The Stacks project

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 $(R^ pf_*\mathcal{F})_{\overline{y}} = 0$. $\square$

slogan

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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