Lemma 30.20.1. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Set $B = \bigoplus _{n \geq 0} I^ n$. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. 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.
30.20 The theorem on formal functions
In this section we study the behaviour of cohomology of sequences of sheaves either of the form $\{ I^ n\mathcal{F}\} _{n \geq 0}$ or of the form $\{ \mathcal{F}/I^ n\mathcal{F}\} _{n \geq 0}$ as $n$ varies.
Here and below we use the following notation. Given a morphism of schemes $f : X \to Y$, a quasi-coherent sheaf $\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Y$ we denote $\mathcal{I}^ n\mathcal{F}$ the quasi-coherent subsheaf generated by products of local sections of $f^{-1}(\mathcal{I}^ n)$ and $\mathcal{F}$. In a formula
\[ \mathcal{I}^ n\mathcal{F} = \mathop{\mathrm{Im}}\left( f^*(\mathcal{I}^ n) \otimes _{\mathcal{O}_ X} \mathcal{F} \longrightarrow \mathcal{F} \right). \]
Note that there are natural maps
\[ f^{-1}(\mathcal{I}^ n) \otimes _{f^{-1}\mathcal{O}_ Y} \mathcal{I}^ m\mathcal{F} \longrightarrow f^*(\mathcal{I}^ n) \otimes _{\mathcal{O}_ X} \mathcal{I}^ m\mathcal{F} \longrightarrow \mathcal{I}^{n + m}\mathcal{F} \]
Hence a section of $\mathcal{I}^ n$ will give rise to a map $R^ pf_*(\mathcal{I}^ m\mathcal{F}) \to R^ pf_*(\mathcal{I}^{n + m}\mathcal{F})$ by functoriality of higher direct images. Localizing and then sheafifying we see that there are $\mathcal{O}_ Y$-module maps
\[ \mathcal{I}^ n \otimes _{\mathcal{O}_ Y} R^ pf_*(\mathcal{I}^ m\mathcal{F}) \longrightarrow R^ pf_*(\mathcal{I}^{n + m}\mathcal{F}). \]
In other words we see that $\bigoplus _{n \geq 0} R^ pf_*(\mathcal{I}^ n\mathcal{F})$ is a graded $\bigoplus _{n \geq 0} \mathcal{I}^ n$-module.
If $Y = \mathop{\mathrm{Spec}}(A)$ and $\mathcal{I} = \widetilde{I}$ we denote $\mathcal{I}^ n\mathcal{F}$ simply $I^ n\mathcal{F}$. The maps introduced above give $M = \bigoplus H^ p(X, I^ n\mathcal{F})$ the structure of a graded $S = \bigoplus I^ n$-module. If $f$ is proper, $A$ is Noetherian and $\mathcal{F}$ is coherent, then this turns out to be a module of finite type.
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 30.19.3 part (1). $\square$
Lemma 30.20.2. Given a morphism of schemes $f : X \to Y$, a quasi-coherent sheaf $\mathcal{F}$ on $X$, and a quasi-coherent sheaf of ideals $\mathcal{I} \subset \mathcal{O}_ Y$. Assume $Y$ locally Noetherian, $f$ proper, and $\mathcal{F}$ coherent. Then is a graded $\mathcal{A} = \bigoplus _{n \geq 0} \mathcal{I}^ n$-module which is quasi-coherent and of finite type.
\[ \mathcal{M} = \bigoplus \nolimits _{n \geq 0} R^ pf_*(\mathcal{I}^ n\mathcal{F}) \]
Proof. The statement is local on $Y$, hence this reduces to the case where $Y$ is affine. In the affine case the result follows from Lemma 30.20.1. Details omitted. $\square$
Lemma 30.20.3. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Then 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 $n \geq c$,
the image of $H^ p(X, I^{n + m}\mathcal{F}) \to H^ p(X, I^ n\mathcal{F})$ is contained in the submodule $I^{m - e} H^ p(X, I^ n\mathcal{F})$ where $e = \max (0, c - n)$ for $n + m \geq c$, $n, m \geq 0$,
we have
for $n \geq c$,
there are maps $I^ nH^ p(X, \mathcal{F}) \to H^ p(X, I^{n - c}\mathcal{F})$ for $n \geq c$ such that the compositions
and
for $n \geq 2c$ are the canonical ones, and
the inverse systems $(H^ p(X, I^ n\mathcal{F}))$ and $(I^ nH^ p(X, \mathcal{F}))$ are pro-isomorphic.
\[ \mathop{\mathrm{Ker}}(H^ p(X, I^ n\mathcal{F}) \to H^ p(X, \mathcal{F})) = \mathop{\mathrm{Ker}}(H^ p(X, I^ n\mathcal{F}) \to H^ p(X, I^{n - c}\mathcal{F})) \]
\[ H^ p(X, I^ n\mathcal{F}) \to I^{n - c}H^ p(X, \mathcal{F}) \to H^ p(X, I^{n - 2c}\mathcal{F}) \]
\[ I^ nH^ p(X, \mathcal{F}) \to H^ p(X, I^{n - c}\mathcal{F}) \to I^{n - 2c}H^ p(X, \mathcal{F}) \]
Proof. Write $M_ n = H^ p(X, I^ n\mathcal{F})$ for $n \geq 1$ and $M_0 = H^ p(X, \mathcal{F})$ so that we have maps $\ldots \to M_3 \to M_2 \to M_1 \to M_0$. Setting $B = \bigoplus _{n \geq 0} I^ n$, then $M = \bigoplus _{n \geq 0} M_ n$ is a finite graded $B$-module, see Lemma 30.20.1. Observe that the products $B_ n \otimes M_ m \to M_{m + n}$, $a \otimes m \mapsto a \cdot m$ are compatible with the maps in our inverse system in the sense that the diagrams
\[ \xymatrix{ B_ n \otimes _ A M_ m \ar[r] \ar[d] & M_{n + m} \ar[d] \\ B_ n \otimes _ A M_{m'} \ar[r] & M_{n + m'} } \]
commute for $n, m' \geq 0$ and $m \geq m'$.
Proof of (1). Choose $d_1, \ldots , d_ t \geq 0$ and $x_ i \in M_{d_ i}$ such that $M$ is generated by $x_1, \ldots , x_ t$ over $B$. For any $c \geq \max \{ d_ i\} $ we conclude that $B_{n - c} \cdot M_ c = M_ n$ for $n \geq c$ and we conclude (1) is true.
Proof of (2). Let $c$ be as in the proof of (1). Let $n + m \geq c$. We have $M_{n + m} = B_{n + m - c} \cdot M_ c$. If $c > n$ then we use $M_ c \to M_ n$ and the compatibility of products with transition maps pointed out above to conclude that the image of $M_{n + m} \to M_ n$ is contained in $I^{n + m - c}M_ n$. If $c \leq n$, then we write $M_{n + m} = B_ m \cdot B_{n - c} \cdot M_ c = B_ m \cdot M_ n$ to see that the image is contained in $I^ m M_ n$. This proves (2).
Let $K_ n \subset M_ n$ be the kernel of the map $M_ n \to M_0$. The compatibility of products with transition maps pointed out above shows that $K = \bigoplus K_ n \subset M$ is a graded $B$-submodule. As $B$ is Noetherian and $M$ is a finitely generated graded $B$-module, this shows that $K$ is a finitely generated graded $B$-module. Choose $d'_1, \ldots , d'_{t'} \geq 0$ and $y_ i \in K_{d'_ i}$ such that $K$ is generated by $y_1, \ldots , y_{t'}$ over $B$. Set $c = \max (d'_ i, d'_ j)$. Since $y_ i \in \mathop{\mathrm{Ker}}(M_{d'_ i} \to M_0)$ we see that $B_ n \cdot y_ i \subset \mathop{\mathrm{Ker}}(M_{n + d'_ i} \to M_ n)$. In this way we see that $K_ n = \mathop{\mathrm{Ker}}(M_ n \to M_{n - c})$ for $n \geq c$. This proves (3).
Consider the following commutative solid diagram
\[ \xymatrix{ I^ n \otimes _ A M_0 \ar[r] \ar[d] & I^ nM_0 \ar[r] \ar@{..>}[d] & M_0 \ar[d] \\ M_ n \ar[r] & M_{n - c} \ar[r] & M_0 } \]
Since the kernel of the surjective arrow $I^ n \otimes _ A M_0 \to I^ nM_0$ maps into $K_ n$ by the above we obtain the dotted arrow and the composition $I^ nM_0 \to M_{n - c} \to M_0$ is the canonical map. Then clearly the composition $I^ nM_0 \to M_{n - c} \to I^{n - 2c}M_0$ is the canonical map for $n \geq 2c$. Consider the composition $M_ n \to I^{n - c}M_0 \to M_{n - 2c}$. The first map sends an element of the form $a \cdot m$ with $a \in I^{n - c}$ and $m \in M_ c$ to $a m'$ where $m'$ is the image of $m$ in $M_0$. Then the second map sends this to $a \cdot m'$ in $M_{n - 2c}$ and we see (4) is true.
Part (5) is an immediate consequence of (4) and the definition of morphisms of pro-objects. $\square$
In the situation of Lemmas 30.20.1 and 30.20.3 consider the inverse system
\[ \mathcal{F}/I\mathcal{F} \leftarrow \mathcal{F}/I^2\mathcal{F} \leftarrow \mathcal{F}/I^3\mathcal{F} \leftarrow \ldots \]
We would like to know what happens to the cohomology groups. Here is a first result.
Lemma 30.20.4. Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. Fix $p \geq 0$. There exists a $c \geq 0$ such that
for all $n \geq c$ we have
the inverse system
satisfies the Mittag-Leffler condition (see Homology, Definition 12.31.2), and
we have
for all $k \geq n + c$.
\[ \mathop{\mathrm{Ker}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) \subset I^{n - c}H^ p(X, \mathcal{F}). \]
\[ \left(H^ p(X, \mathcal{F}/I^ n\mathcal{F})\right)_{n \in \mathbf{N}} \]
\[ \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}/I^ k\mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) = \mathop{\mathrm{Im}}(H^ p(X, \mathcal{F}) \to H^ p(X, \mathcal{F}/I^ n\mathcal{F})) \]
Proof. Let $c = \max \{ c_ p, c_{p + 1}\} $, where $c_ p, c_{p + 1}$ are the integers found in Lemma 30.20.3 for $H^ p$ and $H^{p + 1}$.
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 Lemma 30.20.3 part (2) we see that this is contained in $I^{n - c}H^ p(X, \mathcal{F})$ for $n \geq c$.
Note that part (3) implies part (2) by definition of the Mittag-Leffler systems.
Let us prove part (3). Fix an $n$. 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
\[ \xymatrix{ H^ p(X, \mathcal{F}) \ar[r] & H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \ar[r]_\delta & H^{p + 1}(X, I^ n\mathcal{F}) \ar[r] & H^{p + 1}(X, \mathcal{F}) \\ H^ p(X, \mathcal{F}) \ar[r] \ar[u]^1 & H^ p(X, \mathcal{F}/I^{n + m}\mathcal{F}) \ar[r] \ar[u]^\gamma & H^{p + 1}(X, I^{n + m}\mathcal{F}) \ar[u]^\alpha \ar[r]^-\beta & H^{p + 1}(X, \mathcal{F}) \ar[u]_1 } \]
with exact rows. By Lemma 30.20.3 part (4) the kernel of $\beta $ is equal to the kernel of $\alpha $ for $m \geq c$. By a diagram chase this shows that the image of $\gamma $ is contained in the kernel of $\delta $ which shows that part (3) is true (set $k = n + m$ to get it). $\square$
Theorem 30.20.5 (Theorem on formal functions). Let $A$ be a Noetherian ring. Let $I \subset A$ be an ideal. Let $f : X \to \mathop{\mathrm{Spec}}(A)$ be a proper morphism. Let $\mathcal{F}$ be a coherent sheaf on $X$. 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. This follows from Lemma 30.20.4 as follows. Set $M = H^ p(X, \mathcal{F})$, $M_ n = H^ p(X, \mathcal{F}/I^ n\mathcal{F})$, and denote $N_ n = \mathop{\mathrm{Im}}(M \to M_ n)$. By Lemma 30.20.4 parts (2) and (3) we see that $(M_ n)$ is a Mittag-Leffler system with $N_ n \subset M_ n$ equal to the image of $M_ k$ for all $k \gg n$. It follows that $\mathop{\mathrm{lim}}\nolimits M_ n = \mathop{\mathrm{lim}}\nolimits N_ n$ as topological modules (with limit topologies). On the other hand, the $N_ n$ form an inverse system of quotients of the module $M$ and hence $\mathop{\mathrm{lim}}\nolimits N_ n$ is the completion of $M$ with respect to the topology given by the kernels $K_ n = \mathop{\mathrm{Ker}}(M \to N_ n)$. By Lemma 30.20.4 part (1) we have $K_ n \subset I^{n - c}M$ and since $N_ n \subset M_ n$ is annihilated by $I^ n$ we have $I^ n M \subset K_ n$. Thus the topology defined using the submodules $K_ n$ as a fundamental system of open neighbourhoods of $0$ is the same as the $I$-adic topology and we find that the induced map $M^\wedge = \mathop{\mathrm{lim}}\nolimits M/I^ nM \to \mathop{\mathrm{lim}}\nolimits N_ n = \mathop{\mathrm{lim}}\nolimits M_ n$ is an isomorphism of topological modules1. $\square$
Lemma 30.20.6. 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. 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 30.20.5) in the case of a complete Noetherian base ring. Namely, in this case the $A$-module $H^ p(X, \mathcal{F})$ is finite (Lemma 30.19.2) hence $I$-adically complete (Algebra, Lemma 10.97.1) and we see that completion on the left hand side is not necessary. $\square$
Lemma 30.20.7. Given a morphism of schemes $f : X \to Y$ and a quasi-coherent sheaf $\mathcal{F}$ on $X$. Assume
$Y$ locally Noetherian,
$f$ proper, and
$\mathcal{F}$ coherent.
Let $y \in Y$ be a point. Consider the infinitesimal neighbourhoods
\[ \xymatrix{ X_ n = \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}/\mathfrak m_ y^ n) \times _ Y X \ar[r]_-{i_ n} \ar[d]_{f_ n} & X \ar[d]^ f \\ \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}/\mathfrak m_ y^ n) \ar[r]^-{c_ n} & Y } \]
of the fibre $X_1 = X_ y$ and set $\mathcal{F}_ n = i_ n^*\mathcal{F}$. Then we have
\[ \left(R^ pf_*\mathcal{F}\right)_ y^\wedge \cong \mathop{\mathrm{lim}}\nolimits _ n H^ p(X_ n, \mathcal{F}_ n) \]
as $\mathcal{O}_{Y, y}^\wedge $-modules.
Proof. This is just a reformulation of a special case of the theorem on formal functions, Theorem 30.20.5. Let us spell it out. Note that $\mathcal{O}_{Y, y}$ is a Noetherian local ring. Consider the canonical morphism $c : \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y}) \to Y$, see Schemes, Equation (26.13.1.1). This is a flat morphism as it identifies local rings. Denote momentarily $f' : X' \to \mathop{\mathrm{Spec}}(\mathcal{O}_{Y, y})$ the base change of $f$ to this local ring. We see that $c^*R^ pf_*\mathcal{F} = R^ pf'_*\mathcal{F}'$ by Lemma 30.5.2. Moreover, the infinitesimal neighbourhoods of the fibre $X_ y$ and $X'_ y$ are identified (verification omitted; hint: the morphisms $c_ n$ factor through $c$).
Hence we may assume that $Y = \mathop{\mathrm{Spec}}(A)$ is the spectrum of a Noetherian local ring $A$ with maximal ideal $\mathfrak m$ and that $y \in Y$ corresponds to the closed point (i.e., to $\mathfrak m$). In particular it follows that
\[ \left(R^ pf_*\mathcal{F}\right)_ y = \Gamma (Y, R^ pf_*\mathcal{F}) = H^ p(X, \mathcal{F}). \]
In this case also, 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 30.9.9 we see that
\[ H^ p(X_ n, \mathcal{F}_ n) = H^ p(X, i_{n, *}\mathcal{F}_ n) = H^ p(X, \mathcal{F}/\mathfrak m^ n\mathcal{F}) \]
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 30.20.8. Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume
$Y$ locally Noetherian,
$f$ is proper, and
$f^{-1}(\{ y\} )$ is finite.
Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^ pf_*\mathcal{F})_ y = 0$ for all $p > 0$.
Proof. The fibre $X_ y$ is finite, and by Morphisms, Lemma 29.20.7 it is a finite discrete space. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_ n$ is the same. Hence each of the schemes $X_ n$ is affine according to Schemes, Lemma 26.11.8. Hence it follows that $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > 0$. Hence we see that $(R^ pf_*\mathcal{F})_ y^\wedge = 0$ by Lemma 30.20.7. Note that $R^ pf_*\mathcal{F}$ is coherent by Proposition 30.19.1 and hence $R^ pf_*\mathcal{F}_ y$ is a finite $\mathcal{O}_{Y, y}$-module. By Nakayama's lemma (Algebra, Lemma 10.20.1) if the completion of a finite module over a local ring is zero, then the module is zero. Whence $(R^ pf_*\mathcal{F})_ y = 0$. $\square$
Lemma 30.20.9. Let $f : X \to Y$ be a morphism of schemes. Let $y \in Y$. Assume
$Y$ locally Noetherian,
$f$ is proper, and
$\dim (X_ y) = d$.
Then for any coherent sheaf $\mathcal{F}$ on $X$ we have $(R^ pf_*\mathcal{F})_ y = 0$ for all $p > d$.
Proof. The fibre $X_ y$ is of finite type over $\mathop{\mathrm{Spec}}(\kappa (y))$. Hence $X_ y$ is a Noetherian scheme by Morphisms, Lemma 29.15.6. Hence the underlying topological space of $X_ y$ is Noetherian, see Properties, Lemma 28.5.5. Moreover, the underlying topological space of each infinitesimal neighbourhood $X_ n$ is the same as that of $X_ y$. Hence $H^ p(X_ n, \mathcal{F}_ n) = 0$ for all $p > d$ by Cohomology, Proposition 20.20.7. Hence we see that $(R^ pf_*\mathcal{F})_ y^\wedge = 0$ by Lemma 30.20.7 for $p > d$. Note that $R^ pf_*\mathcal{F}$ is coherent by Proposition 30.19.1 and hence $R^ pf_*\mathcal{F}_ y$ is a finite $\mathcal{O}_{Y, y}$-module. By Nakayama's lemma (Algebra, Lemma 10.20.1) if the completion of a finite module over a local ring is zero, then the module is zero. Whence $(R^ pf_*\mathcal{F})_ y = 0$. $\square$
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