[Paper Review] Rings with Topologies Induced by Spaces of Functions

Răzvan Gelca|ArXiv.org|Apr 10, 1996

Holomorphic and Operator Theory4 references16 citations

TL;DR

This paper establishes a topological duality between closed ideals and closed submodules in Noetherian rings endowed with function-space-induced topologies, proving that closed ideals have closed associated prime ideals and vice versa under a topological Nullstellensatz. It demonstrates this in polynomial rings with topologies from Hardy and Bergman spaces, showing a one-to-one correspondence between closed ideals of finite codimension and closed submodules preserving primary decompositions.

ABSTRACT

The paper contains the proof, in dimension 2, of a conjecture of R. G. Douglas and V. Paulsen concerning the characterization of the ideals of polynomials which are closed in the relative topology induced by the Hardy space of the polydisk.

Motivation & Objective

• To investigate the topological properties of ideals in Noetherian rings endowed with topologies induced by spaces of holomorphic functions.
• To resolve a question from [5] about whether associated primes of closed ideals are themselves closed.
• To establish a topological version of the Nullstellensatz for rings of polynomials and analytic functions.
• To show a one-to-one correspondence between closed ideals of finite codimension and closed submodules of finite codimension in such rings.
• To extend results to Hardy spaces on the polydisc, Bergman spaces on the unit ball, and Reinhardt domains.

Proposed method

• Uses the topology induced by Hilbert spaces of holomorphic functions (e.g., Hardy, Bergman) on polynomial or analytic function rings.
• Applies separate continuity of multiplication and continuity of addition to analyze closure properties of ideals.
• Employs the radical of an ideal and inductive continuity arguments to prove that the radical of a closed ideal is closed.
• Uses primary decomposition and the ideal quotient (I:f) to show that associated primes of closed ideals are closed.
• Introduces the set C of maximal ideals whose powers are dense, and uses J-adic topologies to characterize closed ideals.
• Applies Krull’s Theorem and the condition that J + P_i ≠ R to verify closedness in J-adic topologies.

Experimental results

Research questions

• RQ1 If an ideal is closed in a ring with a function-space-induced topology, are all its associated prime ideals also closed?
• RQ2 Under what conditions does the topological Nullstellensatz hold, ensuring that closed ideals are characterized by closed associated primes?
• RQ3 What is the relationship between closed ideals of finite codimension and closed submodules of finite codimension in such rings?
• RQ4 How do the topologies induced by Hardy spaces, Bergman spaces, and Reinhardt domain L^2 norms affect the closedness of ideals?
• RQ5 For which Reinhardt domains is the ring of polynomials with the L^2-topology over the domain such that maximal ideals are either dense or have weaker J-adic topologies?

Key findings

• If an ideal I is closed in a Noetherian ring with a function-space-induced topology, then every prime ideal associated to I is also closed.
• The converse holds if the topological Nullstellensatz is satisfied: if all associated primes of an ideal are closed, then the ideal itself is closed.
• The topological completion of the ring is a module, and there exists a one-to-one correspondence between closed ideals of finite codimension and closed submodules of finite codimension that preserves primary decompositions.
• For the ring of polynomials in two variables with the topology induced by the Hardy space on the polydisc, the topological Nullstellensatz holds, and the correspondence between closed ideals and submodules is valid.
• For the ring of analytic functions in a neighborhood of the unit ball with the topology induced by the Bergman space, the maximal ideals corresponding to boundary points are dense, satisfying the conditions for the topological Nullstellensatz.
• On Reinhardt domains Ω_{p,q} = { |z₁|^p + |z₂|^q < 1 }, the L^2-topology on C[z₁,z₂] satisfies the hypothesis of the main theorem, with maximal ideals dense if the point lies on the boundary.