Is every noetherian Hopf algebra Artin–Schelter Gorenstein? (the Brown–Goodearl Gorenstein question)

A guiding question in the homological theory of Hopf algebras, first posed by Brown and Goodearl in 1997 and repeated by Brown in 1998, asks whether every noetherian Hopf algebra has the good homological properties familiar from commutative Gorenstein rings — specifically, whether it is Artin–Schelter (AS-)Gorenstein and Auslander–Gorenstein, i.e. has finite injective dimension with a one-dimensional top self-Ext group for the trivial module. Affirmative answers are known for many classes — algebras satisfying a polynomial identity, enveloping algebras, quantised enveloping and function algebras, group algebras of polycyclic-by-finite groups, Hopf algebras with a suitable filtration (Lu–Wu–Zhang 2009), connected Hopf algebras of finite Gelfand–Kirillov dimension, and more — but the general case remains open. As Brown's May 2026 status report states, the question 'remains open after 30 years'. The AS-Gorenstein half of the question is equivalent, for noetherian Hopf algebras, to the Van den Bergh condition, i.e. to the existence of a rigid dualizing complex concentrated in one degree; twisted Calabi–Yau duality corresponds instead to the stronger AS-regular property. A positive answer would also force the antipode to be bijective.

Background

A Hopf algebra over a field $K$ is simultaneously an algebra and a coalgebra with a compatible multiplication, comultiplication, antipode $S$, and counit, so that the category of its modules is a tensor category; standard examples are group algebras, enveloping algebras $U(\mathfrak{g})$ of Lie algebras, quantised enveloping algebras, and quantised function algebras $\mathcal{O}_q(G)$. A Hopf algebra is noetherian if it is left and right noetherian as a ring. Commutative noetherian Hopf algebras are affine and well understood (Cartier–Gabriel–Kostant type theorems), but beyond the commutative, cocommutative, and polynomial identity (PI) settings no general structure theory exists, and even basic finiteness questions are unresolved: it is not known whether every noetherian Hopf algebra is affine.

The homological notions under discussion generalize classical properties of commutative rings. A noetherian $K$-algebra $A$ has finite injective dimension if $\operatorname{injdim}({}_A A)$ and $\operatorname{injdim}(A_A)$ are finite; Zaks showed these integers then coincide, so one writes $\operatorname{injdim}(A)=d$. Following Levasseur (after Artin–Schelter), $A$ is Artin–Schelter Gorenstein (AS-Gorenstein) if $\operatorname{injdim}(A)=d<\infty$ and, for every finite-dimensional simple module $V$, $\operatorname{Ext}^i_A(V,A)=0$ for $i\ne d$ while $\operatorname{Ext}^d_A(V,A)$ is a nonzero finite-dimensional simple right $A$-module (and symmetrically with left and right interchanged); if moreover the global dimension is finite, $A$ is AS-regular. The grade of a module $M$ is $j(M)=\inf\{n:\operatorname{Ext}^n_A(M,A)\ne 0\}$; $A$ satisfies the Auslander condition if $j(N)\ge i$ for every submodule $N$ of $\operatorname{Ext}^i_A(M,A)$, and $A$ is Auslander–Gorenstein if it has finite injective dimension and all finitely generated modules satisfy the Auslander condition. For a commutative affine domain these conditions collapse to classical Gorenstein-ness. When $H$ is a Hopf algebra the AS-Gorenstein condition can be checked on the trivial (counit) module $K$ alone: by a lemma of Brown–Zhang it suffices that $\operatorname{Ext}^d_H(K,H)$ be one-dimensional, and this space $\int^l_H$ is called the homological left integral of $H$ (Lu–Wu–Zhang), generalizing integrals of finite-dimensional Hopf algebras.

A deeper reformulation uses dualizing complexes. For a noetherian algebra $A$ with enveloping algebra $A^e=A\otimes A^{\mathrm{op}}$, a dualizing complex is a bimodule complex $R$ inducing a duality between the bounded derived categories of finite left and right $A$-modules; $R$ is rigid (Van den Bergh) when

$$\operatorname{RHom}_{A^e}\left(A,\,R\otimes_A^{\mathbf{L}}R^{\mathrm{op}}\right)\cong A.$$

Brown–Zhang (2008) proved that a noetherian AS-Gorenstein Hopf algebra with bijective antipode has a rigid dualizing complex of the form $R\cong{}^{\nu}H^1[d]$, where ${}^{\nu}H^1$ is $H$ as a bimodule with left action twisted by the Nakayama automorphism (the twisting automorphism measured by the top self-Ext group) $\nu=S^2\circ\tau^\ell_\chi$, i.e. the antipode squared composed with a winding automorphism — an algebra automorphism defined by a character, here built from the homological integral; conversely such a complex forces AS-Gorenstein-ness. Reyes–Rogalski–Zhang and Lü–Oh–Wang–Yu completed this circle: for a noetherian Hopf algebra, AS-Gorenstein is equivalent to the Van den Bergh condition (existence of such a rigid dualizing complex), while twisted Calabi–Yau duality (a noncommutative analogue of Poincaré duality for the algebra, twisted by an automorphism) is equivalent to the stronger AS-regular property; AS-Gorenstein-ness implies that the antipode is bijective (a property conjectured by Skryabin for all right noetherian Hopf algebras and still open in general).

The driving question traces to Brown–Goodearl (1997, §1.14–1.15) and Brown (1998, Question A): is every noetherian (affine) Hopf algebra AS-Gorenstein and Auslander–Gorenstein? Affirmative answers cover: affine noetherian PI Hopf algebras (Wu–Zhang 2003, who proved Auslander–Gorenstein, AS-Gorenstein and GK-Cohen–Macaulay); enveloping algebras of finite-dimensional Lie algebras; quantised enveloping algebras; group algebras of polycyclic-by-finite groups; Hopf algebras with an ascending filtration whose associated graded ring is connected graded noetherian with enough normal elements — the filtration hypothesis $(F)$ of Lu–Wu–Zhang (2009), proved via rigid dualizing complexes; quotient Hopf algebras of generic quantised function algebras (Yakimov); connected Hopf algebras of finite Gelfand–Kirillov dimension over an algebraically closed field of characteristic zero (Zhuang); Hopf algebras that are connected graded as algebras (Brown–Gilmartin–Zhang); and noetherian cocommutative Hopf algebras of known type. (Gelfand–Kirillov dimension, GK-dimension, measures growth of finitely generated algebras; GK-Cohen–Macaulay is the duality $\operatorname{GKdim}(M)+j(M)=\operatorname{GKdim}(A)$.) No example of a noetherian Hopf algebra of infinite injective dimension is known, and no argument covers the general case.

Problem Statement

Let $K$ be a field and let $H$ be a noetherian Hopf $K$-algebra. Is $H$ necessarily Artin–Schelter Gorenstein and Auslander–Gorenstein? Concretely, must $\operatorname{injdim}({}_H H)=\operatorname{injdim}(H_H)=d<\infty$ with $\operatorname{Ext}^i_H(K,H)=0$ for $i\ne d$ and $\dim_K\operatorname{Ext}^d_H(K,H)=1$ (where $K$ is the trivial module via the counit), together with the Auslander condition for all finitely generated $H$-modules? By the equivalences established in the literature, the AS-Gorenstein half of the question is the same as asking whether every noetherian Hopf algebra satisfies the Van den Bergh condition, i.e. admits a rigid dualizing complex of the form ${}^{\nu}H^1[d]$ for an automorphism $\nu$ (then the Nakayama automorphism $\nu=S^2\circ\tau^\ell_\chi$); the Auslander condition is a separate requirement, not known to follow from the Van den Bergh condition. The historical form of the question (Brown–Goodearl 1997; Brown 1998; Brown–Zhang 2008) restricts to affine $H$; the unrestricted form subsumes it, since it is open whether every noetherian Hopf algebra is affine. The question is asked in full generality: no polynomial-identity, filtration, graded, connected, pointed, finite-GK-dimension, or cocommutativity hypothesis is assumed. A negative answer would be an explicit noetherian Hopf algebra of infinite injective dimension, of finite injective dimension failing the Ext condition, or satisfying these Gorenstein conditions yet failing the Auslander condition for some finitely generated module.

References

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