A Hopf algebra that is noetherian is conjectured to always be Artin–Schelter (AS) Gorenstein: it should have finite injective dimension on both sides, with the Ext modules against the trivial module concentrated in a single degree and one-dimensional there. Posed by Brown and Goodearl in 1997 and reiterated in surveys by Brown, Goodearl, and Brown–Zhang (who label it the Brown–Goodearl Conjecture), the question extends the Larson–Sweedler Frobenius theorem for finite-dimensional Hopf algebras to infinite dimensions. All known classes of noetherian Hopf algebras satisfy it, and a January 2026 paper still records it as open; no counterexample and no general proof is known.

Background

A Hopf algebra HH over a field kk is simultaneously an associative algebra and a coalgebra (with comultiplication Δ:H→H⊗kH\Delta:H\to H\otimes_k H, counit ε:H→k\varepsilon:H\to k, and an antipode SS intertwining the two structures so that the category of HH-modules carries a tensor product). Standard examples include group algebras kGkG, universal enveloping algebras U(g)U(\mathfrak g) of Lie algebras, coordinate rings O(G)\mathcal O(G) of affine algebraic groups, and their quantum deformations. The algebra HH is called noetherian if it is both left and right noetherian, and affine if it is finitely generated as a kk-algebra; it satisfies a polynomial identity (PI) if a single nonzero polynomial vanishes on all substitutions from HH. For affine algebras the Gelfand–Kirillov (GK) dimension measures the polynomial growth rate of the span of words in a finite generating set, giving a noncommutative analogue of Krull dimension.

The counit makes kk a one-dimensional HH-module, the trivial module. A Hopf algebra HH is Artin–Schelter (AS) Gorenstein of dimension dd if: (AS1) the left regular module HH_H H has finite injective dimension d<∞d<\infty; (AS2) Ext⁡Hi(k,H)=0\operatorname{Ext}^i_H(k,H)=0 for i≠di\ne d and dim⁡kExt⁡Hd(k,H)=1\dim_k\operatorname{Ext}^d_H(k,H)=1; and (AS3) the right-sided analogues hold. If moreover the global dimension is finite (and equals dd), HH is AS-regular. A stronger homological finiteness condition is the Auslander property: HH has finite injective dimension and satisfies the Auslander condition, namely for every finitely generated left (respectively right) module MM, every i≥0i\ge 0, and every finitely generated right (respectively left) submodule NN of Ext⁡Hi(M,H)\operatorname{Ext}^i_H(M,H), the homological grade j(N)=min⁡{l:Ext⁡Hl(N,H)≠0}j(N)=\min\{l:\operatorname{Ext}^l_H(N,H)\ne 0\} is at least ii; HH is GK-Cohen–Macaulay if the homological grade of every nonzero finitely generated module MM satisfies j(M)+GKdim⁡M=GKdim⁡Hj(M)+\operatorname{GKdim} M=\operatorname{GKdim} H. Auslander–Gorenstein together with GK-Cohen–Macaulay implies AS-Gorenstein (Brown–Zhang), and for a noetherian group algebra of a polycyclic-by-finite group (a group built up from the trivial group by finitely many extensions by cyclic groups and then by a finite group) the Auslander–Gorenstein property holds while GK-Cohen–Macaulay can fail, so Cohen–Macaulay behaves as a companion property rather than a universal expectation.

The question is the infinite-dimensional analogue of a classical finite-dimensional fact: by a theorem of Larson and Sweedler, every finite-dimensional Hopf algebra is a Frobenius algebra, hence self-injective (injective dimension 00). Whether the homological regularity encoded by AS-Gorenstein survives in the noetherian infinite-dimensional world is the content of the Brown–Goodearl question, first posed in Brown and Goodearl's 1997 study of noetherian PI Hopf algebras as their Question 1.15 and repeated by Brown and Zhang in 2008. It is known affirmatively under extra hypotheses: Brown–Goodearl and Wu–Zhang proved that every affine noetherian PI Hopf algebra (in particular every noetherian Hopf algebra that is a finite module over its center) is Auslander–Gorenstein, Cohen–Macaulay, and semiprime; Brown–Zhang verified AS-Gorenstein — equipping these algebras with a rigid dualizing complex (the noncommutative analogue of a dualizing sheaf, required to be isomorphic to its own double dual) and a Nakayama automorphism (the automorphism measuring the failure of that complex to be symmetric) — for enveloping algebras of finite-dimensional Lie algebras, quantised enveloping algebras (qq-deformations of U(g)U(\mathfrak g)), quantised function algebras of semisimple groups (qq-deformations of coordinate rings O(G)\mathcal O(G)), and noetherian group algebras; Lu–Wu–Zhang proved it for Hopf algebras with a filtration whose associated graded algebra is connected graded noetherian with enough normal elements. Noetherian Hopf algebras of GK-dimension at most two are accessible through classification results and satisfy the conjecture. What is missing is any argument, or any counterexample, in the generality of an arbitrary noetherian Hopf algebra with no PI, filtration, graded, or finite-over-center assumption.

Problem Statement

Brown–Goodearl question (Brown–Goodearl Conjecture). Let HH be a Hopf algebra over a field kk that is noetherian (left and right). Must HH be Artin–Schelter Gorenstein — that is, must the injective dimension of HH_H H be a finite number dd with Ext⁡Hi(k,H)=0\operatorname{Ext}^i_H(k,H)=0 for i≠di\ne d and dim⁡kExt⁡Hd(k,H)=1\dim_k\operatorname{Ext}^d_H(k,H)=1, together with the corresponding right HH-module conditions? In particular, must every noetherian Hopf algebra have finite injective dimension (already entailed by the AS-Gorenstein demand), and — as a separate, stronger companion sub-question — must it satisfy the Auslander (Auslander–Gorenstein) condition, which yields AS-Gorenstein only together with GK-Cohen–Macaulay? The question is posed both for arbitrary noetherian Hopf algebras and for the class of noetherian affine Hopf algebras, the latter being a subclass of the former. Its affirmative side comprises two sub-questions: the main AS-Gorenstein sub-question (which subsumes finite injective dimension) and the separate, stronger companion Auslander–Gorenstein sub-question. A proof for the general class also covers the nested affine class, and a counterexample to either sub-question disproves that claim outright. Known affirmative results all impose additional hypotheses (PI or module-finiteness over the center, a suitable filtration with well-behaved associated graded algebra, membership in the classical quantum-group and group-algebra classes, or GK-dimension at most two via classification), and none of these techniques currently reaches the general case.

References

  1. 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), no. 1, 240–265. DOI: 10.1006/jabr.1997.7109, https://doi.org/10.1006/jabr.1997.7109
  2. Q.-S. Wu and J. J. Zhang, 'Noetherian PI Hopf algebras are Gorenstein', Transactions of the American Mathematical Society 355 (2003), no. 3, 1043–1066. DOI: 10.1090/S0002-9947-02-03106-9, https://doi.org/10.1090/S0002-9947-02-03106-9
  3. D.-M. Lu, Q.-S. Wu, and J. J. Zhang, 'Homological Integral of Hopf Algebras', Transactions of the American Mathematical Society 359 (2007), no. 10, 4945–4975. DOI: 10.1090/S0002-9947-07-04159-1, arXiv ID: math/0510646, https://arxiv.org/abs/math/0510646
  4. K. A. Brown and J. J. Zhang, 'Dualising complexes and twisted Hochschild (co)homology for noetherian Hopf algebras', Journal of Algebra 320 (2008), no. 5, 1814–1850. DOI: 10.1016/j.jalgebra.2007.03.050, arXiv ID: math/0603732, https://arxiv.org/abs/math/0603732
  5. D.-M. Lu, Q.-S. Wu, and J. J. Zhang, 'Hopf algebras with rigid dualizing complexes', Israel Journal of Mathematics 169 (2009), no. 1, 89–108. DOI: 10.1007/s11856-009-0005-1, https://doi.org/10.1007/s11856-009-0005-1
  6. K. R. Goodearl and J. J. Zhang, 'Noetherian Hopf algebra domains of Gelfand-Kirillov dimension two', arXiv:0905.0621 (2009). arXiv ID: 0905.0621, https://arxiv.org/abs/0905.0621
  7. K. R. Goodearl, 'Noetherian Hopf algebras', Glasgow Mathematical Journal 55 (2013), no. A, 75–87. DOI: 10.1017/S0017089513000517, arXiv ID: 1201.4854, https://arxiv.org/abs/1201.4854
  8. K. Brown and J. J. Zhang, 'Survey on Hopf algebras of GK-dimension 1 and 2', in Hopf Algebras, Tensor Categories and Related Topics, Contemporary Mathematics 771, American Mathematical Society, 2021, pp. 43–62. DOI: 10.1090/conm/771/15507, arXiv ID: 2003.14251, https://arxiv.org/abs/2003.14251
  9. D. Rogalski, R. Won, and J. J. Zhang, 'Homological Integrals for Weak Hopf Algebras', Journal of Algebra 692 (2026), 173–204. DOI: 10.1016/j.jalgebra.2025.12.009, arXiv ID: 2504.03057, https://arxiv.org/abs/2504.03057
  10. R. Zhu, 'Artin-Schelter Gorenstein property of Hopf Galois extensions', arXiv:2501.02828 (2025). arXiv ID: 2501.02828, https://arxiv.org/abs/2501.02828
  11. W. Ren and R. Zhu, 'Gorenstein homological invariants and monoidal model categories of Hopf algebras', arXiv:2601.20249 (2026). arXiv ID: 2601.20249, https://arxiv.org/abs/2601.20249