The Stacks project

52.7 The theorem on formal functions

We interrupt the flow of the exposition to talk a little bit about derived completion in the setting of quasi-coherent modules on schemes and to use this to give a somewhat different proof of the theorem on formal functions. We give some pointers to the literature in Remark 52.7.4.

Lemma 52.6.19 is a (very formal) derived version of the theorem on formal functions (Cohomology of Schemes, Theorem 30.20.5). To make this more explicit, suppose $f : X \to S$ is a morphism of schemes, $\mathcal{I} \subset \mathcal{O}_ S$ is a quasi-coherent sheaf of ideals of finite type, and $\mathcal{F}$ is a quasi-coherent sheaf on $X$. Then the lemma says that

52.7.0.1
\begin{equation} \label{algebraization-equation-formal-functions} Rf_*(\mathcal{F}^\wedge ) = (Rf_*\mathcal{F})^\wedge \end{equation}

where $\mathcal{F}^\wedge $ is the derived completion of $\mathcal{F}$ with respect to $f^{-1}\mathcal{I} \cdot \mathcal{O}_ X$ and the right hand side is the derived completion of $Rf_*\mathcal{F}$ with respect to $\mathcal{I}$. To see that this gives back the theorem on formal functions we have to do a bit of work.

Lemma 52.7.1. Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $K$ be a pseudo-coherent object of $D(\mathcal{O}_ X)$ with derived completion $K^\wedge $. Then

\[ H^ p(U, K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ p(U, K)/I^ nH^ p(U, K) = H^ p(U, K)^\wedge \]

for any affine open $U \subset X$ where $I = \mathcal{I}(U)$ and where on the right we have the derived completion with respect to $I$.

Proof. Write $U = \mathop{\mathrm{Spec}}(A)$. The ring $A$ is Noetherian and hence $I \subset A$ is finitely generated. Then we have

\[ R\Gamma (U, K^\wedge ) = R\Gamma (U, K)^\wedge \]

by Remark 52.6.21. Now $R\Gamma (U, K)$ is a pseudo-coherent complex of $A$-modules (Derived Categories of Schemes, Lemma 36.9.2). By More on Algebra, Lemma 15.86.4 we conclude that the $p$th cohomology module of $R\Gamma (U, K^\wedge )$ is equal to the $I$-adic completion of $H^ p(U, K)$. This proves the first equality. The second (less important) equality follows immediately from a second application of the lemma just used. $\square$

Lemma 52.7.2. Let $X$ be a locally Noetherian scheme. Let $\mathcal{I} \subset \mathcal{O}_ X$ be a quasi-coherent sheaf of ideals. Let $K$ be an object of $D(\mathcal{O}_ X)$. Then

  1. the derived completion $K^\wedge $ is equal to $R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n)$.

Let $K$ is a pseudo-coherent object of $D(\mathcal{O}_ X)$. Then

  1. the cohomology sheaf $H^ q(K^\wedge )$ is equal to $\mathop{\mathrm{lim}}\nolimits H^ q(K)/\mathcal{I}^ nH^ q(K)$.

Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module1. Then

  1. the derived completion $\mathcal{F}^\wedge $ is equal to $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/\mathcal{I}^ n\mathcal{F}$,

  2. $\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F}$,

  3. $H^ p(U, \mathcal{F}^\wedge ) = 0$ for $p \not= 0$ for all affine opens $U \subset X$.

Proof. Proof of (1). There is a canonical map

\[ K \longrightarrow R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n), \]

see Remark 52.6.13. Derived completion commutes with passing to open subschemes (Remark 52.6.14). Formation of $R\mathop{\mathrm{lim}}\nolimits $ commutes with passsing to open subschemes. It follows that to check our map is an isomorphism, we may work locally. Thus we may assume $X = U = \mathop{\mathrm{Spec}}(A)$. Say $I = (f_1, \ldots , f_ r)$. Let $K_ n = K(A, f_1^ n, \ldots , f_ r^ n)$ be the Koszul complex. By More on Algebra, Lemma 15.86.1 we have seen that the pro-systems $\{ K_ n\} $ and $\{ A/I^ n\} $ of $D(A)$ are isomorphic. Using the equivalence $D(A) = D_{\mathit{QCoh}}(\mathcal{O}_ X)$ of Derived Categories of Schemes, Lemma 36.3.5 we see that the pro-systems $\{ K(\mathcal{O}_ X, f_1^ n, \ldots , f_ r^ n)\} $ and $\{ \mathcal{O}_ X/\mathcal{I}^ n\} $ are isomorphic in $D(\mathcal{O}_ X)$. This proves the second equality in

\[ K^\wedge = R\mathop{\mathrm{lim}}\nolimits \left( K \otimes _{\mathcal{O}_ X}^\mathbf {L} K(\mathcal{O}_ X, f_1^ n, \ldots , f_ r^ n) \right) = R\mathop{\mathrm{lim}}\nolimits (K \otimes _{\mathcal{O}_ X}^\mathbf {L} \mathcal{O}_ X/\mathcal{I}^ n) \]

The first equality is Lemma 52.6.9.

Assume $K$ is pseudo-coherent. For $U \subset X$ affine open we have $H^ q(U, K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ q(U, K)/\mathcal{I}^ n(U)H^ q(U, K)$ by Lemma 52.7.1. As this is true for every $U$ we see that $H^ q(K^\wedge ) = \mathop{\mathrm{lim}}\nolimits H^ q(K)/\mathcal{I}^ nH^ q(K)$ as sheaves. This proves (2).

Part (3) is a special case of (2). Parts (4) and (5) follow from Derived Categories of Schemes, Lemma 36.3.2. $\square$

Lemma 52.7.3. Let $A$ be a Noetherian ring and let $I \subset A$ be an ideal. Let $X$ be a Noetherian scheme over $A$. Let $\mathcal{F}$ be a coherent $\mathcal{O}_ X$-module. Assume that $H^ p(X, \mathcal{F})$ is a finite $A$-module for all $p$. Then there are short exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathcal{F})^\wedge \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

of $A$-modules where $H^ p(X, \mathcal{F})^\wedge $ is the usual $I$-adic completion. If $f$ is proper, then the $R^1\mathop{\mathrm{lim}}\nolimits $ term is zero.

Proof. Consider the two spectral sequences of Lemma 52.6.20. The first degenerates by More on Algebra, Lemma 15.86.4. We obtain $H^ p(X, \mathcal{F})^\wedge $ in degree $p$. This is where we use the assumption that $H^ p(X, \mathcal{F})$ is a finite $A$-module. The second degenerates because

\[ \mathcal{F}^\wedge = \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F} \]

is a sheaf by Lemma 52.7.2. We obtain $H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F})$ in degree $p$. Since $R\Gamma (X, -)$ commutes with derived limits (Injectives, Lemma 19.13.6) we also get

\[ R\Gamma (X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\Gamma (X, R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) = R\mathop{\mathrm{lim}}\nolimits R\Gamma (X, \mathcal{F}/I^ n\mathcal{F}) \]

By More on Algebra, Remark 15.80.6 we obtain exact sequences

\[ 0 \to R^1\mathop{\mathrm{lim}}\nolimits H^{p - 1}(X, \mathcal{F}/I^ n\mathcal{F}) \to H^ p(X, \mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n\mathcal{F}) \to \mathop{\mathrm{lim}}\nolimits H^ p(X, \mathcal{F}/I^ n\mathcal{F}) \to 0 \]

of $A$-modules. Combining the above we get the first statement of the lemma. The vanishing of the $R^1\mathop{\mathrm{lim}}\nolimits $ term follows from Cohomology of Schemes, Lemma 30.20.4. $\square$

Remark 52.7.4. Here are some references to discussions of related material the literature. It seems that a “derived formal functions theorem” for proper maps goes back to [Theorem 6.3.1, lurie-thesis]. There is the discussion in [dag12], especially Chapter 4 which discusses the affine story, see More on Algebra, Section 15.84. In [Section 2.9, G-R] one finds a discussion of proper base change and derived completion using (ind) coherent modules. An analogue of (52.7.0.1) for complexes of quasi-coherent modules can be found as [Theorem 6.5, HL-P]

[1] For example $H^ q(K)$ for $K$ pseudo-coherent on our locally Noetherian $X$.

Comments (2)

Comment #531 by Keenan Kidwell on

In the statement of Lemma 0A0L, the should be .


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