Zariski cancellation for Sklyanin algebras of dimension at least three (the PI and four-dimensional cases)

The Zariski cancellation problem for a $k$-algebra $A$ asks whether an isomorphism $A[t]\cong B[t]$ forces $A\cong B$. Lezama, Wang and Zhang (Math. Z. 2018), developing cancellation theory for noncommutative algebras that need not be domains, posed as Question 5.4(3) the problem of proving or disproving that the Sklyanin algebras of dimension at least three — in particular the PI (polynomial identity) Sklyanin algebras — are cancellative. Later work of Tang–Venegas Ramírez–Zhang (Pacific J. Math. 2021) and Lu–Wu–Zhang (Canad. J. Math. 2020) resolved the non-PI three-dimensional case affirmatively in characteristic zero, but the PI three-dimensional Sklyanin algebras and the four-dimensional Sklyanin algebras (PI and non-PI) remain unsettled: every known cancellation criterion requires a center of Gelfand–Kirillov dimension at most one, an artinian center, or the non-PI hypothesis in global dimension three. The problem remains open as of 2026, with this precise surviving core.

Background

The Zariski cancellation problem (ZCP) asks whether an algebra is determined by its polynomial extension. An algebra $A$ over a field $k$ is called cancellative if every $k$-algebra isomorphism $A[t]\cong B[t]$, for another $k$-algebra $B$, forces $A\cong B$; it is strongly cancellative if the same conclusion holds from an isomorphism $A[t_1,\dots,t_n]\cong B[t_1,\dots,t_n]$ with several commuting indeterminates. In the commutative case, polynomial rings in one and two variables are cancellative, Gupta constructed counterexamples for $k[x_1,\dots,x_n]$ with $n\geq 3$ in positive characteristic, and the characteristic-zero case for $n\geq 3$ remains open. Bell and Zhang re-introduced the problem for noncommutative algebras, proving for instance that the first Weyl algebra in characteristic zero is cancellative; their method controls the Makar–Limanov invariant, the subalgebra generated by the kernels of all locally nilpotent derivations (a derivation $d$ is locally nilpotent if every element is killed by some power of $d$).

Lezama, Wang and Zhang developed the theory for algebras that need not be domains. Their tools are retractability and detectability of the center $Z(A)$ — conditions that allow one to pull an isomorphism $A[t]\cong B[t]$ back to an isomorphism of the centers — together with the theorem that a strongly Hopfian algebra (one all of whose polynomial extensions are Hopfian, i.e. every surjective self-map is injective) with artinian center is cancellative. In the closing section of that paper the authors ask which important families of noncommutative algebras are cancellative; Question 5.4(3) names 'the Sklyanin algebras of dimension $\geq 3$, or the PI Sklyanin algebras of dimension $\geq 3$'.

Sklyanin algebras are the central examples of Artin–Schelter regular algebras: connected graded algebras ($A=\bigoplus_{i\geq 0}A_i$ with $A_0=k$) of finite global dimension satisfying a Poincaré-duality-type condition on Ext groups, which play the role of noncommutative polynomial rings in noncommutative projective geometry. The three-dimensional Sklyanin algebra $S(a,b,c)$ is generated by $x,y,z$ subject to three quadratic relations parametrized by a point $[a:b:c]$ of $\mathbb{P}^2$ off the diagonals; its geometry is encoded by a triple $(E,L,\sigma)$ with $E$ an elliptic curve and $\sigma$ a translation automorphism of $E$, and it is a noetherian domain of global dimension $3$. The four-dimensional Sklyanin algebras, on four generators with six quadratic relations, are noetherian domains of global dimension $4$. A Sklyanin algebra satisfies a polynomial identity (is PI) exactly when $\sigma$ has finite order $n$; in that case it is a finite module over its center and has PI degree $n$ — the square root of the maximal dimension of its irreducible representations — while for infinite order it is not PI. The centers are known: a non-PI three-dimensional Sklyanin algebra has center $k[g]$ with $g$ central of degree $3$ (Artin–Schelter–Tate); a non-PI four-dimensional one has center $k[g_1,g_2]$, a polynomial ring on two central elements of degree $2$; a PI four-dimensional one has center $k[z_0,z_1,z_2,z_3,g_1,g_2]/(F_1,F_2)$ of Krull dimension $4$ (see Walton–Wang–Yakimov). Gelfand–Kirillov dimension (GK dimension) is the standard growth measure of a finitely generated algebra.

After the source paper appeared, part of Question 5.4(3) was resolved: Tang–Venegas Ramírez–Zhang proved that every noetherian connected graded Artin–Schelter regular algebra of global dimension three, generated in degree one, that is not PI, is cancellative over a characteristic-zero field — this covers all non-PI three-dimensional Sklyanin algebras — and Lu–Wu–Zhang independently proved these algebras are strongly cancellative and Morita cancellative (cancellable already at the level of module categories). What remains open is exactly the regime the known criteria cannot reach. The center-based theorems require $\operatorname{GKdim} Z(A)\leq 1$ or an artinian center: PI Sklyanin algebras are module-finite over centers of GK dimension at least $3$, and the four-dimensional Sklyanin algebras have centers of GK dimension $2$ (non-PI) or $4$ (PI). The other main tool, the discriminant criterion, requires an effective discriminant (the discriminant is a canonical central element whose vanishing locates the central characters at which the irreducible representations degenerate; it is effective when it determines the isomorphism type of the pair $(A,Z(A))$) together with representation-theoretic trace maps available only in characteristic zero, and no effective discriminant has been established for PI Sklyanin algebras; their Poisson structures and Azumaya loci (the Azumaya locus is the set of central characters at which the algebra is a full matrix algebra over the residue field, its best-behaved representation-theoretic region) have been computed, but this infrastructure has not been converted into a cancellation criterion.

Problem Statement

Let $k$ be a field and let $A$ be a Sklyanin algebra of dimension at least three over $k$ — concretely, a three-dimensional or four-dimensional Sklyanin algebra as described in the background — including in particular the PI Sklyanin algebras, those for which the associated automorphism $\sigma$ of the elliptic curve has finite order.

Prove or disprove that $A$ is cancellative, i.e. that

$A[t]\cong B[t]\ \Rightarrow\ A\cong B$

for every $k$-algebra $B$, all isomorphisms being isomorphisms of $k$-algebras.

A complete answer settles each of the three cases — PI three-dimensional Sklyanin algebras, PI four-dimensional Sklyanin algebras, and non-PI four-dimensional Sklyanin algebras — either by proving the algebras in that case cancellative (a proof of strong cancellativity is stronger and also accepted) or by exhibiting an explicit pair of algebras $A$ and $B$, with $A$ in that case, together with an isomorphism $A[t]\cong B[t]$ and a proof that $A\not\cong B$. These three cases are to be settled over the standard setting, an algebraically closed base field of characteristic zero, where the known partial results hold; results over other fields, in particular of positive characteristic, are admissible as additional, field-specific contributions but do not by themselves settle a case, and such settings are currently unexplored. The non-PI three-dimensional characteristic-zero case, contained in the source question, was resolved affirmatively by later work (see the final paragraph of Background) and need not be re-proved.

References

1. O. Lezama, Y.-H. Wang, J. J. Zhang, Zariski cancellation problem for non-domain noncommutative algebras, Mathematische Zeitschrift 292 (2018), no. 3-4, 1269-1290. DOI: 10.1007/s00209-018-2153-7. arXiv:1711.08071, https://arxiv.org/abs/1711.08071 (source of Question 5.4(3))
2. Xin Tang, Helbert J. Venegas Ramírez, James J. Zhang, Cancellation problem for AS-regular algebras of dimension three, Pacific Journal of Mathematics 312 (2021), no. 1, 233-256. DOI: 10.2140/pjm.2021.312.233. arXiv:1904.07281, https://arxiv.org/abs/1904.07281
3. D.-M. Lu, Q.-S. Wu, J. J. Zhang, A Morita Cancellation Problem, Canadian Journal of Mathematics 72 (2020), no. 3, 708-731. DOI: 10.4153/s0008414x1900004x. arXiv:1804.01888, https://arxiv.org/abs/1804.01888
4. Hongdi Huang, Xin Tang, Xingting Wang, A survey on Zariski cancellation problems for noncommutative and Poisson algebras, Contemporary Mathematics 801 (2024), 125-141. DOI: 10.1090/conm/801/16085. arXiv:2304.05914, https://arxiv.org/abs/2304.05914
5. Kenneth Chan, Jason Gaddis, Robert Won, James J. Zhang, Ozone groups of Artin-Schelter regular algebras satisfying a polynomial identity, Mathematische Zeitschrift 310 (2025), no. 4, Paper No. 69. DOI: 10.1007/s00209-025-03777-9. arXiv:2312.13014, https://arxiv.org/abs/2312.13014
6. Chelsea Walton, Xingting Wang, Milen Yakimov, Poisson geometry and representations of PI 4-dimensional Sklyanin algebras, Selecta Mathematica (N.S.) 27 (2021), no. 5, Paper No. 99. DOI: 10.1007/s00029-021-00713-7. arXiv:1802.06487, https://arxiv.org/abs/1802.06487
7. Chelsea Walton, Xingting Wang, Milen Yakimov, Poisson geometry of PI three-dimensional Sklyanin algebras, Proceedings of the London Mathematical Society 118 (2019), no. 6, 1471-1500. DOI: 10.1112/plms.12220, https://doi.org/10.1112/plms.12220
8. Michael Artin, William Schelter, John Tate, The centers of 3-dimensional Sklyanin algebras, in: Barsotti Symposium in Algebraic Geometry, Academic Press, San Diego, 1994, pp. 1-10. DOI: 10.1016/b978-0-12-197270-7.50006-1, https://doi.org/10.1016/b978-0-12-197270-7.50006-1
9. Jason Bell, James J. Zhang, Zariski cancellation problem for noncommutative algebras, Selecta Mathematica (N.S.) 23 (2017), no. 3, 1709-1737. DOI: 10.1007/s00029-017-0317-7. arXiv:1601.04625, https://arxiv.org/abs/1601.04625
10. Jason Bell, Maryam Hamidizadeh, Hongdi Huang, Helbert Venegas, Noncommutative analogues of a cancellation theorem of Abhyankar, Eakin, and Heinzer, Beiträge zur Algebra und Geometrie / Contributions to Algebra and Geometry 62 (2021), no. 2, 295-315. DOI: 10.1007/s13366-020-00518-3. arXiv:1909.04023, https://arxiv.org/abs/1909.04023
11. J.-F. Lü, X.-F. Mao, J. J. Zhang, Nakayama automorphism and applications, Transactions of the American Mathematical Society 369 (2017), no. 4, 2425-2460. DOI: 10.1090/tran/6718. arXiv:1408.5761, https://arxiv.org/abs/1408.5761
12. Jesse Levitt, Milen Yakimov, Quantized Weyl algebras at roots of unity, Israel Journal of Mathematics 225 (2018), no. 2, 681-719. DOI: 10.1007/s11856-018-1675-3. arXiv:1606.02121, https://arxiv.org/abs/1606.02121
13. Neena Gupta, On Zariski's Cancellation Problem in positive characteristic, Advances in Mathematics 264 (2014), 296-307. DOI: 10.1016/j.aim.2014.07.012, https://doi.org/10.1016/j.aim.2014.07.012
14. César F. Venegas R., Helbert J. Venegas R., Cancellation problem via locally nilpotent derivations, arXiv:2512.15590 (2025), https://arxiv.org/abs/2512.15590
15. Hongdi Huang, Zahra Nazemian, Yanhua Wang, James J. Zhang, Relative Cancellation, arXiv:2503.10083 (2025), https://arxiv.org/abs/2503.10083