The Brown–Goodearl question: is every noetherian affine Hopf algebra Artin–Schelter Gorenstein?

A foundational open problem in the homological theory of noetherian Hopf algebras, first posed by Brown and Goodearl in 1997: is every affine noetherian Hopf algebra over an algebraically closed field (the standing convention in the sources) Artin–Schelter (AS) Gorenstein — that is, of finite injective dimension $d$ on the left with $Ext^i_A(k,A)=0$ for $i≠d$ and $dim_k Ext^d_A(k,A)=1$ for the trivial module $k$ (with the mirror right-sided conditions)? The answer is known positively for all previously studied classes — finite-dimensional Hopf algebras, affine PI Hopf algebras, enveloping algebras, quantised enveloping and function algebras, noetherian group algebras — and Ren–Zhu (2026) proved the property is invariant under monoidal Morita–Takeuchi equivalence in the noetherian case, yet the general question remains open; Brown's own 2026 review of the topic records that the question 'remains open after 30 years'. A positive answer would extend the rigid dualising complex, Nakayama automorphism, and twisted Poincaré duality framework of Brown–Zhang (2008) to every noetherian affine Hopf algebra; a counterexample would be a foundational exception steering the classification programme.

Background

A Hopf algebra over a field $k$ is an algebra $A$ carrying a compatible coalgebra structure — a comultiplication $Δ:A→A⊗A$ and a counit $ε:A→k$ , coassociative and multiplicative — together with an antipode $S:A→A$ , a linear map satisfying convolution identities against the multiplication and comultiplication that gives finite-dimensional $A$ -modules natural duals; group algebras, universal enveloping algebras of Lie algebras, and quantised enveloping algebras are the standard examples. Via the counit, $A$ acts on a distinguished one-dimensional trivial module $k$ . Call $A$ noetherian if it is left and right noetherian as a ring, and affine if it is finitely generated as a $k$ -algebra. Following Artin and Schelter, a Hopf algebra $A$ is AS Gorenstein (of dimension $d$ ) if ${}_{A}A$ has finite injective dimension $d$ , $Ext^i_A(k,A)=0$ for all $i≠d$ , and $Ext^d_A(k,A)$ is one-dimensional over $k$ — its natural right $A$ -module structure being in general a twist of the trivial module by the winding automorphism, as in the Brown–Zhang framework described below — together with the mirror-image right-sided conditions; if moreover the global dimension equals $d$ , $A$ is AS regular, and $A$ is AS Cohen–Macaulay when the injective dimension of every non-zero finitely generated module equals its grade, the smallest degree in which its cohomology against the trivial module is non-zero — the noncommutative analogue of the commutative Gorenstein–Macaulay dichotomy. The question originates in Brown and Goodearl's 1997 study of noetherian Hopf algebras satisfying a polynomial identity (PI algebras: algebras in which some fixed noncommutative polynomial vanishes under every substitution, the noncommutative analogue of commutativity), where such homological duality could be established, and was repeated by Brown in 1998; it is usually posed over an algebraically closed base field. A positive answer is known for: finite-dimensional Hopf algebras (Frobenius algebras — finite-dimensional algebras self-dual as modules — by the Larson–Sweedler theorem); affine noetherian PI Hopf algebras (Wu–Zhang 2002, who prove the AS Cohen–Macaulay property as well); enveloping algebras of finite-dimensional Lie algebras, quantised enveloping algebras, quantised function algebras of semisimple groups, and noetherian group algebras (Brown–Zhang 2008). Brown and Zhang showed what a positive answer would buy: a noetherian AS Gorenstein Hopf algebra with bijective antipode admits a rigid dualising complex $R≅{}^{ν}A^{1}[d]$ whose twist $ν$ , the Nakayama automorphism (noncommutative analogue of the Frobenius invariant of a self-injective algebra), satisfies $ν=S^2∘Ξ$ with $Ξ$ the winding automorphism determined by a homological integral (a generalisation of the Haar integral on a compact group), yielding a Radford-type formula $S^4=γ∘φ∘Ξ^{-1}$ . The question therefore decides whether this entire duality framework applies to the whole class of noetherian affine Hopf algebras. Two adjacent open difficulties frame it. First, most homological machinery assumes a bijective antipode: Skryabin conjectured (2006) that every noetherian Hopf algebra has bijective antipode, known for semiprime algebras (those with no non-zero nilpotent two-sided ideals) and affine PI noetherian Hopf algebras, and for further sufficient homological and ring-theoretic conditions (Lü–Oh–Wang–Yu 2018), but open in general. Second, Ren–Zhu (2026) proved that the AS Gorenstein property and the Gorenstein global dimension are invariants of monoidal Morita–Takeuchi equivalence — equivalence of the monoidal categories of comodules, the Hopf analogue of Morita equivalence — for noetherian Hopf algebras; this transfers the property within equivalence classes but cannot decide it for a class as a whole, and Ren–Zhu explicitly record the Brown–Goodearl question as remaining open. Brown's own review of the topic (2026) confirms that the question, now nearly three decades old, remains open.

Problem Statement

Over an algebraically closed base field $k$ (the standing convention in the sources), is every affine noetherian Hopf $k$ -algebra $A$ Artin–Schelter Gorenstein — that is, does ${}_{A}A$ have finite injective dimension $d$ such that $Ext^i_A(k,A)=0$ for all $i≠d$ and $Ext^d_A(k,A)$ is one-dimensional over $k$ , where $k$ is the trivial module induced by the counit, and do the corresponding right-sided conditions hold for $A_A$ ? The question must be answered without auxiliary hypotheses: no bijectivity of the antipode, no polynomial identity, no pointedness (every simple subcoalgebra being one-dimensional), no connectedness (the coradical, the sum of all finite-dimensional subcoalgebras, being just $k$ ), and no finite Gelfand–Kirillov dimension assumption (Gelfand–Kirillov dimension being the standard growth measure of an affine algebra) may be imposed. A complete proof that every noetherian affine Hopf algebra satisfies the stated conditions resolves it positively; an explicit noetherian affine Hopf algebra failing at least one of the stated conditions resolves it negatively.

References

K. A. Brown and K. R. Goodearl, Homological aspects of Noetherian PI Hopf algebras and irreducible modules of maximal dimension, Journal of Algebra 198 (1997) 240–265. DOI: 10.1006/jabr.1997.7109, https://doi.org/10.1006/jabr.1997.7109
K. A. Brown, Representation theory of Noetherian Hopf algebras satisfying a polynomial identity, in Trends in the Representation Theory of Finite-Dimensional Algebras, Contemporary Mathematics 229, Amer. Math. Soc. (1998) 49–79. DOI: 10.1090/conm/229/03310, https://doi.org/10.1090/conm/229/03310
Q.-S. Wu and J. J. Zhang, Noetherian PI Hopf algebras are Gorenstein, Transactions of the American Mathematical Society 355 (2002) 1043–1066. DOI: 10.1090/S0002-9947-02-03106-9, https://doi.org/10.1090/S0002-9947-02-03106-9
S. Skryabin, New results on the bijectivity of antipode of a Hopf algebra, Journal of Algebra 306 (2006) 622–633. DOI: 10.1016/j.jalgebra.2006.04.017, https://doi.org/10.1016/j.jalgebra.2006.04.017
K. A. Brown and J. J. Zhang, Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras, Journal of Algebra 320 (2008) 1814–1850. DOI: 10.1016/j.jalgebra.2007.03.050, preprint arXiv:math/0603732, https://arxiv.org/abs/math/0603732
J. Lü, S.-Q. Oh, X. Wang, X. Yu, A note on the bijectivity of the antipode of a Hopf algebra and its applications, Proceedings of the American Mathematical Society 146 (2018), no. 11, 4619–4631. DOI: 10.1090/proc/14140, https://doi.org/10.1090/proc/14140
R. Zhu, Artin-Schelter Gorenstein property of Hopf Galois extensions, arXiv:2501.02828 (2025), https://arxiv.org/abs/2501.02828
W. Ren and R. Zhu, Gorenstein homological invariants and monoidal model categories of Hopf algebras, arXiv:2601.20249 (2026). DOI: 10.48550/arXiv.2601.20249, https://arxiv.org/abs/2601.20249
J. Bichon, Monoidal invariance of the cohomological dimension of Hopf algebras: the finite case, arXiv:2602.12731 (2026), https://arxiv.org/abs/2602.12731
K. A. Brown, Report on $AS$ -Gorenstein Hopf algebras, arXiv:2605.00774 (2026), https://arxiv.org/abs/2605.00774