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Restricted Steinberg modules for general linear groups admit explicit unimodular generators and Bykovskii presentations, and compute the relative homology of partial Borel–Serre compactifications.

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2026-07-12 00:58 UTC pith:JGQAXOPX

load-bearing objection Solid technical paper that cleanly extends classical Steinberg presentations to local restrictions and ties them to partial Borel–Serre homology; useful for the author’s cocycle program and for anyone working with modular symbols beyond GL2.

arxiv 2607.03642 v1 pith:JGQAXOPX submitted 2026-07-03 math.RT math.KTmath.NT

Presenting restricted Steinberg modules of general linear groups

Peter Xu This is my paper
classification math.RT math.KTmath.NT MSC 20G1011F7555U10
keywords Steinberg modulesTits buildingsBorel–Serre compactificationmodular symbolsmatroidsOrlik–Solomon algebraBykovskii presentationarithmetic groups
verification ladder T0 review T1 audit T2 compute T3 formal T4 reserved

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper supplies generators and relations for local restrictions of Steinberg modules of GL_n over a field. These restricted modules are the top homology of Tits buildings that avoid a prescribed subspace modulo a prime ideal (or a pair of such conditions). The same modules are shown to be generated by unimodular apartment classes and to admit Bykovskii-type quadratic presentations whenever the residue characteristic is not 2. The geometric payoff is that the coinvariants of these modules recover the relative homology of the corresponding partial Borel–Serre compactifications of the locally symmetric space. The constructions extend the classical modular-symbol formalism from GL_2(Q) to higher rank and to number fields, and they clarify that circuit presentations and Bykovskii presentations are equivalent once homogeneity is imposed.

Core claim

For a torsion-free arithmetic subgroup stabilizing local data S and T of residue characteristic not 2, the relative homology of the partially compactified locally symmetric space is canonically the coinvariants of the restricted Steinberg module. That module is generated by R-unimodular apartment symbols subject to antisymmetry and the Bykovskii (equivalently circuit) relations.

What carries the argument

The restricted Tits buildings S T_n(F) and T T_n(F) (and their doubly restricted variants), together with their Whitney-homology resolutions and the duality involution that interchanges upward and downward restrictions; these supply both the connectivity needed for partial compactifications and the matroidal presentations that produce the generators and relations.

Load-bearing premise

The residue characteristic of the ideals used to impose the local restrictions must not be 2; otherwise the quadratic Bykovskii relations need not generate all circuit relations.

What would settle it

Compute the restricted Steinberg module for n=3 over F_2 with a concrete subspace S that admits no length-3 circuits; if its top homology is not spanned by the symbols that satisfy only the Bykovskii relations, the presentation claim fails in characteristic 2.

Watch this falsifier — get emailed when new claim-graph text bears on it.

If this is right

  • Partial modular symbols with local restrictions become available for arithmetic subgroups of GL_n over number fields, not merely for GL_2(Q).
  • Relative homology of partial Borel–Serre compactifications is computable from explicit generators and quadratic relations once residue characteristic is odd.
  • The equivalence of circuit and Bykovskii presentations holds for all singly-restricted Steinberg modules over fields of odd residue characteristic.
  • Koszulity of the Steinberg VB-algebra up to degree 2 is equivalent to the existence of quadratic presentations for the unrestricted modules.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The same restricted modules can be used as coefficient systems for explicit Eisenstein cocycles valued in motivic cohomology of elliptic schemes, as foreshadowed by the author’s related work.
  • Function-field analogues via partial compactifications of Bruhat–Tits buildings would give geometric interpretations of Drinfeld modular symbols free of p-power torsion discrepancies.
  • When the residue characteristic is 2 the full circuit presentation remains valid, so one may still compute the modules by retaining higher-arity relations.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit.

Referee Report

0 major / 5 minor

Summary. The paper constructs generators and relations for restricted Steinberg modules of GL_n over fields (and certain rings), obtained by imposing local linear-algebraic conditions at finitely many primes. These modules are identified with the relative homology of partial Borel–Serre compactifications of the associated locally symmetric spaces for arithmetic subgroups that stabilize the local data (Theorem 1.2). The main technical results are Cohen–Macaulay connectivity of the restricted Tits buildings (Propositions 3.5, 3.7), generation by R-unimodular apartment classes under mild hypotheses on R (Proposition 3.8), and the equivalence of circuit-type and Bykovskii-type presentations for the restricted modules when the residue characteristic is not 2 (Propositions 3.10–3.11). An appendix relates these presentations to Koszulity of the Steinberg VB-algebra.

Significance. The work supplies a clean, usable generators-and-relations package for the restricted Steinberg modules that appear in the author’s (and collaborators’) constructions of Eisenstein cocycles and motivic modular symbols. Extending the classical modular-symbol formalism of Ash–Rudolph and Manin from GL_2(Q) to higher rank and to partial compactifications is a natural and useful step; the careful comparison of circuit versus Bykovskii presentations and the isolation of the characteristic-2 obstruction are genuine clarifications. The geometric identification with partial Borel–Serre homology is standard but correctly executed, and the appendix places the results in the modern language of VB-algebras and Koszulity. The paper is therefore a solid foundational contribution that will be cited by anyone working with these modules.

minor comments (5)
  1. [Theorem 1.2 / §3.4] Definition 3.4 is cited in Theorem 1.2 and in the abstract but never appears as a numbered definition; the generators and relations are only described informally in the theorem statement and later in §3.4. A single numbered definition collecting the restricted generators and the two families of relations would improve readability.
  2. [Proposition 3.11] In the statement of Proposition 3.11 the second bullet still refers to “containing S mod I” while describing the downward-restricted module TSt; this appears to be a slip for T.
  3. [Proposition 3.8] The phrase “or two non-archimedean places” in Proposition 3.8 is slightly ambiguous; the intended meaning is presumably “a Dedekind domain with at least two non-archimedean places that are inverted or real.” A short clarification would help.
  4. [Throughout] Several typographical inconsistencies appear (e.g., “Bykowskii” vs. “Bykovskii”, “homologicallyCohen–Macaulay”, missing spaces after commas in displayed maps). A light copy-edit pass would remove them.
  5. [Appendix A / Introduction] The appendix is interesting but somewhat loosely connected to the main theorems; a sentence or two in the introduction explaining why the Koszulity perspective is included would orient the reader.

Circularity Check

0 steps flagged

No circularity: generators/presentations and partial-compactification identification are derived from external connectivity and matroid results, not from self-referential inputs.

full rationale

The paper is a pure-mathematics generators-and-relations note. Theorem 1.2 rests on two independent pillars proved in the text: (i) the spectral-sequence identification of relative homology of the partial Borel–Serre compactification with Γ-coinvariants of the restricted Steinberg module (Corollary 4.3 + Hochschild–Serre, §4.1), which is the standard Borel–Serre argument applied to a downward-closed subposet of parabolics; and (ii) generation by R-unimodular apartments (Proposition 3.8, bootstrapped from Scalamandre via the Hatcher–Vogtmann deletion lemma) together with the circuit/Bykovskii presentation of the restricted modules (Propositions 3.10–3.11, obtained from Björner’s Cohen–Macaulayness of matroid order complexes and an explicit inductive reduction of circuit relations that isolates the characteristic-2 obstruction). All load-bearing citations (Solomon–Tits, Orlik–Solomon, Björner, Scalamandre, Church–Putman, Borel–Serre) are external. The author’s own forthcoming papers appear only in the motivation paragraph and are not invoked in any proof. There are no fitted parameters, no uniqueness theorems imported from the author, and no renaming of known empirical patterns. The derivation is therefore self-contained against its stated external inputs; ordinary literature dependence does not constitute circularity.

Axiom & Free-Parameter Ledger

0 free parameters · 5 axioms · 1 invented entities

The work is pure algebra/topology; it imports standard theorems on buildings, matroids and Borel–Serre compactifications and introduces only the restricted complexes and modules needed for the statements. No free parameters or physical entities appear.

axioms (5)
  • standard math Tits buildings of GL_n over quotients/localizations of Dedekind domains are Cohen–Macaulay (Solomon–Tits / Björner / Scalamandre).
    Invoked as Theorem 3.1; used to obtain connectivity of unrestricted and restricted buildings.
  • standard math Orlik–Solomon algebra of a matroid is isomorphic to the top homology of its lattice of flats via the apartment map (Orlik–Solomon / Björner).
    Proposition 3.3; supplies circuit presentations for matroidal restricted modules.
  • standard math Borel–Serre compactification of the symmetric space is homotopy-equivalent to the Tits building, and partial compactifications inherit the homotopy type of the corresponding subcomplex.
    Theorems 4.1–4.2 and Corollary 4.3; used for the relative-homology identification in Theorem 1.2.
  • domain assumption Residue characteristic of R/I (resp. R/J) is not 2.
    Required in Proposition 3.11 so that pairwise sums can be reordered to avoid S; fails in characteristic 2 (Remark 3.12).
  • domain assumption R is a PID that is semi-local or a Dedekind domain with a real place (or two non-archimedean places).
    Hypothesis of Proposition 3.8 guaranteeing generation by R-integral apartment classes.
invented entities (1)
  • Restricted Tits complexes S T_n(R), T T_n(R) and their top homologies (restricted Steinberg modules) no independent evidence
    purpose: Encode local linear-algebraic conditions at finite places so that partial compactifications and restricted modular symbols can be defined uniformly.
    Defined in §3.1.1; the entire paper studies their generators, relations and geometric meaning. They are natural subcomplexes rather than free inventions, but are new as named objects.

pith-pipeline@v1.1.0-grok45 · 23192 in / 2962 out tokens · 20462 ms · 2026-07-12T00:58:30.951567+00:00 · methodology

0 comments
read the original abstract

We give generators and presentations for various local restrictions of Steinberg modules over fields and relate them to \emph{partial} Borel--Serre compactifications of locally symmetric spaces in the case of number fields, extending the existing theory of partial modular symbols for $\mathrm{GL}_2(\Q)$. Along the way, we clarify the relationship between ``circuit'' and ``Bykovskii''-type presentations for such modules. In an appendix, we relate the existence of such presentations to Koszulity properties of the Steinberg $\mathrm{VB}$-algebra.

discussion (0)

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Works this paper leans on

159 extracted references · 2 linked inside Pith

  1. [1]

    Aravind Asok and Frédéric Déglise and Jan Nagel , title=

  2. [2]

    Romyar Sharifi and Peter Xu , title=

  3. [3]

    Ahmed Abbes , title=

  4. [4]

    Inventiones mathematicae

    Avner Ash and Lee Rudolph , title=. Inventiones mathematicae. , volume = 55, year = 1979, pages =

    1979

  5. [5]

    Nicolas Bergeron and Pierre Charollois and Luis Garcia , title=

  6. [6]

    Eisenstein cohomology classes for

    Bergeron, Nicolas and Charollois, Pierre and Garc. Eisenstein cohomology classes for. Journal f

  7. [7]

    Lea Beneish and Henri Darmon and Lennart Gehrmann , title=

  8. [8]

    Automorphic Forms and Galois Representations , series =

    Bertolini, Massimo and Castella, Francesc and Darmon, Henri and Dasgupta, Samit and Prasanna, Kartik and Rotger, Victor , title =. Automorphic Forms and Galois Representations , series =. 2014 , pages =

    2014

  9. [9]

    arXiv preprint arXiv:2508.09915 , year =

    Bachmann, Tom and Elmanto, Elden and Morrow, Matthew , title =. arXiv preprint arXiv:2508.09915 , year =

    Pith/arXiv arXiv

  10. [10]

    Borisov and Paul E

    Lev A. Borisov and Paul E. Gunnells , title=. Journal für die reine und angewandte Mathematik , volume = 539, year = 2001, pages =

    2001

  11. [11]

    Communications in Mathematical Physics

    Jean-Michel Bismut and Henri Gillet and Christophe Soulé , title=. Communications in Mathematical Physics. , volume = 115, number = 1, year = 1988, pages =

    1988

  12. [12]

    Mathematische Annalen

    Siegfried Bosch and Werner Lütkebohmert , title=. Mathematische Annalen. , volume=295, pages=

  13. [13]

    Transactions of the American Mathematical Society

    Kenneth Brown , title=. Transactions of the American Mathematical Society. , volume=186, pages=

  14. [14]

    Mathematische Annalen , volume = 371, year = 2018, pages =

    Alexander Beilinson and Guido Kings and Andrey Levin , title=. Mathematische Annalen , volume = 371, year = 2018, pages =

    2018

  15. [15]

    François Brunault and Michael Neururer , title=

  16. [16]

    Matroid Applications

    Anders Bj\"orner , title=. Matroid Applications

  17. [17]

    Journal of Algebraic Geometry

    Spencer Bloch , title=. Journal of Algebraic Geometry. , volume=3, year=1994, pages=

    1994

  18. [18]

    Advances in Mathematics

    Spencer Bloch , title=. Advances in Mathematics. , volume=61, year=1986, pages=

    1986

  19. [19]

    Lecture Notes in Mathematics

    Spencer Bloch , title=. Lecture Notes in Mathematics. , volume=341, year=1973, pages=

    1973

  20. [20]

    Duke Mathematical Journal , volume=97, year=1999, pages=

    Richard Borcherds , title=. Duke Mathematical Journal , volume=97, year=1999, pages=

    1999

  21. [21]

    Quantum Field Theory: Perspective and Prospective

    Raoul Bott , title=. Quantum Field Theory: Perspective and Prospective. , publisher=

  22. [22]

    Acta Arithmetica , volume=134, year=2008, pages=

    François Brunault , title=. Acta Arithmetica , volume=134, year=2008, pages=

    2008

  23. [23]

    2020 , url =

    Brunault, François , title =. 2020 , url =

    2020

  24. [24]

    Raoul Bott and Loring Tu , title=

  25. [25]

    Algebraic K -theory II

    Hyman Bass and John Tate , title=. Algebraic K -theory II. , publisher=

  26. [26]

    Mathematische Annalen , volume=372, year=2018, pages=

    Bryden Cais , title=. Mathematische Annalen , volume=372, year=2018, pages=

    2018

  27. [27]

    Compositio Mathematica , volume=154, year=2018, pages=

    Bryden Cais , title=. Compositio Mathematica , volume=154, year=2018, pages=

    2018

  28. [28]

    Colloque tenu à Bruxelles , year=1953, pages=

    Henri Cartan , title=. Colloque tenu à Bruxelles , year=1953, pages=

    1953

  29. [29]

    Séminaire Bourbaki : volume 2002/2003, exposés 909-923 , series =

    Colmez, Pierre , title =. Séminaire Bourbaki : volume 2002/2003, exposés 909-923 , series =. 2004 , mrnumber =

    2002

  30. [30]

    Functional Analysis and its Applications , volume=10, year=1976, pages=

    Vladimir Drinfeld , title=. Functional Analysis and its Applications , volume=10, year=1976, pages=

    1976

  31. [31]

    Clément Dupont , title =

  32. [32]

    Séminaire Bourbaki , year=1971, number=13, pages=

    Pierre Deligne , title=. Séminaire Bourbaki , year=1971, number=13, pages=

    1971

  33. [33]

    Annals of Mathematics , volume=174, year=2011, pages=

    Henri Darmon and Samit Dasgupta and Robert Pollack , title=. Annals of Mathematics , volume=174, year=2011, pages=

    2011

  34. [34]

    Journal fur reine und angewandte Mathematik

    Christopher Deninger and Jaap Murre , title=. Journal fur reine und angewandte Mathematik. , volume=422, year=1991, pages=

    1991

  35. [35]

    Mathematische Annalen , volume=379, year=2021, pages=

    Henri Darmon and Alice Pozzi and Jan Vonk , title=. Mathematische Annalen , volume=379, year=2021, pages=

    2021

  36. [36]

    Journal of the European Mathematical Society , volume=26, number=10, pages=

    Henri Darmon and Alice Pozzi and Jan Vonk , title=. Journal of the European Mathematical Society , volume=26, number=10, pages=

  37. [37]

    Henri Darmon and Victor Rotger , title=

  38. [38]

    Duke Math Journal , volume=170, number=1, pages=

    Henri Darmon and Jan Vonk , title=. Duke Math Journal , volume=170, number=1, pages=

  39. [39]

    Henri Darmon and Jan Vonk , title=

  40. [40]

    Jean Dieudonné and Alexander Grothendieck , title=

  41. [41]

    Samuel Eilenberg , title=

  42. [42]

    Samuel Eilenberg and Saunders MacLane , title=

  43. [43]

    Kyoto Journal of Mathematics , volume=64, number=2, pages=

    Takako Fukaya and Kazuya Kato , title=. Kyoto Journal of Mathematics , volume=64, number=2, pages=

  44. [44]

    Annales Scientifiques de l'École Normale Supérieure

    Eric Friedlander and Andre Suslin , title=. Annales Scientifiques de l'École Normale Supérieure. , volume=4, year=2002, pages=

    2002

  45. [45]

    Alexander Goncharov , title=

  46. [46]

    Geometric and Functional Analysis , volume=17, year=2008, pages=

    Alexander Goncharov , title=. Geometric and Functional Analysis , volume=17, year=2008, pages=

    2008

  47. [47]

    Journal of the American Mathematical Society , volume=18, year=2005, pages=

    Alexander Goncharov , title=. Journal of the American Mathematical Society , volume=18, year=2005, pages=

    2005

  48. [48]

    Mathematische Zeitschrift

    Thomas Geisser , title=. Mathematische Zeitschrift. , volume = 248, year = 2004, pages =

    2004

  49. [49]

    Roger Godemont , title=

  50. [50]

    Inventiones mathematicae

    Thomas Geisser and Marc Levine , title=. Inventiones mathematicae. , volume = 139, year = 2000, pages =

    2000

  51. [51]

    Mathematische Annalen , volume=278, year=1987, pages=

    Benedict Gross and Winfried Kohnen and Don Zagier , title=. Mathematische Annalen , volume=278, year=1987, pages=

    1987

  52. [52]

    , title =

    Gross, Benedict H. , title =

  53. [53]

    de Shalit, Ehud , title =

  54. [54]

    Inventiones , volume=85, pages=

    Haruzo Hida , title=. Inventiones , volume=85, pages=

  55. [55]

    Journal fur die reine und angewandte Mathematik

    Helmut Hamm , title=. Journal fur die reine und angewandte Mathematik. , volume=338, pages=

  56. [56]

    Allen Hatcher , title=

  57. [57]

    Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg , year=1927, volume=5, pages=

    Erich Hecke , title=. Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg , year=1927, volume=5, pages=

    1927

  58. [58]

    Sigurdur Helgason , title=

  59. [59]

    Advanced Studies in Pure Mathematics , year=2002, volume=35, pages=

    Klaus Hulek and Gregory Sankaran , title=. Advanced Studies in Pure Mathematics , year=2002, volume=35, pages=

    2002

  60. [60]

    Automorphic forms, representations, and L -functions (Proceedings of Symposia in Pure Mathematics)

    Roger Howe , title=. Automorphic forms, representations, and L -functions (Proceedings of Symposia in Pure Mathematics). , volume = "XXXIII", number = 1, year = 1977, pages =

    1977

  61. [61]

    Annette Huber and Guido Kings , title=

  62. [62]

    Astérisque

    Kazuya Kato , title=. Astérisque. , volume = 295, year = 2004, pages =

    2004

  63. [63]

    Advanced Studies in Contemporary Mathematics

    Kazuya Kato , title=. Advanced Studies in Contemporary Mathematics. , volume = 1, year = 1999, pages =

    1999

  64. [64]

    K-Theory

    Matt Kerr , title=. K-Theory. , volume = 29, year = 2003, pages =

    2003

  65. [65]

    Moritz Kerz , title =

  66. [66]

    Mathematische Annalen

    Robion Kirby and Paul Melvin , title=. Mathematische Annalen. , volume = 299, year = 1994, pages =

    1994

  67. [67]

    Compositio Mathematica , volume=160, number=10, pages=

    Romyar Sharifi and Akshay Venkatesh , title=. Compositio Mathematica , volume=160, number=10, pages=

  68. [68]

    1981 , isbn =

    Modular Units , author =. 1981 , isbn =

    1981

  69. [69]

    The Bloch–Kato Conjecture for the Riemann Zeta Function , editor =

    Guido Kings , title =. The Bloch–Kato Conjecture for the Riemann Zeta Function , editor =

  70. [70]

    Special Values of Rankin L series , year=2001, venue=

    Stephen Kudla , title=. Special Values of Rankin L series , year=2001, venue=

    2001

  71. [71]

    Steven Landsburg , title=

  72. [72]

    Astérisque

    Marc Levine , title=. Astérisque. , volume=226, year=1994, pages=

    1994

  73. [73]

    Journal of Algebraic Geometry

    Marc Levine , title=. Journal of Algebraic Geometry. , volume=10, year=2001, pages=

    2001

  74. [74]

    Morten Lüders , title=

  75. [75]

    Annales Scientifiques d'École Normale Supérieure

    Alberto Minguez , title=. Annales Scientifiques d'École Normale Supérieure. , volume = 41, year = 2008, pages =

    2008

  76. [76]

    Commentarii Mathematici Helvetici , volume=

    Corners and arithmetic groups , author=. Commentarii Mathematici Helvetici , volume=. 1973 , publisher=

    1973

  77. [77]

    Geometry and Topology Monographs

    William Lickorish , title=. Geometry and Topology Monographs. , volume=2, pages=

  78. [78]

    Manuscripta Mathematica

    Florence Lecomte and Nathalie Wach , title=. Manuscripta Mathematica. , volume=141, pages=

  79. [79]

    Annals of Mathematics

    John Milnor , title=. Annals of Mathematics. , volume=63, issue=3, pages=

  80. [80]

    Topology

    Varghese Mathai and Daniel Quillen , title=. Topology. , volume=25, issue=1, pages=

Showing first 80 references.