de Bartolomeis, Paolo

Tomassini, Giuseppe

Elsevier - Publisher Connector 

ADVANCES IN MATHEMATICS 46, 162-170 (1982) 
Finitely Generated Ideals in A”(D) 
PAOLO DE BARTOLOMEIS 
Istituto di Matematica Applicata, Universitti di Firenre, Florence, Italy 
AND 
GIUSEPPE TOMASSINI 
Department of Mathematics, Scuola Normale Superiore Piss, Pisa, Ita1.v 
Let DE C” be a bounded pseudoconvex domain with P-smooth 
boundary bD and let V be an analytic subvariety of a neighborhood of 0. 
Let KV denote the sheaf of ideals of V. From the general theory of Oka- 
Cartan-Serre it follows that if f, ,...,fk E r(D, &) generate KVy.; at every 
point z E D, then they generate T(D, &) over 4(D) (=T(D, @)). 
Let A be a subalgebra (or more generally a vector subspace) of P’(D) 
containing f, ,..., fk. It is a natural question to ask whether f, ,...,fJ generate 
T(D, &) nA over A. There are several results in this direction in the special 
case V is a point and A is the algebra of holomorphic functions satisfying 
some regularity condition at the boundary [3, 6, 7, 10, 141. 
The situation becomes much more complicated when V has positive 
dimension, as the following elementary example shows: the holomorphic 
function f (2,) z2) = z2( 1 - z,)- 1’4 is continuous up to the boundary on the 
unit ball B in Cz but its factor in z2 is not even bounded (and it can be easily 
proved that the ideal of the complex line z2 = 0 is not finitely generated over 
A O(B), the algebra e(B) n C’(g)). 
The aim of the present paper is to prove that similar phenomena disappear 
in a high regularity situation, i.e., for the algebra A”(D) = e(D) n Cm(~). 
Namely, we prove that if D is strictly pseudoconvex, bD and V are 
“regularly separated” (see Section 1) and V is smooth near bD then f, ,..., fk 
generate P( I’) = I’(D, &) n P(D) over A “O(D). Some of these results were 
announced in [5]. 
162 
0001-8708/82/l 10162-O9$05.OO/O 
Copyright ‘i;, 1982 by Academic Press, Inc. 
All rights of reproduction in any form reserved. 
IDEALS IN Am(D) 163 
1. THE LOCAL AND SEMI-LOCAL CASES 
1. From now on D will denote a bounded domain in C” with C-smooth 
boundary bD, D’ a neighborhood of D and V’ c D’ a closed complex 
analytic subset, such that Sing V’ n bD = 0. 
Let &, be the ideal of germs of holomorphic functions vanishing on V’. 
A complete defining system for V’ is a finite set of holomorphic functions 
on D', f,,...& such that for every z E D the germs f,,, ,..., fk.: generate g,.,.;. 
It is well known that such a system always exists if D’ is a Stein 
neighborhood of 0. 
Let V = V’ n D # 0. For every relative open subset U of D we set 
Pp’(U) = (j-E F(l$j-E Cm@J (bD n Or))}, 
Pp’(Un v) = (j-E F(lh V)ZJ-=$~~~~,~E C”(UU (bD c-7 U))}; 
UHP~‘(U) and Un VwPp)(Un v> define two sheaves on D and v 
respectively, Pbai”’ and ?k?) such that 
Pgy, = FD and Pgy,. = PI,. 
Let E-p) c Pkm’ be the ideal of germs vanishing on I? In particular we 
have 
A m(D) = Z-@, Pp’), Am(V) = Z-(v, Pbm’), I”(V) = T(V, Fp’). 
We say that V’ and bD are regularly separated at p E bD n V’ if there 
exist NE N, c > 0 and a neighborhood U of p such that 
dist(z, UfT bD n V’).” < C dist(z, Un V’) (*I 
for every z E U n bD. 
We say V’ and bD are regularly separated if in the previous definition U 
can be chosen in such a way as to be a neighborhood of bD. 
Note that if bD is real analytic (*) is the well known tojasiewicz’s 
inequality [ 121. It is worth noting also that (*) does not follow from strict 
pseudoconvexity of D. 
Suppose now V’ is smooth near bD and let d be its complex dimension. 
We say that V’ is transversal to bD at z E bD if V’ n bD is smooth of real 
dimension 2d - 1 and 
Z-F< V’ n bD) = T;( V’) n T;(bD) 
(where TF(.) is the complex tangent space at z). 
It is a very simple matter to prove that if V’ and bD are transversal at z 
then they are regularly separated at z. 
164 DE BARTOLOMEIS ANDTOMASSINI 
2. We shall consider first the case where V’ is a linear subspace: our first 
statement is the following: 
PROPOSITION 1. Let D be pseudoconvex and let V’ = {zk+ , = es* = 
z, = 0). Assume V = V’ f~ D # 0 and V’ and bD are regularly separated. 
Then for every f E A*(D) such that f Ic. = 0 there exist &+, ,..., ,I,, E A”O(D) 
such that 
f = s AjZj. 
j=k+ 1 
We prove first the following result in codimension 1: 
LEMMA 2. Let D c C” be a bounded domain with ?-smooth boundary 
and let V’ be the hyperplane z, = 0. Assume V = V’ n D # 0 and V’ and 
bD are regularly separated. Then: 
(i) every f E A*(D) such that f Iy = 0 can be written as f = gz,, , 
where g E A O”(D), 
(ii) if D is pseudoconvex the restriction homomorphism A”O(D) + 
A m(V) is onto. 
Proof We shall divide the proof into several steps. 
(a) If f E P(o) satisfies f IV = 0 then f =gz, + hYn, where 
g, h E P(D). 
Using the fact that V’ and bD are regularly separated, via Whitney’s 
extension theorem we can find a ball B containing D and FE C”O@) such 
that FJ Bny, = 0, F,-=f Fix z E B and set q(t) = F(z, ,..., z,-, , tz,). We have 
F(Z ~,-.,z~)=q(l)= ‘*dt=z I o dt n I 
‘~(zl,...,z,&z,)dt 
0 az, 
i 
’ aF 
+ z;, -(z, ,..., znplr tz,) dt. 
0 az;, 
(b) Iff E C”(B) is ii-flat on V (i.e., (amf/&r)IY = 0 V m > 0) then for 
any k E N there exist g, h E P(o) such that f = gz, t hli. 
By induction on k. Assume it is true for k; we have 
a”f akg 
at=szntk!h 
n n 
from the E-flatness it follows that h IV = 0 and so by step (a) h = z,u t mv, 
where u, v E A O”@) and f = z,J g t Ft u) + Yj’ ‘v. 
(c) If f E Cm@) is %-flat on V then z; ‘fE P(D). 
IDEALSINACC(D) 165 
For any k E N we have z;‘f=g + 5i”z;‘h E C?(o). Now iffE A”‘(D) 
and f/,. = 0 then f is ti-flat on V and so setting g = fz; ’ we have f = gz,,, 
where g E A”(D): this proves (i). 
Now let rr: C”- V’ be the natural projection: for every fE A”(V) let 
f’: V’ + C be a C-smooth extension of f and let F=ro 7~1~. We have 
FE C”(D) and aF= xi”=, b,id5j. where b, ,..., b, are ii-flat on I’. Then 
a = z, -’ ~~ is C%mooth on D so that we can lind u E C’=(B) such that 
& = (r ] 111. Then F = F - Z,U belongs to A”(D) and it is the required 
extension off (cf. also IS]). 
We are now in a position to prove Proposition 1. 
Proof of Proposition 1. Let V; be defined by zktIfp=...=z,=O, 
1 <p<n-k- 1, and set D,=Dn Vi. 
We may assume (possibly after a linear transformation of coordinates) 
that VL n bD is C-smooth and VI, and Vi,, n bD are regularly separated. 
Let f EA”(D) be such that flc = 0. In view of Lemma 2 we have f /,j1 = 
Zk+lgk+l, where g,, , is the restriction to D, of a function in A”(D,). 
It follows that f - zk+, gk+ 1 vanishes on D, so that on D, we have 
f-g ktlZk+l =gktZZkt2r etc. 
With the same notations, as a consequence of Proposition 1, we get the 
following: 
COROLLARY 3. Assume D is pseudoconvex, V’ and bD are regularly 
separated and Sing V’ ~7 bD = 0. Let f,,...,fk be a complete defining system 
for V’. Then the sheaf homomorphisms 
(i) ((dyp))Ok + Kp’ (given b-y (A, ,..., A,) H CT=, lj&), 
(ii) /“p) -+ c”k?) (given by “restriction to V’) are onto. 
2. THE GENERAL CASE 
1. In view of Corollary 3 the problem we are dealing with is locally 
solvable. 
In order to get the global result, in the case V’ is a global complete inter- 
section (i.e., k = n-dim,: V’) cohomological techniques can be employed 
proving that the sheaf 9 of ccPba’-relations between f, ,..., fk actually satisfies 
H’(D,9)=0 [l]. 
In the genera1 case here we use a construction which reduces the problem 
to the linear case, via an extension theorem. 
THEOREM 4. Assume D is strongly pseudoconvex and let Vi, Vi be 
analytic subvarieties of D’ such that if we set Vj = Vi n Vi we have: 
166 DE BARTOLOMEIS AND TOMASSINI 
(i) Sing VjnbD=a,j= 1,2,3, 
(ii) V; and bD are regularly separated, j = 1, 2, 3. 
Assume also that Vi and Vi intersect transversally along bD and let Z’ = 
v; u v;, Vi = Vi’ n D, j = 1, 2, 3. Then the restriction homomorphism 
A”O(D) + A”O(Z) is onto, where Z = Z’ CID. 
ProoJ (1) First of all we prove that the restriction homomorphism 
F@’ -+ Fp’“’ is onto. D 
For th;s let p E bD and U be a small neighborhood of p. Let 
f~ A”O(UfIZ): in view of Corollary 3(ii) we may assume flYlnu= 0. 
Assume for the moment we may choose complex coordinates z, ,..., z, in 
such a way that 
V~nU={zEU:z,=...=z,=O}, 
v;nu= {ZE u:zs=... =z,=O,s<k+ l}. 
Let.6 =flvjnu; from Corollary 3(i) we get 
f, = f! h,z,, 
j=k+l 
h,EA”O(V,nU), k+ l<j<m, 
and in view of part (ii) we can extend hj as Hj E AOC(Un 0). If we set F, = 
Cjm,k+l Hjzj we have F, E Am(Un D) and F,/,,, = 0. 
In the same way we can construct F, E A*(U n D) extending fi and 
vanishing on V, n U, The function F = F, + F, satisfies FIznu =J This 
shows that the sequence of sheaves 
0 -+ a”’ + ep + c”p + 0 
Z 
is exact and so in order to conclude the proof we have to show that the 
group H’(fi, Wgm’) is actually zero. This amounts to proving the following 
claim: let F be a (0, 1)-form, P-smooth up to bD, klosed and such that 
FI, = 0; then there exists u E Cm@) such that $U = F and u], = 0. 
(2) The proof of this claim is based on a construction on the theme of 
the “bumps lemma” of Andreotti and Grauert 12, p. 2371 (cf. also [4] and 
191). 
2. Consider a finite covering {Bj}, 1 <j,< q, of bD where Bj = B(Cj,p) is 
the open ball of radius p centered at rj E bD, in such a way that {B(rj, p/2)}, 
1 <j < q, is also a covering and Z n Bj is holomorphically equivalent to a 
plane crossing, 1 Q j < q. In view of the bumps lemma we can find an 
increasing family of pseudoconvex domains {Dj), 0 <j < q, with Coo-smooth 
boundary, such that D, = D, DjP, U {cj} c Dj c Dj~, U B(rj, p/2) and 
DCD,CD~. 
Let a, E Cm(D) be such that 8a, = F; in particular on Z\Sing Z we have 
IDEALS IN A “O(D) 167 
go, = 0 and so in view of a result of Malgrange [ 13 ] a, Iz is holomorphic. 
From part (1) we deduce that there exists g, E A “‘(D fl B,) such that 
g, lz’ml = a, ; it follows that U, = a, -g, E P(m), &, = F on D f’ B, 
and U, = 0 on- Z’ n B,. Let q, E CF(B,) be such that 11, = 0 on B(i, ,p/2_) 
and set F, =a[(1 - ~7,) u,]. We have the following: F, is C%mooth on D, 
aF, = 0 and F, = 0 on Z’ f? D,. Moreover on D we have F, = F- $3,, 
where /I, = q1 u1 E Coc(fi) and /J, Iz = 0. By iteration we get a (0, I)-form F,, 
P-smooth on 09 such that 3F, =O, Fq= 0 on Z’ n D, and on L?, F, = 
F - ;58,, where /I, E P(D) and /3& =-0. 
Now let y E CcsI(Dq) be such that 8y = F, ; then Y]~,~,,, is holomorphic. 
Let G be holomorphic on D, and such that G = y on Z’ n D, and set 
u=y-G~+~.WehaveuEC~(~),&=FanduvanishesonZ. 
Thus in order to establish Theorem 4 we only have to prove the following: 
LEMMA 5. Let V, , V, be germs of analytic submanifolds at p E C” such 
that 
(i) V, n VZ is smooth, 
(ii) T,“( V, n V,) = T,“( V,) n T,“( V,). 
Then we can choose complex coordinates z,,..., z, in a neighborhood I/ of p 
in such a way that 
Proof We shall consider first the case where dimc(T,“( VI) + Ti( V,)) = n. 
Let V, be defined (in a neighborhood U of p) by zi = . . . = z,-~ = 0 and 
V, by f,= ..a = fn-k = 0; it follows that V, n V, has dimension d + k - n 
and 3fl A ..a A i?fnpk A az, A ... A ... A&_,(p) # 0. Then there is a 
system of local (holomorphic) coordinates cl,..., <, such that cj = zj for 
l<j<kandc k+, =f,w c, =fn-k* 
In order to reduce the general case to the previous one we only have to 
check that under our assumptions Tt(V,) + TF(V,) is the Zariski tangent 
space c(V, U V,) of V, U V, at p. We have TF( V,) + TF( Vz) c 
c(V, U V,): in order to prove the opposite inclusion we must prove the 
following: given a C-linear map L: C” -+ C vanishing on Ti(V,) + TF(V,) 
there is a local holomorphic function g such that g = 0 on V, U V, and 
ag(P) = L. 
With the above notations we may suppose that on U 
V, = {z E U: zj = 0, 1 <j < k}, 
v,= {ZE U:&(z)= ... =f,,,(z)=O,s<k+ 1) 
168 DE BARTOLOMEIS AND TOMASSINI 
and 
where 
afs A . . . r\af,(p)#o, aZ,r\ . . . Aaz,Aaf,+,r\...Aaf,(p)#o 
and dim,(c(V,) + c(V,)) = n + s -k - 1. 
In particular 
v, n v2= {z E v,:fk+l(z)= .** =f&>=O}. 
Let g’ be a local holomorphic function such that &f(p) = L and glyz = 0; 
set g, =g(,,,. 
Then g, = Cm , k+I ajfj, where oj E P( I’,), and 
4+5(p)=an4 CW,)=w+W3=0 
a,, being the &operator for V, and so ai = 0, k + 1 <j < m. Extend aj by 
Aj (holomorphically), k + 1 <j< m, and set G = Cjm_k+, A,&; we have 
GI,, = g,, GI,, = 0, aG(p) = 0. The function we are looking for is now 
g=g-G. 
3. We are now in a position to prove our main theorem. 
Let D c C” be a bounded domain with P-smooth boundary bD and let 
V’ be a complex analytic subvariety of an open neighborhood D’ of D such 
that V = V’ f7 D # 0. Let f, ,..., fk b e a complete defining system for V’. 
THEOREM 6. Assume 
(i) D is strongly pseudoconvex, 
(ii) Sing V’ ~7 bD = 0 and V’, bD are regularly separated. 
Then every f E A”(D) vanishing on V can be written as 
where h 1 ,..., h, E Am(D). 
Proof. Consider the holomorphic map F: D’ + Ck given by F(z) = 
(f,(z),...,fk(z)) and let I’ be its graph. Consider in D’ x Ck a bounded 
domain B with P-smooth boundary, strongly pseudoconvex and such that 
B f-l (D’ x {0)) = D and r intersects bB transversally. From the fact 
IDEALS IN Am(D) 169 
(f,,...,&} is a complete defining system for V’ it follows that the jacobian 
matrix of F has rank n-dim, V’ near bD and in particular r intersects D’ 
transversally along bB. Let 2 = (D’ U r) n B and let fE A”(D) be such 
thatf= 0 on V. 
Let f be defined by: T=f on D and f = 0 on rn B; because of the 
transversality, f’is holomorphic on Z and TE A “(Z). 
By Theorem 4 applied to Z we can find G E A m(B) such that GI, =r In 
particular G = 0 on rfI B. Now r n B is holomorphically equivalent to a 
plane section and thus, using Proposition 1 we can find h; ,..., h;, E A”(B) 
such that G =-JJ;= r h(fj - wj) (w, ,..., wk complex coordinates in C”). By 
restriction to D we get f = CT=, hjfi, where hj = I&. This concludes the 
proof of our main theorem. 
COROLLARY 7. In the above hypothesis assume f E A”O( V) vanishes on 
V of order q > 0. Then f can be written as 
f= 1 hj,.. .j,f{’ “‘fk? 
j, + . . tjk=9 
where hj,. .jk E A”O(D),j, + ... tj, = q. 
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Finitely generated ideals in A∞(D) 

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Finitely generated ideals in A∞(D)

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