The Stacks project

52.1 Introduction

This chapter continues the study of formal algebraic geometry and in particular the question of whether a formal object is the completion of an algebraic one. A fundamental reference is [SGA2]. Here is a list of results we have already discussed in the Stacks project:

  1. The theorem on formal functions, see Cohomology of Schemes, Section 30.20.

  2. Coherent formal modules, see Cohomology of Schemes, Section 30.23.

  3. Grothendieck's existence theorem, see Cohomology of Schemes, Sections 30.24, 30.25, and 30.27.

  4. Grothendieck's algebraization theorem, see Cohomology of Schemes, Section 30.28.

  5. Grothendieck's existence theorem more generally, see More on Flatness, Sections 38.28 and 38.29.

Let us give an overview of the contents of this chapter.

Let $X$ be a scheme and let $\mathcal{I} \subset \mathcal{O}_ X$ be a finite type quasi-coherent sheaf of ideals. Many questions in this chapter have to do with inverse systems $(\mathcal{F}_ n)$ of quasi-coherent $\mathcal{O}_ X$-modules such that $\mathcal{F}_ n = \mathcal{F}_{n + 1}/\mathcal{I}^ n\mathcal{F}_{n + 1}$. An important special case is where $X$ is a scheme over a Noetherian ring $A$ and $\mathcal{I} = I \mathcal{O}_ X$ for some ideal $I \subset A$. In Section 52.2 we prove some elementary results on such systems of coherent modules. In Section 52.3 we discuss additional results when $I = (f)$ is a principal. In Section 52.4 we work in the slightly more general setting where $\text{cd}(A, I) = 1$. One of the themes of this chapter will be to show that results proven in the case $I = (f)$ also hold true when we only assume $\text{cd}(A, I) = 1$.

In Section 52.6 we discuss derived completion of modules on a ringed site $(\mathcal{C}, \mathcal{O})$ with respect to a finite type sheaf of ideals $\mathcal{I}$. This section is the natural continuation of the theory of derived completion in commutative algebra as described in More on Algebra, Section 15.91. The first main result is that derived completion exists. The second main result is that for a morphism $f$ if ringed sites derived completion commutes with derived pushforward:

\[ (Rf_*K)^\wedge = Rf_*(K^\wedge ) \]

if the ideal sheaf upstairs is locally generated by sections coming from the ideal downstairs, see Lemma 52.6.19. We stress that both main results are very elementary in case the ideals in question are globally finitely generated which will be true for all applications of this theory in this chapter. The displayed equality is the “correct” version of the theorem on formal functions, see discussion in Section 52.7.

Let $A$ be a Noetherian ring and let $I, J$ be two ideals of $A$. Let $M$ be a finite $A$-module. The next topic in this chapter is the map

\[ R\Gamma _ J(M) \longrightarrow R\Gamma _ J(M)^\wedge \]

from local cohomology of $M$ into the derived $I$-adic completion of the same. It turns out that if we impose suitable depth conditions this map becomes an isomorphism on cohomology in a range of degrees. In Section 52.8 we work essentially in the generality just mentioned. In Section 52.9 we assume $A$ is a local ring and $J = \mathfrak m$ is a maximal ideal. We encourage the reader to read this section before the other two in this part of the chapter. Finally, in Section 52.10 we bootstrap the local case to obtain stronger results back in the general case.

In the next part of this chapter we use the results on completion of local cohomology to get a nonexhaustive list of results on cohomology of the completion of coherent modules. More precisely, let $A$ be a Noetherian ring, let $I \subset A$ be an ideal, and let $U \subset \mathop{\mathrm{Spec}}(A)$ be an open subscheme. If $\mathcal{F}$ is a coherent $\mathcal{O}_ U$-module, then we may consider the maps

\[ H^ i(U, \mathcal{F}) \longrightarrow \mathop{\mathrm{lim}}\nolimits H^ i(U, \mathcal{F}/I^ n\mathcal{F}) \]

and ask if we get an isomorphism in a certain range of degrees. In Section 52.11 we work out some examples where $U$ is the punctured spectrum of a local ring. In Section 52.12 we discuss the general case. In Section 52.14 we apply some of the results obtained to questions of connectedness in algebraic geometry.

The remaining sections of this chapter are devoted to a discussion of algebraization of coherent formal modules. In other words, given an inverse system of coherent modules $(\mathcal{F}_ n)$ on $U$ as above with $\mathcal{F}_ n = \mathcal{F}_{n + 1}/I^ n\mathcal{F}_{n + 1}$ we ask whether there exists a coherent $\mathcal{O}_ U$-module $\mathcal{F}$ such that $\mathcal{F}_ n = \mathcal{F}/I^ n\mathcal{F}$ for all $n$. We encourage the reader to read Section 52.16 for a precise statement of the question, a useful general result (Lemma 52.16.10), and a nontrivial application (Lemma 52.16.11). To prove a result going essentially beyond this case quite a bit more theory has to be developed. Please see Section 52.22 for the strongest results of this type obtained in this chapter.

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