1.2 A Simple Problem 3 determines a maximal ideal MA = Ker{/ -+ /(A) : H(U) - C} and, in fact, every maximal ideal with C as quotient field has this form. To see this, observe that if M is a maximal ideal with quotient field C, cj) : H(U) C is the quotient homomorphism modulo M, and A = j(z), then z- X belongs to M. But, for each / G H{U), it follows from the power series expansion of / at A that f{z) /(A) is divisible by z A. That is, ^ has a removable singularity at A and so it defines an h G T~i{U) z A such that f(z) - /(A) = (z - X)h(z). Thus, / - /(A) also belongs to M and cj)(f) - /(A) = /(/ - /(A)) = 0. This means that (j) agrees with the evaluation homomorphism / /(A) and M = M\. Is every maximal ideal of H(U) of the form M\l No. Let {an} c U be a sequence of points which has no limit point in U. Let / be the set of functions in H(U) which vanish at all but finitely many points of {an}. Then J is a proper ideal of H(U). However, there is no single point at which all the functions in / vanish. This follows from the fact that, given any discrete set S of points of £/, there is a function in H(U) which vanishes exactly on S (a corollary of the Weierstrass theorem, which we will prove later in the chapter). Thus, no maximal ideal that contains / can be of the form M\. The following is true, however: 1.2.1 Theorem. Each finitely generated maximal ideal of 7i(U) is of the form M\ for some A G U. If a maximal ideal M of H{U) is finitely generated, say by {gi,..., gn}, and if these generators all vanish at some point A G £/, then all the functions in M vanish at A and we have M C M\. This, of course, implies that M = MA, since M is a maximal ideal. Thus, Theorem 1.2.1 will be proved if we can show that: if a finite set of functions {#i,. . . , gn} does not have a common zero, then it does not generate a proper ideal. That is the content of the following proposition. 1.2.2 Proposition. If a finite set of functions {gi,. . . , gn} c H(U) has no common zero in U 7 then the equation (1.2.1) fl9l + '" + fn9n = l has a solution for / i , . . . , fn G H{U). The proof of this result will occupy most of the chapter. Along the way, we shall prove a number of important results from the theory of holomor- phic functions of one complex variable that are often not covered in a first
Previous Page Next Page