A Prime-Generated Formalization of Nagata's Factoriality Theorem in Lean 4
This work provides a formal verification of a key theorem in commutative algebra, addressing a gap in proof assistant libraries for mathematicians and formal methods researchers.
The authors formalized Nagata's factoriality theorem in Lean 4, proving that a noetherian domain is a unique factorization domain (UFD) under specific conditions, and applied it to show that polynomial rings over noetherian UFDs are UFDs, including iterated cases like R[X][Y].
We present a Lean 4 Mathlib formalization of Nagata's factoriality theorem: if R is a noetherian domain and S <= R is a prime-generated submonoid such that S^{-1}R is a UFD, then R itself is a UFD. The prime-generated hypothesis -- every element of S is a finite product of primes belonging to S -- replaces a superficially cleaner but degenerate prime-or-unit condition that the formalization effort exposed. The development packages the theorem both for the concrete type Localization S and through abstract IsLocalization formulations. As applications, we formalize two Nagata-based proofs that R[X] is a UFD whenever R is a noetherian UFD: one via Laurent-polynomial localization at powers of X, and one via localization at the constant primes and identification with Frac(R)[X]. Reusing the same package, we also obtain the iterated polynomial corollary R[X][Y]. No public formalization of this result is known to us in Lean, Coq, or Isabelle.