On Landweber`s unique factorization problem

Technical Summary: On Landweber's Unique Factorization Problem

Problem Statement
The paper addresses a long-standing question in commutative algebra regarding the preservation of the unique factorization domain (UFD) property under the formation of formal power series rings. Specifically, it investigates whether $R[[t]]$ is a UFD when $R$ is a regular UFD. While the Noetherian case was settled positively by Samuel and Buchsbaum (proving that if $R$ is a regular Noetherian UFD, then $R[[t]]$ is a UFD), the non-Noetherian case remained open. In 1974, Landweber raised the specific question of whether $R[[t]]$ is a UFD when $R = A[x_1, x_2, \dots]$ is a polynomial ring in countably many variables over a regular UFD $A$. This problem was highlighted in subsequent surveys by Anderson and Gilmer but remained unsolved.

Methodology
The authors employ a strategy that reduces the global problem to local properties and finite approximations. The proof is structured around three main theoretical pillars:

1. Finite Irreducibility Theorem (Theorem B): The core technical contribution is proving that if $R$ is a Krull domain and $f \in R[[t]]$ is irreducible, then $f$ is "irreducible of finite height." That is, there exists an integer $n \ge 1$ such that $f$ remains irreducible modulo $t^n$. The proof proceeds by contraposition: assuming $f$ is reducible modulo arbitrarily high powers of $t$, the authors construct a proper factorization of $f$ in $R[[t]]$. This construction utilizes a Kőnig's lemma argument (Lemma 3.1) to extract a convergent sequence of partial factorizations.

   • To ensure the compatibility of these partial factorizations, the authors establish a "primality of finite height" property for power series over Discrete Valuation Rings (DVRs). This involves defining $C$-prime elements and proving quantitative bounds on factorization lengths (Sections 4 and 5).
   • The result is extended from DVRs to general Krull domains using a "local-global" argument (Section 6), leveraging the fact that a Krull domain is the intersection of its localizations at height-1 primes.

2. Finite-Stage Retraction Criterion (Theorem C): The authors establish a general criterion (Theorem 7.2) stating that if $R$ is a Krull domain such that for every finite subset $E \subset R$, there exists a subring $S \subseteq R$ containing $E$ and a retraction $\rho: R \to S$ where $S[[t]]$ is a UFD, then $R[[t]]$ is a UFD. This theorem bridges the gap between the finite-variable case (where unique factorization is known) and the infinite-variable case.

3. Synthesis: The proof of the main result combines the finite irreducibility theorem with the retraction criterion. Since $R[[t]]$ is a Krull domain (and thus atomic), proving that every irreducible element is prime is sufficient to establish the UFD property. The finite irreducibility theorem ensures that irreducibles have finite height, and the retraction criterion allows the primality of these elements to be deduced from their behavior in finite-variable subrings.

Key Results

• Theorem A (Theorem 7.3): Let $A$ be a regular UFD, $I$ any set, and $R = A[x_i \mid i \in I]$. Then $R[[t]]$ is a UFD. This provides a complete affirmative answer to Landweber's question, covering the case of polynomial rings in countably many variables over a field or any regular UFD.
• Theorem B (Theorem 6.4): Let $R$ be a Krull domain. If $f \in R[[t]]$ is irreducible, then $f$ is irreducible modulo $t^n$ for some $n \ge 1$. This generalizes a result by Bayart (who proved it for characteristic-zero UFDs where all integers are units) to the general context of Krull domains.
• Theorem C (Theorem 7.2): A sufficient condition for $R[[t]]$ to be a UFD based on the existence of retractions to finite-stage subrings that are UFDs.

Significance and Scope
The paper resolves Landweber's 1974 problem in full generality. The authors note that while the Noetherian case was understood, the non-Noetherian case (specifically involving infinite variables) presented significant difficulties, particularly in establishing irreducibility results for the full ring rather than just graded subrings.

The significance of the work lies in:

1. Completing the Classification: It settles the question for regular UFDs in the non-Noetherian setting, a gap that had persisted for decades.
2. Generalizing Primality Properties: The proof of Theorem B extends the understanding of irreducibility in power series rings beyond the specific cases handled by Artin's strong approximation theorem or Bayart's work, applying to all Krull domains.
3. Methodological Contribution: The introduction of the "finite irreducibility" concept and the "primality of finite height" property provides new tools for analyzing factorization in power series rings over non-Noetherian domains.

The authors explicitly state that the conclusion of Theorem 6.4 does not characterize Krull domains (as it holds for some non-Krull domains like $F_q[x^2, x^3]$) and that the question of whether $R[[t]]$ is a UFD for all Noetherian domains remains open. Furthermore, the paper notes that the related question raised by Bayart—whether $S[[t]]$ is a UFD if $S = R[[x]]$ is a UFD—remains unresolved.