# A Weak Coherence Theorem and Remarks to the Oka Theory

## TL;DR

This paper introduces a Weak Coherence Theorem for several complex variables that avoids traditional tools like Weierstrass' Preparation Theorem, simplifying proofs of key complex analysis problems using only power series expansions.

## Contribution

It formulates and proves a new Weak Coherence Theorem relying solely on power series, enabling simpler proofs of fundamental problems in complex analysis.

## Key findings

- Proves a Weak Coherence Theorem without Weierstrass' Preparation
- Provides elementary proofs of approximation and Cousin problems
- Simplifies the proof of Levi's problem in complex analysis

## Abstract

The proofs of K. Oka's Coherence Theorems are based on Weierstrass' Preparation (division) Theorem. Here we formulate and prove a Weak Coherence Theorem without using Weierstrass' Preparation Theorem, but only with power series expansions: The proof is almost of linear algebra. Nevertheless, this simple Weak Coherence Theorem suffices to give other proofs of the Approximation, Cousin I/II, and Levi's (Hartogs' Inverse) Problems even in simpler ways than those known, as far as the domains are non-singular; they constitute the main basic part of the theory of several complex variables. The new approach enables us to complete the proofs of those problems in quite an elementary way without Weierstrass' Preparation Theorem or the cohomology theory of Cartan--Serre, nor L2-dbar method of Hoermander.