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
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
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
by Remark 52.6.21. Now R\Gamma (U, K) is a pseudo-coherent complex of A-modules (Derived Categories of Schemes, Lemma 36.10.2). By More on Algebra, Lemma 15.94.4 we conclude that the pth 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
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
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
the derived completion \mathcal{F}^\wedge is equal to \mathop{\mathrm{lim}}\nolimits \mathcal{F}/\mathcal{I}^ n\mathcal{F},
\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F} = R\mathop{\mathrm{lim}}\nolimits \mathcal{F}/I^ n \mathcal{F},
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
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 passing 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 15.94.1 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
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
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 15.94.4. 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
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
By More on Algebra, Remark 15.87.6 we obtain exact sequences
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 15.91. 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]
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