Closed finitely generated ideals in algebras of holomorphic functions and smooth to the boundary in strictly pseudoconvex domains (Q789571)

Let D be a bounded, strictly pseudoconvex domain in \({\mathbb{C}}^ n\) with \(C^{\infty}\)-boundary and let \(A^ m\) be the algebra of holomorphic functions in D, of class \(C^ m\) in \(\bar D\), \(m=0,1,2,...,\infty\). We prove: Theorem. The finitely generated ideals in \(A^ m\), \(0\leq m<\infty\), which are closed are exactly those whose zero-variety is a finite set of points in D, and in this case they are characterized by their local ideals. In particular, there are no closed, finitely generated ideals in \(A^ m\), \(0\leq m<\infty\), with less than n generators. - Theorem. A necessary condition for a finitely generated ideal I in \(A^{\infty}\) to be closed is that I does not have a zero of infinite order at \(\partial D\). - We also prove some flatness theorems for certain ideals of \(A^ m\).