## 66.21 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.

Situation 66.21.1. 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 $X$.

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 66.21.2. In Situation 66.21.1. 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 66.20.4. $\square$

Lemma 66.21.3. In Situation 66.21.1. For every $p \geq 0$ there exists an integer $c \geq 0$ such that

1. 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

2. 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 66.21.2 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 66.21.4. In Situation 66.21.1. Fix $p \geq 0$.

1. There exists a $c_1 \geq 0$ such that for all $n \geq c_1$ we have

$\mathop{\mathrm{Ker}}( H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}) ) \subset I^{n - c_1}H^ p(X, \mathcal{F}).$
2. The inverse system

$\left(H^ p(X, \mathcal{F}/I^ n\mathcal{F})\right)_{n \in \mathbf{N}}$

satisfies the Mittag-Leffler condition (see Homology, Definition 12.31.2).

3. In fact for any $p$ and $n$ there exists a $c_2(n) \geq n$ such that

$\mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}/I^ k\mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) = \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F}))$

for all $k \geq c_2(n)$.

Proof. Let $c_1 = \max \{ c_ p, c_{p + 1}\}$, where $c_ p, c_{p +1}$ are the integers found in Lemma 66.21.3 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 10.50.3) 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 66.21.5 (Theorem on formal functions). In Situation 66.21.1. Fix $p \geq 0$. The system of maps

$H^ p(X, \mathcal{F})/I^ nH^ p(X, \mathcal{F}) \longrightarrow H^ p(X, \mathcal{F}/I^ n\mathcal{F})$

define an isomorphism of limits

$H^ p(X, \mathcal{F})^\wedge \longrightarrow \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F})$

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 10.95. Moreover, this is in fact a homeomorphism for the limit topologies.

Proof. In fact, this follows immediately from Lemma 66.21.4. 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 66.21.4 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 66.21.4 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 66.21.6. 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

$H^ p(X, \mathcal{F}) = \mathop{\mathrm{lim}}\nolimits _ n H^ p(X, \mathcal{F}/I^ n\mathcal{F})$

for all $p \geq 0$.

Proof. This is a reformulation of the theorem on formal functions (Theorem 66.21.5) in the case of a complete Noetherian base ring. Namely, in this case the $A$-module $H^ p(X, \mathcal{F})$ is finite (Lemma 66.20.3) hence $I$-adically complete (Algebra, Lemma 10.96.1) and we see that completion on the left hand side is not necessary. $\square$

Lemma 66.21.7. 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

1. $Y$ locally Noetherian,

2. $f$ proper, and

3. $\mathcal{F}$ 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 66.21.5. Let us spell it out. Note that $\mathcal{O}_{Y, \overline{y}}$ is a Noetherian local ring, see Properties of Spaces, Lemma 63.24.4. 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 66.11.2. 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 66.12.9 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 66.21.8. 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

1. $Y$ locally Noetherian,

2. $f$ is proper, and

3. $X_{\overline{y}}$ 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 66.21.7 we set $X_{\overline{y}} = X_1 = \mathop{\mathrm{Spec}}(\kappa (\overline{y})) \times _ Y X$. By Morphisms of Spaces, Lemma 64.34.8 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 64.50.2. 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 66.17.1 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 66.21.7. Note that $R^ pf_*\mathcal{F}$ is coherent by Lemma 66.20.2 and hence $R^ pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma 10.96.1 this implies that $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$. $\square$

Lemma 66.21.9. 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

1. $Y$ locally Noetherian,

2. $f$ is proper, and

3. $\dim (X_{\overline{y}}) = d$.

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 66.21.7 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 66.10.1. Hence we see that $(R^ pf_*\mathcal{F})_{\overline{y}}^\wedge = 0$ by Lemma 66.21.7 for $p > d$. Note that $R^ pf_*\mathcal{F}$ is coherent by Lemma 66.20.2 and hence $R^ pf_*\mathcal{F}_{\overline{y}}$ is a finite $\mathcal{O}_{Y, \overline{y}}$-module. By Algebra, Lemma 10.96.1 this implies that $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$. $\square$

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