A Remark to a Division Algorithm in the Proof of Oka s First Coherence Theorem

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1 A Remark to a Division Algorithm in the Proof of Oka s First Coherence Theorem Junjiro Noguchi 1 The University of Tokyo/Tokyo Institute of Technology, Emeritus Abstract The problem is the locally finite generation of a relation sheaf R(τ 1,..., τ q ) in O C n. After τ j reduced to Weierstrass polynomials in z n, it is the key for applying an induction on n to show that elements of R(τ 1,..., τ q ) are expressed as a finite linear sum of z n -polynomiallike elements of degree at most p = max j τ j over O C n. In that proof one is used to use a division by τ j of the maximum degree, τ j = p (Oka 48, Cartan 50, L. Hörmander 66, R. Narasimhan 66, T. Nishino 96,...). Here we shall confirm that the division above works by making use of τ k of the minimum degree, min j τ j. This proof is naturally compatible with the simple case when some τ j is a unit, and gives some improvement in the degree estimate of generators. 1 Introduction and results It will be of no necessity to mention the importance of Oka s First Coherence Theorem that the sheaf O C n (also denoted simply by O n ) of germs of holomorphic functions over n-dimensional complex vector space C n (Oka [7], [8]) 2. Let Ω C n be an open set and let τ j O(Ω) := Γ(Ω, O n ), 1 j q. Oka s First Coherence Theorem claims that the relation sheaf R(τ 1,..., τ q ) defined by f 1 τ 1z + + f q τ qz = 0, f j O n,z, z Ω. is locally finite in Ω, where z stands for the germ at z. The problem is local, so that we consider in a neighborhood of a point a Ω; further we may assume a = 0 with complex coordinate system (z 1,..., z n ). 1 Research supported in part by Grant-in-Aid for Scientific Research (B) There are some differences in these two versions of Oka VII.

2 By Weierstrass Preparation Theorem τ j are reduced to Weierstrass polynomials P j O(P n 1 )[z n ] about 0, where P n 1 is a small polydisk in z = (z 1,..., z n 1 ) C n 1. Set R = R(P 1,..., P q ), p = max 1 j q deg z n P j, p = min 1 j q deg z n P j. We call f O n 1,b [z n ] (resp. f O(P n 1 )[z n ]) a z n -polynomial-like germ (resp. function) and denote by f its degree in variable z n ; for convention, f < 0 means f = 0. We also call an element (f j ) (O n,(b,b n)) q (resp. (f j ) (O(P n 1 C))) q ) with f j O P n 1,b [z n] (resp. f j O(P n 1 )[z n ]) a z n -polynomial-like element (resp. section), and (f j ) = max j f j the degree of (f j ). The proof of the local finiteness of R relies on the induction on n, and the key which makes the induction to work is: Lemma A. Every element of R b at b = (b, b n ) with b P n 1 is expressed as a finite linear sum of z n -polynomial-like elements of R b of degree at most p with coefficients in O b. There is some structure in the generator system with respect to the degree in z n. For 1 i < j q there are sections of R given by T i,j = (0,..., 0, i-th P j, 0,..., 0, j-th P i, 0,..., 0), which we call the trivial solutions, and are z n -polynomial-like sections of T i,j p. Without loss of generality we may assume that p 1 = p, p q = p, and set T j = T 1,j, 2 j q. 2

3 In the proof of Lemma A a division algorithm is applied; in the original proof of Oka as well as in many references such as H. Cartan [1], R. Narasimhan [4], L. Hörmander [3], T. Nishino [5], J. Noguchi [6],... etc., the division algorithm by P q of the maximum degree is used to conclude the existence of a finite generator system consisting of T i,q of degree p, 1 i q 1, and a finite number of z n -polynomial-like elements α of degree < p. In case p = 0, it is immediate that the trivial solutions T j with 2 j q form already a generator system, while by the original proof one still needs elements α of degree < p. The aim of this note is to confirm that Oka s original proof still works with the division algorithm by P 1 of the minimum degree in z n : Lemma 1.1. Let the notation be as above. Then an element of R b is written as a finite linear sum of the trivial solutions, T j, 2 j q, and z n - polynomial-like elements α = (α 1, α 2,..., α q ) of R b with coefficients in O n,b such that (1.2) α 1 p 1, α j p 1, 2 j q. N.B. If p = 0, then there is no term of α, and if p = 1. α j are constants for 2 j q. To decrease p 1 in (1.2) one needs to transform the relation sheaf R(P 1, P 2,..., P q ) with dividing P j (2 j q) by P 1 (here we use an idea from Hironaka s proof, cf. [2]). Set P j = Q j P 1 + R j, Q j, R j O n 1 (P n 1 )[z n ], R j p 1, 2 j q. 3

4 Then for (f j ) (O n,z ) q we have ( (1.3) f j P jz = f 1 + j=1 = h 1 P 1z + ) f j Q jz P 1z + j=2 f j R jz, j=2 f j R jz where h 1 = f 1 + q j=2 f jq jz. Thus the locally finite generation of R(P 1,..., P q ) is equivalent to that of R(P 1, R 2,..., R q ). Let j=2 T j = (R j, 0,..., 0, j-th P 1, 0,..., 0), 2 j q be the trivial solutions of R(P 1, R 2,..., R q ), which are z n -polynomial-like sections of T j = p. Lemma 1.4. Set R := R(P 1, R 2,..., R q ) be as above. Then an element of R b is written as a finite linear sum of the trivial solutions, T j, 2 j q, of degree p and z n -polynomial-like elements α = (α 1, α 2,..., α q ) of R b with coefficients in O n,b such that (1.5) α 1 p 2, α j p 1, 2 j q. N.B. If p = 0, then there is no term of α, and if p = 1. then α 1 = 0 and α j are constants for 2 j q. 2 Proofs of Lemmas (1)(Lemma 1.1) By making use of Weierstrass Preparation Theorem at b = (b, b n ) with b P n 1 we decompose P 1 to a unit u and a Weierstrass polynomial Q: P 1 (z, z n ) = u Q(z, z n b n ), Q = d p 1. 4

5 Here and in the sequel we abbreviate Q z to Q for the sake of notational simplicity; there will be no confusion. It follows that u O n 1,b [z n ], and then (2.1) u = p 1 d. Take an arbitrary f = (f 1,..., f q ) R b. By Weierstrass Preparation Theorem we divide f i by Q: (2.2) f i =c i Q + β i, 1 i q, c i O n,b, β i O n 1,b [z n ], β i < d. Since u O n,b is a unit, with c i := c i u 1 we get the division of f i by P 1 : (2.3) f i = c i P 1 + β i, 1 i q. By making use of this we have (2.4) (f 1,..., f q ) + c 2 T c q T q = ( c 1 P 1 + β 1, c 2 P 1 + β 2,..., c q P 1 + β q ) + ( c 2 P 2, c 2 P 1, 0,..., 0) + + ( c q P q, 0,..., 0, c q P 1 ) ( = i=1 c i P i + β 1, β 2,..., β q ) = (g 1, β 2,..., β q ). Here we put g 1 = q i=1 c ip i +β 1 O n,b. Note that β i O n 1,b [z n ], 2 i q. Since (g 1, β 2,..., β q ) R b, (2.5) g 1 P 1 = β 2 P 2 β q P q O n 1,b [z n ]. 5

6 It should be noticed that if p 1 = 0, then P 1 = 1, β i = 0, 1 i q, and hence g 1 = 0; the proof is finished in this case. In general, it follows from the expression of the above right-hand side of (2.5) that g 1 P 1 O n 1,b [z n ] and g 1 P 1 max 2 i q deg z n β i + max 2 i q deg z n P i d + p 1. On the other hand, g 1 P 1 = g 1 uq and Q is a Weierstrass polynomial at b. We see that (2.6) α 1 := g 1 u O n 1,b [z n ], α 1 = g 1 P 1 Q d + p 1 d = p 1. Set α i = uβ i for 2 i q. Then, by (2.1) and (2.2) we have (2.7) α i p 1 d + d 1 = p 1 1 = p 1, 2 i q, and by (2.9) that (2.8) f = c i T i + u 1 (α 1, α 2,..., α q ). i=2 (2)(Lemma 1.4) First note that (f 1,..., f q ) and (h 1, f 2,..., f q ) with h 1 = f 1 + q j=2 f jq j as defined in (1.3) are related by h 1 1 Q 2 Q q f 1 f 2. = f , f q f q f 1 1 Q 2 Q q h 1 f 2. = f f q f q 6

7 Therefore, the locally finite generation of R is equivalent to that of R. The proof is similar to the above except for some degree estimates. Now we have for (f j ) (O n,b ) q (2.9) (f 1,..., f q ) + c 2 T c q T q ( ) = c 1 P 1 + β 1 + c i R i, β 2,..., β q = (h 1, β 2,..., β q ). i=2 Here we put h 1 = c 1 P 1 + β 1 + q i=2 c ir i O n,b. In stead of (2.5) we have (2.10) h 1 P 1 = β 2 R 2 β q R q O n 1,b [z n ]. From this we obtain h 1 P 1 d 1 + p 1 = d + p 2. With α 1 := h 1 u we have h 1 P 1 = h 1 uq = α 1Q and so α 1 d + p 2 d = p 2. For α i = uβ i, 2 i q we have the same estimate: α i p 1. With the above defined we have f = c i T i + u 1 (α 1, α 2,..., α q ). i=2 References [1] H. Cartan, Idéaux et modules de fonctions analytiques de variables complexes Bull. Soc. Math. France 78 (1950),

8 [2] H. Hironaka and T. Urabe, Introduction to Analytic Spaces (in Japanese), Asakura Shoten, Tokyo, [3] L. Hörmander, Introduction to Complex Analysis in Several Variables, First Edition 1966, Third Edition, North-Holland, [4] R. Narasimhan, Introduction to the Theory of Analytic Spaces, Lecture Notes in Math. 25, Springer-Verlag, [5] T. Nishino, Function Theory in Several Complex Variables (in Japanese), The University of Tokyo Press, Tokyo, 1996; English translation by N. Levenberg and H. Yamaguchi, Amer. Math. Soc. Providence, R.I., [6] J. Noguchi, Analytic Function Theory of Several Variables (in Japanese), Asakura Shoten, Tokyo, [7] K. Oka, Sur les fonctions analytiques de plusieurs variables: VII Sur quelques notions arithmétiques, Iwanami Shoten, Tokyo, [8] K. Oka, Sur les fonctions analytiques de plusieurs variables: VII Sur quelques notions arithmétiques, Bull. Soc. Math. France 78 (1950), Graduate School of Mathematical Sciences The University of Tokyo Komaba, Meguro-ku,Tokyo