is coherent by Lemma 69.20.2 and hence $R^ pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma 10.97.1 this implies that $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$. $\square$
Lemma 69.22.9. Let $S$ be a scheme. Let $f : X \to Y$ be a morphism of algebraic spaces over $S$. Let $\overline{y}$ be a geometric point of $Y$. Assume
$Y$ locally Noetherian,
$f$ is proper, and
$\dim (X_{\overline{y}}) = d$.
Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$ for all $p > d$.
Proof. Let $\kappa (\overline{y})$ be the residue field of the local ring of $\mathcal{O}_{Y, \overline{y}}$. As in Lemma 69.22.7 we set $X_{\overline{y}} = X_1 = \mathop{\mathrm{Spec}}(\kappa (\overline{y})) \times _ Y X$. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_ n$ is the same as that of $X_{\overline{y}}$. Hence $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > d$ by Lemma 69.10.1. Hence we see that $(R^ pf_*\mathcal{F})_{\overline{y}}^\wedge = 0$ by Lemma 69.22.7 for $p > d$. Note that $R^ pf_*\mathcal{F}$ is coherent by Lemma 69.20.2 and hence $R^ pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma 10.97.1 this implies that $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$. $\square$
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