# On formal complex spaces

*1978*

#### Translator’s note

*This page is a translation into English of the following:*

Bingener, J. “Über formale komplexe Räume.” *manuscripta mathematica* **24** (1978), 253–293. DOI: 10.1007/BF01167833.

*The translator (Tim Hosgood) takes full responsibility for any errors introduced, and claims no rights to any of the mathematical content herein.*

Version: `94f6dce`

Formal complex spaces, introduced by Krasnov [24] and independently by the author, are the analytic analogues of the formal schemes of Zariski and Grothendieck. Special cases are the formal completions of complex spaces along analytic sets, see Banica [3]. The technique of formal complex spaces has proved to be a useful tool in analytic geometry and allows even applications to purely algebraic problems, see [24], [4] and [7]. Here the basic theory of these spaces is developed: coherence of the structure sheave, description of the coherent modules, Grauert’s coherence theorem for proper maps…. We further study the question of exactness of the formal Dolbeault and de Rham complexes.

# Introduction

In 1958, Grothendieck introduced formal schemes in algebraic geometry, following on from earlier ideas by Zariski. Since then, the theory of formal schemes has become an important tool in algebraic geometry, cf. e.g. [2,13,20]. Formal structures appeared in (global) analytic geometry for the first time in Grauert’s comparison theorem; this is clearly expressed in the proof given in [3]. In [24], Krasnov then explicitly introduced formal complex spaces, and, in particular, formal complex manifolds, and used this to prove theorems about modifications of complex manifolds.

In the present article, we first develop the basic theory of formal complex spaces.
These are introduced in §1 as inductive limits of a suitable system of complex spaces.
We obtain special formal complex spaces if we consider the formal completions of complex spaces along analytical subsets.
Of course, every complex space is also a formal complex space.
The structure sheaf **!!TO-DO: intersection rings?!!** of compact Stein subsets of **!!TO-DO: number spaces?!!**, cf. (1.7).

Formal complex manifolds, i.e. formal complex spaces whose stalks are all regular, are studied in §2.
Every point of a formal manifold has an open neighbourhood that is isomorphic to the formal completion of an open subspace of

Let

The fact that

An analogous statement can be made for the formal de Rham complex. In fact, the following more general statement (cf. (2.11)) holds:

Let

In the special case where

In §3, proper maps between formal complex spaces are considered. For such maps, Grauert’s coherence law (cf. (3.1)) applies. In §4 we show how the most important statements of the relative comparison theory between algebraic and analytic geometry [5,15] can be transferred to the case where the base is a formal complex space.

In §5 we use the results of §4 to study formal meromorphic functions. We mention here the following statement (cf. (5.2)):

Let

In addition, we determine the ring of meromorphic functions on the product of a normal formal complex space

Some results from the present article have already been used in [4] and [7].

# 1 Formal complex spaces

Let

A locally ringed space *formal complex space* if the following conditions are satisfied:

X_n\coloneqq (X,{\mathcal{O}}_X/{\mathscr{I}}_X^{n+1}) is a complex space for alln\in\mathbb{N} ; and- the canonical homomorphism
{\mathcal{O}}_X\to\varinjlim_n{\mathcal{O}}_X/{\mathscr{I}}_X^{n+1} is bijective.

Formal complex spaces form a category, with

As usual, we say that a subset of a complex space is Stein compact if it is a compact, semi-analytic subset that has an neighbourhood system of open Stein sets.
This definition can be extended in a trivial way to formal complex spaces: A subset *Stein compact* if

The following lemma is fundamental for what follows.

Let

B_K\coloneqq\varprojlim_n\Gamma(K,{\mathcal{O}}_{X_n}) is a Noetherian ring, which is further separated and complete with respect to the topology defined by the ideal{\mathfrak{b}}_K\coloneqq\operatorname{Ker}(B_K\to\Gamma(K,{\mathcal{O}}_{X_0})) .- If
L\subseteq X is another Stein compact subset such thatK\subseteq L , then the canonical homomorphismB_L\to B_K is flat. {\mathcal{O}}_X is a coherent sheaf of rings.

*Proof*. We write

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