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

摘要

The problem is the locally finite generation of a relation sheaf R(tau(1),..., tau(q)) in O-Cn. After tau(j) reduced toWeierstrass' polynomials in z(n), it is the key for applying an induction on n to show that elements of R(tau(1),..., tau(q)) are expressed as a finite linear sum of z(n)-polynomial- like elements of degree at most p = max(j) deg(zn) tau(j) over O-Cn. In that proof one is used to use a division by tau(j) of the maximum degree, deg(zn) tau(j) = p (Oka (1948); Cartan (1950); Hormander (1966); Narasimhan (1966); Nishino (1996), etc.) Here we shall confirm that the division above works by making use of tau(k) of the minimum degree, minj deg(zn) tau(j), 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 tau(j) is a unit, and gives some improvement in the degree estimate of generators.