PI Artin--Schelter regular algebras of dimension 3 are unique factorization rings

Abstract: We prove that all noetherian PI Artin--Schelter regular algebras of dimension $3$ are unique factorization rings. In a certain sense, this result is a noncommutative analogue to the fact that regular local rings of dimension 3 are UFDs. The fact constitutes a crucial component in the proof of the assertion that all regular local rings are UFDs, known as the Auslander--Buchsbaum theorem.

• The paper proves that every noetherian PI Artin–Schelter regular algebra of dimension at most 3 is a unique factorization ring, so each height-one prime is principal.
• The authors reduce unique factorization to projectivity of height-one graded primes, then establish projectivity in dimension 3 through Ext vanishing and local cohomology over the center.
• The results distinguish UFRs from stronger UFDs, extend factorization to skew polynomial and selected Ore extensions, and leave higher-dimensional and non-PI cases open.

Overview

This paper, by Silu Liu and Quanshui Wu, establishes that every noetherian PI Artin–Schelter (AS) regular algebra of dimension at most 3 is a noetherian unique factorization ring (UFR) in the sense of Chatters and Jordan. The result is explicitly framed as a noncommutative analogue of the Auslander–Buchsbaum theorem: Nagata reduced the statement that regular local rings are UFDs to dimension 3, and Auslander–Buchsbaum settled that case. Here the analogous statement holds for the class of connected graded algebras widely regarded as noncommutative polynomial rings, restricted to the PI (polynomial identity) setting.

The proof is homological, mirroring the strategy in the commutative case, but it must contend with the fact that localization techniques and Nagata's dimension induction are unavailable in the connected graded context. The authors instead deploy a noncommutative class group to reduce the problem to height-one graded prime ideals, then prove projectivity of those ideals via local cohomology computations over the center.

Background and the UFR condition

A noetherian prime ring $A$ is a noetherian UFR if every nonzero prime ideal contains a nonzero principal prime ideal. For noetherian prime rings satisfying a descending chain condition on prime ideals — in particular PI rings and algebras of finite GK-dimension — this is equivalent to requiring that all height-one prime ideals be principal. The authors note that the older Chatters UFD condition (height-one primes completely prime and principal) fails to be stable under polynomial extensions precisely because completely primeness is lost; the UFR notion repairs this and is closed under polynomial and matrix extensions. A commutative noetherian domain is a noetherian UFR exactly when it is a classical UFD.

The paper works under the hypothesis (H) that $A$ is a prime PI ring whose noetherian normal center $Z$ makes $A$ a maximal $Z$-order. By Posner's theorem and work of Stafford–Zhang, every noetherian PI AS-regular algebra satisfies this hypothesis: it is a domain, a finitely generated module over a noetherian normal center, and a maximal order. Under (H), a theorem of Akasaki identifies the UFR condition with triviality of the normal class group $\mathrm{NCl}(A)$, the quotient of the group $\mathrm{Div}(A)$ of reflexive fractional ideals (free abelian on height-one primes) by the subgroup of principal normal ideals.

Reduction to the graded case

The first structural result is that for a graded domain satisfying (H), the normal class group coincides with its graded counterpart: $\mathrm{NCl}(A) \cong \mathrm{NCl}^{gr}(A)$. The proof localizes at the homogeneous nonzero elements of the center, shows that the induced map kills exactly the graded divisor group (using a graded Posner theorem and the fact that the localized center becomes a UFD over which $A_{\mathcal{S}}$ is Azumaya, forcing the localized class groups to vanish), and concludes via the snake lemma applied to the diagram of divisor groups. Consequently, for a connected graded algebra satisfying (H), $A$ is a noetherian UFR if and only if every height-one graded prime ideal is projective as both a left and right $A$0-module — such an ideal is automatically free of rank 1 on each side, hence principal.

As a companion result, the paper identifies the central Picard group $A$1 with the ozone group $A$2 of center-fixing automorphisms, generalizing the classical fact that connected graded noetherian normal domains have trivial Picard group. This identification is of independent interest in the study of invertible bimodules over graded PI algebras.

The homological argument

The key technical input is a graded analogue, adapted from the commutative literature, of a lemma relating Ext groups to local cohomology: if $A$3, $A$4 are finitely generated graded modules over a noetherian connected graded algebra finite over its center, with $A$5 of finite length over $A$6 for $A$7 and $A$8, then $A$9 for $Z$0, with an injection at the boundary degree. This is proved by splicing short exact sequences from a free resolution of $Z$1 and applying $Z$2 and its derived functors.

Noetherian PI AS-regular algebras of dimension $Z$3 are homologically homogeneous (all simple modules share projective dimension $Z$4) and are Gorenstein orders; consequently $Z$5 for primes of the center. The proof of the main theorem then proceeds for $Z$6 as follows:

• Reflexivity of height-one primes gives $Z$7.
• Localizing at any graded prime of the center other than the graded maximal ideal, the global dimension of $Z$8 is at most 2, so $Z$9 is projective; hence $A$0 is a finite-length $A$1-module.
• The Ext–local cohomology lemma, combined with the AS-Gorenstein vanishing $A$2, yields $A$3.
• A minimal-resolution argument shows that a module of projective dimension at most 1 with $A$4 for some nonzero finitely generated $A$5 is projective; applied to $A$6, this proves projectivity.

For $A$7, all reflexive modules are already projective and the result is immediate. The paper also records a corollary relevant to automorphism theory: the $A$8-discriminant (and its $A$9 refinement) exists for all such algebras, since homologically homogeneous noetherian UFRs with affine center admit reflexive hull discriminants.

UFD versus UFR, and examples beyond PI

The paper is careful to distinguish UFR from the stronger Chatters UFD condition. It exhibits the algebra $Z$0 over a field of characteristic not 2: a noetherian PI AS-regular algebra of dimension 2, hence a UFR, but not a UFD. The central element $Z$1 is shown to be irreducible among normal elements, so its principal ideal is prime of height one, yet it factors as $Z$2 and is therefore not completely prime. This example demonstrates that principal height-one primes need not be completely prime even in the regular graded setting.

Beyond the PI case, the paper proves that all skew polynomial rings $Z$3 are noetherian UFRs, by induction on dimension using the Ore extension stability theorem of Chatters–Jordan together with a lemma showing that every nonzero $Z$4-prime ideal of an $Z$5-dimensional skew polynomial ring contains a $Z$6-stable principal prime. The proof rests on the observation that homogeneous components of normal elements in a skew polynomial ring are themselves normal with the same associated automorphism. It also verifies that all noetherian AS-regular algebras of dimension 2 — skew polynomial rings and the algebras $Z$7, realized as $Z$8 — are UFRs. Additionally, an Ore extension $Z$9 of a noetherian PI AS-regular algebra by a finite-order automorphism preserves the UFR property, using the finite-order result for ozone automorphisms.

Limitations and open questions

The paper is explicit about the boundaries of its result. Nagata's dimension induction, which would extend the dimension-3 statement to all dimensions in the commutative setting, relies on localization of local rings and does not transfer to connected graded algebras; whether all noetherian PI AS-regular algebras, regardless of dimension, are UFRs is posed as an open question. A second question asks whether the PI hypothesis can be dropped in dimension 3 — all noetherian AS-regular algebras of dimension 3 are known to be domains, so the UFR framework applies, but projectivity of height-one graded primes is unverified without the PI structure. More fundamentally, it remains open whether noetherian AS-regular algebras are always domains, a conjecture outstanding for over thirty years; a positive answer would be a prerequisite for formulating the UFR question in full generality. The proof also depends on the finite GK-dimension and maximality of the order structure guaranteed by the PI hypothesis, and no strategy is offered for non-PI algebras beyond dimension 2 and the specific families treated.

Conclusion

The paper establishes a clean homological criterion — triviality of the normal class group, equivalently projectivity of height-one graded primes — and verifies it for noetherian PI AS-regular algebras of dimension at most 3 via local cohomology over the center. The result confirms the conjecture of Braun and Hajarnavis for the algebras they studied, provides the existence of reflexive hull discriminants in dimension 3, and extends the UFR property to all skew polynomial rings and all dimension-2 noetherian AS-regular algebras. The principal open problem left by the paper is whether the UFR property holds for noetherian PI AS-regular algebras in arbitrary dimension, and whether it extends beyond the PI hypothesis in dimension 3.