A remark to a division algorithm in the proof of Oka's first coherence theorem

Abstract

The problem is the locally finite generation of a relation sheaf ℛ(τ1,…,τq) in 𝒪Cn. After τj reduced to Weierstrass' polynomials in zn, it is the key for applying an induction on n to show that elements of ℛ(τ1,…,τq) are expressed as a finite linear sum of zn-polynomial-like elements of degree at most p = max j deg zn τj over 𝒪Cn. In that proof one is used to use a division by τj of the maximum degree, deg zn τj = p (Oka (1948); Cartan (1950); Hörmander (1966); Narasimhan (1966); Nishino (1996), etc.) Here we shall confirm that the division above works by making use of τk of the minimum degree, min j deg zn τj, and show that there is a degree structure in the locally finite generator system. This proof is naturally compatible with the simple case when some τj is a unit, and gives some improvement in the degree estimate of generators.