Terminal orders on arithmetic surfaces beyond prime degree: classification and étale-local structure

Chan and Ingalls (Algebra & Number Theory 18 (2024) 2027–2046) classified the étale-local structure of terminal orders on arithmetic surfaces only under restrictive hypotheses: the order has prime degree $m>5$ , and the residue field contains a primitive $m$ -th root of unity and has trivial Brauer group. Under these assumptions every terminal order is given by an explicitly constructed triangular matrix algebra over a symbol and has global dimension two. The open problem is to extend this local theory to the tame setting without these restrictions: classify the terminal localised Brauer classes for Brauer classes of composite order (and primes $m\le5$ ), and prove explicit structure theorems — models, Morita uniqueness, and the determination of the global dimension — over residue fields that need not contain the relevant roots of unity or have trivial Brauer group, e.g. finite fields $\mathbb{F}_q$ with $q\not\equiv 1\pmod m$ .

Background

An arithmetic surface, in the sense used here, is a normal excellent two-dimensional scheme with finite residue fields, such as the spectrum of the ring of integers of a number field or a curve over a finite field; its étale-local model at a closed point is a Hensel local domain (a local ring satisfying Hensel's lemma, so that factorisations of polynomials over the residue field lift to factorisations over the ring itself). Let $R$ be such a local ring with fraction field $K$ . An $R$ -order is a subring $\Lambda$ of a central simple $K$ -algebra $Q$ (a finite-dimensional noncommutative analogue of a field extension, classified up to Morita equivalence by its Brauer class $\beta\in\mathrm{Br}\,K$ , where $\mathrm{Br}\,K$ is the Brauer group of $K$ ) that is a finitely generated $R$ -module with $K\Lambda=Q$ ; it plays the role of a model of $Q$ over $R$ , and a maximal order is one maximal under inclusion. An order is Azumaya at a prime if it is there a twisted form of a matrix algebra; the failure is measured by ramification maps $a_{\mathfrak{p}}:\mathrm{Br}\,K\to H^1_{\mathrm{\acute{e}t}}(\kappa(\mathfrak{p}),\mathbb{Q}/\mathbb{Z})$ at codimension-one primes $\mathfrak{p}$ , where $\kappa(\mathfrak{p})$ is the residue field. Since maximality is not stable under étale localisation, Chan–Ingalls work with normal orders: reflexive $R$ -orders whose localisation at each height-one prime has a principal radical, and to these one attaches a localised Brauer class $(\beta,g_{\mathfrak{p}})$ , where the integers $g_{\mathfrak{p}}$ record how far the order is from maximal at $\mathfrak{p}$ . The minimal model program (MMP) for orders, initiated by Chan–Ingalls (2005), enriches the surface $X$ by $\beta$ through a canonical divisor $K_{X,\beta}=K_X+\sum_C(1-1/e_C)C$ , where $C$ runs over the ramified curves and $e_C$ is the order of the ramification; discrepancies and hence the notions of terminal and canonical order then follow the commutative MMP, replacing the usual discrepancy by the b-discrepancy. In the commutative case a terminal surface is smooth, so terminal orders are the noncommutative analogue of smooth points, and one expects them to be homologically regular, i.e. to have global dimension two. Over an algebraically closed residue field the picture is complete (Chan–Ingalls 2005, with canonical and klt orders treated by Chan–Hacking–Ingalls 2009): every terminal order is Morita equivalent to a matrix algebra over a symbol algebra $R\langle y,z\rangle/(y^m-u,z^m-v,zy-\zeta yz)$ , where $\zeta$ is a primitive $m$ -th root of unity and $u,v$ form a regular system of parameters, or to a triangular modulo- $z$ matrix algebra over such symbols. Over finite residue fields new phenomena appear: ramification on a single multiplicity-one curve, and centres that are Hirzebruch–Jung singularities (cyclic quotient surface singularities, characterised by a minimal resolution whose exceptional locus is a chain of projective lines over the residue field). Chan–Ingalls (arXiv:2108.03105, published in Journal of Algebra 676 (2025) 475–511) developed the MMP for this arithmetic setting and classified the terminal localised Brauer classes, but only for Brauer classes of prime index $p>5$ with order prime to the residue characteristics (the tame assumption). Building on this, Chan–Ingalls (Algebra & Number Theory 18 (2024) 2027–2046, Theorem 1.1) proves the structure theorem: if $R$ is an excellent noetherian two-dimensional normal Hensel local domain, $\Lambda$ is a terminal $R$ -order of prime degree $m>5$ , the residue field $\kappa$ contains a primitive $m$ -th root of unity, and $\mathrm{Br}\,\kappa=0$ , then $\Lambda$ has global dimension two, and all such $\Lambda$ are explicitly constructed as triangular modulo- $z$ matrix algebras over symbols, with Morita uniqueness: any normal order with the same ramification data is Morita equivalent to the constructed model. What the theorem does not cover is precisely the generic arithmetic situation: composite degree, the primes $m\le5$ , and residue fields such as $\mathbb{F}_q$ with $q\not\equiv 1\pmod m$ , which contain no primitive $m$ -th root of unity, as well as residue fields with nontrivial Brauer group.

Problem Statement

Keep the tame framework of Chan–Ingalls (2024) (the order of the Brauer class is prime to the residue characteristic) and let $R$ be an excellent noetherian normal two-dimensional Hensel local domain with fraction field $K$ and residue field $\kappa$ ; the motivating arithmetic setting has $\kappa$ finite. Chan–Ingalls (2024, Theorem 1.1) proved: if $\Lambda$ is a terminal $R$ -order of prime degree $m>5$ , $\kappa$ contains a primitive $m$ -th root of unity, and $\mathrm{Br}\,\kappa=0$ , then $\Lambda$ has global dimension two and every such $\Lambda$ is Morita equivalent to an explicitly constructed triangular modulo- $z$ matrix algebra over a symbol; the terminal localised Brauer classes $(\beta,g_{\mathfrak{p}})$ entering this theorem were classified for prime index $p>5$ by Chan–Ingalls (Journal of Algebra 676 (2025) 475–511). Determine the corresponding theory when the restrictions on degree and residue field are removed. Specifically: (i) classify the terminal localised Brauer classes $(\beta,g_{\mathfrak{p}})$ on such $R$ for Brauer classes of arbitrary order $m$ prime to the residue characteristic, including composite $m$ and the primes $m\le5$ , extending the prime $p>5$ classification; and (ii) for each such terminal class, construct explicit étale-local models of the corresponding terminal orders, prove Morita uniqueness (any normal order with the same ramification data is Morita equivalent to the constructed model), and decide whether these orders have global dimension two — in particular over residue fields that need not contain a primitive $m$ -th root of unity (such as $\mathbb{F}_q$ with $q\not\equiv1\pmod m$ ) and may have nontrivial Brauer group. The common objective is a structure theorem of the same structural form as the 2024 Theorem 1.1 (explicit models and Morita uniqueness), valid without its prime-degree, root-of-unity, and trivial-Brauer hypotheses, together with the determination of the global dimension.

References

Daniel Chan and Colin Ingalls, Terminal orders on arithmetic surfaces, Algebra & Number Theory 18 (2024), 2027–2046. DOI: 10.2140/ant.2024.18.2027 (arXiv:2211.05948). https://doi.org/10.2140/ant.2024.18.2027
Daniel Chan and Colin Ingalls, The minimal model program for arithmetic surfaces enriched by a Brauer class, Journal of Algebra 676 (2025), 475–511. DOI: 10.1016/j.jalgebra.2025.03.047 (arXiv:2108.03105). https://doi.org/10.1016/j.jalgebra.2025.03.047
Daniel Chan and Colin Ingalls, The minimal model program for orders over surfaces, Inventiones mathematicae 161 (2005), no. 2, 427–452. DOI: 10.1007/s00222-005-0438-z. https://doi.org/10.1007/s00222-005-0438-z
Daniel Chan, Paul Hacking, and Colin Ingalls, Canonical singularities of orders over surfaces, Proceedings of the London Mathematical Society 98 (2009), no. 1, 83–115. DOI: 10.1112/plms/pdn025. https://doi.org/10.1112/plms/pdn025
Daniel Chan and Sean B. Lynch, Zeta functions of orders on surfaces, The Quarterly Journal of Mathematics 76 (2025), 589–607. DOI: 10.1093/qmath/haaf020 (arXiv:2410.16832). https://doi.org/10.1093/qmath/haaf020
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Daniel Chan and Colin Ingalls, Some terminal orders on 3-folds, arXiv:2607.02950 (2026). https://arxiv.org/abs/2607.02950
Daniel Larsson, Arithmetic Geometry of Non-Commutative Spaces with Large Centres, Annales mathématiques du Québec (2026). DOI: 10.1007/s40316-026-00272-3 (arXiv:2308.14579). https://doi.org/10.1007/s40316-026-00272-3