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].