Applications of functional analysis to the solution of power series equations

Abstract

In the study of local rings arising in algebraic geometry one often puts a topology on these rings -the Krull topology, in which a neighborhood base of 0 consists of all powers of the maximal ideal. The Krull topology plays an important role in the study of many topics which appear to be purely algebraic rather than topological. The most basic tools for the study and application of the Krull topology are probably the Artin-Rees lemma, which asserts that if R is a local Noetherian ring and N, M are finitely generated R-modules with N C M, then the natural topology on N is equivalent to the topology on N induced from M; and the Chevalley subspace theorem, which asserts that if R is a complete local Noetherian ring, then the Krull topology is the weakest Hausdorff ring topology on R which as a countable neighborhood base of 0 consisting of ideals.In the study of differential operators (higher order tangent vectors) on analytic varieties, there is another topology of interest: the Frechet topology of simple convergence of each coefficient on the ring ~ of formal power series over IF (see [10] for example). This topology is strictly weaker than the Krull topology on o~ ; however, these two topologies admit the same continuous linear functionals.When one considers the ring of convergent power series, one may wish to use either the Krull topology or the simple topology, inherited from the ring of formal power series. It is frequently useful, however, to use yet another topology, referred to by Grauert and Remmert [16] as the "Folgentopologie"; it is the direct limit topology arising when the ring of convergent power series is regarded in the natural way as the direct limit of Banach algebras. This topology is stronger than the simple topology, and incomparable with the Krull topology.In this paper we use topological techniques to investigate the question raised by Artin [3] and Grothendieck [17]: Is the completion of a injective local algebra homomorphism q5 of ¢ analytic rings, also an injection, i.e. is q~ open in the Krull topology?In the spirit of the Artin-Rees lemma and the Chevally subspace theorem, we prove in this paper a number of uniqueness results for the above topologies, using

Relations among analytic functions. I

“…In fact, by [4] and [30], (ii) is equivalent to the regularity of 0 at a. (i) Ker H^ is generated by Ker Y^ .…”

Solving power series equations. II. Change of ground field

“…where m is the maximal ideal of S. By the rank of (p we mean the rank of The proof of all the unnumbered implications has already been given in [9] or theorem 2.1.…”

“…In [9] we continued the investigation of the properties of local algebra homomorphisms begun in [2,12,14,18], and showed that if (p : R -> S is an algebra homomorphism of reduced analytic rings over C which is closed in the Krull topology, then (p is open in the Krull topology. This can be restated as follows : If (p is injective and cp(R) n S = (p(R), then (p : R -> S is injective, where hat denotes completion.…”

“…/T^O with /((p)=0] and [V formal g with g(n>} conv., 3 conv. / with fW=9W]} ^ {3 formal/^O with/((p)=0}, the proof given in [9] uses topological techniques, and relies heavily on the completeness of the complex numbers. Since the above statement is purely algebraic, it is natural to ask whether it holds for series over a valued field k of characteristic zero.…”

“…We recall first the main result of [9]. Let (p : R -> S be local C algebra homomorphism of reduced analytic rings.…”