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Auslander regularity of $p$-adic Banach algebras via almost mathematics

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Pith's one-line read This paper proves that several Banach algebras over non-discretely valued $p$-adic fields — the completed enveloping algebra of a Lie algebra, the Tate-Weyl algebras, and the norm-completed distribution algebras of compact $p$-adic Lie…

desk verdict A genuinely new lifting theorem with a load-bearing gap in the appendix that needs fixing before the sharp dimension claims are accepted. read the letter →

arxiv 2502.01119 v1 pith:XEEGRCOJ submitted 2025-02-03 math.RT math.NTmath.RA

Andreas Bode This is my paper · ORCID
classification math.RTmath.NTmath.RA MSC 16E6516E1016W7046S1022E50
keywords Auslanderregularp-adicBanachalgebrasalmostmathematicsTate-Weyldistributioncompletedenvelopingglobaldimensionnon-discretelyvaluedfields
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

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

The reading

This paper proves that several Banach algebras over non-discretely valued $p$-adic fields are Auslander regular: the completed enveloping algebra of a finite-dimensional Lie algebra, the Tate-Weyl algebras of the polydisk, and the norm-completed distribution algebra of a compact $p$-adic Lie group (Theorem 1.1). The method introduces an 'almost' version of Auslander regularity for rings over $R/\pi$ and shows that this almost regularity lifts to genuine Auslander regularity after inverting $\pi$ (Theorem 1.2). This is the missing tool for the non-discrete case, since the reduction $R/\pi$ is not Noetherian and classical filtered arguments fail. The key input is that $R/\pi[x_1,\ldots,x_m]$ is almost Auslander regular, which makes the examples reduce to polynomial rings modulo $\pi$. Auslander regularity is what guarantees a well-behaved dimension theory for modules, so the result brings non-discretely valued locally analytic representation theory into the same framework as the discrete case.

What carries the argument

The load-bearing object is the almost Auslander regular $R/\pi$-algebra: an almost Noetherian ring satisfying the Auslander condition and a finite global-dimension bound up to modules killed by the maximal ideal $\mathfrak{m}$. The argument develops filtered and graded techniques in the almost setting — $\epsilon$-strict morphisms, $\epsilon$-approximations, good filtrations, and Rees rings — to lift statements from $\operatorname{gr}(A)$ to $A$ and from $A/\pi A$ to $A$. The central lifting theorem (Theorem 3.10) proves that if $A/\pi A$ is almost Auslander regular of almost global dimension $d$, then $A = A\otimes_R K$ is Auslander regular of global dimension at most $d$. Almost regularity of polynomial rings is established by induction using Theorem 4.12, and the appendix adapts the Ardakov–Wadsley proof of Bernstein's inequality to force the global dimension of $\mathcal{A}_{m,n}$ to be exactly $m$, rather than merely at most $2m$.

What would settle it

Produce a proper direct summand $Z$ of $W$ for which $\widehat{R_\omega\langle W\rangle}_n$ is almost finitely generated over $\widehat{R_\omega\langle Z\rangle}_n$, which would invalidate the proof of the lower bound $\operatorname{gl.dim}\mathcal{A}_{m,n} = m$; or construct a nonzero finitely generated module $M$ over $\mathcal{A}_{m,n}$ with $\operatorname{Ext}^j(M, \mathcal{A}_{m,n}) = 0$ for all $j \le m$, contradicting the proved Bernstein inequality.

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Extended reading notes

Core claim

The central claim, stated as Theorem 1.1, is that for every complete nonarchimedean field $K$ of mixed characteristic $(0,p)$, the completed enveloping algebra $U_n$ of a finite-dimensional Lie algebra has finite global dimension equal to $\dim_K\mathfrak{g}$ and is Auslander regular; the Tate-Weyl algebra $\mathcal{A}_{m,n}$ is Auslander regular with global dimension exactly $m$; and for every $1/p \le s < 1$ there is an $r \ge s$ such that the Banach completion $D_r(G,K)$ of the distribution algebra is Auslander regular with global dimension $\dim_L G$. The discovery is that an almost version of Auslander regularity over $R/\pi$ is the right notion to bridge from characteristic $p$ to characteristic zero: a $\pi$-adically complete, $\pi$-torsionfree $R$-algebra whose reduction mod $\pi$ is almost Auslander regular yields an Auslander regular $K$-algebra after applying $\otimes_R K$ (Theorem 1.2). The polynomial ring $R/\pi[x_1,\ldots,x_m]$ is shown to be almost Auslander regular (Theorem 1.3), and each of the three families of algebras reduces modulo $\pi$ to such a polynomial ring, so the regularity statements follow uniformly.

Load-bearing premise

The sharp statement that the Tate-Weyl algebra has global dimension exactly $m$ depends on an unproved assertion that a completed Weyl algebra is never almost finitely generated over a proper direct-summand subalgebra; if that assertion fails, the lower bound on the global dimension collapses.

Editorial extensions

If this is right

  • The Fréchet–Stein algebras $\overline{U}(\mathfrak{g})$, $\widetilde{\mathcal{D}}_X(X)$, and $D(G,K)$ now admit a well-behaved dimension theory via the Schneider–Teitelbaum framework, because each is an inverse limit of the Auslander regular algebras covered by Theorem 1.1.
  • The three families of algebras are Auslander regular over any complete nonarchimedean field of mixed characteristic, not just over discretely valued fields, extending the theorems of Ardakov–Wadsley and Schmidt.
  • The almost Auslander condition is closed under adjoining a polynomial variable under a mild almost Noetherian hypothesis, so the same induction can produce many more almost regular algebras whose generic fibres are Auslander regular.
  • Bernstein's inequality for Tate-Weyl algebras over non-discretely valued fields implies that any nonzero finitely generated module has grade at most $m$, giving a non-discrete analogue of the classical Bernstein inequality.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • The same lifting mechanism should apply to any $K$-Banach algebra whose unit ball reduces modulo $\pi$ to a polynomial ring, such as rings of $p$-adic differential operators on smooth affinoid spaces; the author indicates a follow-up via Kashiwara equivalence, but the core mechanism is already here.
  • One could try to define characteristic varieties for modules over non-discretely valued Tate-Weyl algebras using the algebraic version of Bernstein's inequality proved in Appendix A; the paper deliberately avoids constructing them, but the inequalities it proves are exactly what such a theory would need.
  • A testable extension is whether the almost global dimension of $R/\pi[x_1,\ldots,x_m]$ remains exactly $m$ under weaker almost-Noetherian hypotheses on $A[x,y]$; if the inductive step in Theorem 4.12 can be refined, the same conclusion would follow for larger classes of almost regular algebras.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

2 major / 5 minor

Summary. The paper develops an almost-mathematics analogue of Auslander regularity for R/π-algebras over a non-discrete valuation ring R of mixed characteristic (0, p), and proves a lifting theorem (Theorem 1.2): if A is a π-adically complete, π-torsionfree R-algebra such that A/πA is almost Auslander regular, then A ⊗_R K is Auslander regular. It then proves Theorem 1.3 that R/π[x_1, ..., x_m] is almost Auslander regular, and combines these results to establish Auslander regularity with sharp global dimension for completed enveloping algebras U_n, Tate–Weyl algebras A'_{m,n}, and norm-completed distribution algebras D_r(G, K). The proofs use almost Noetherian ring theory, filtered/graded techniques, and an appendix proving Bernstein's inequality for Tate–Weyl algebras.

Significance. If the proof is completed, the paper settles a natural open case: it extends the Auslander-regularity results of Ardakov–Wadsley and Schmidt from discretely valued to non-discretely valued base fields, with new applications to completed enveloping algebras and distribution algebras. The almost-Auslander framework and the lifting theorem are of independent interest. The main caveats are the unproved almost-finite-generation assertion in Appendix A and the reliance on the author's preprint [6] for flatness and Spencer-resolution inputs; both are local and fixable.

major comments (2)
  1. [Appendix A (proof of Theorem A.11)] The proof of Theorem A.11 uses the assertion, stated without proof, that R_ω⟨W⟩_n is not almost finitely generated over R_ω⟨Z⟩_n whenever Z is a proper direct summand of W. This assertion is load-bearing: it is needed to conclude that each Ann_{K⟨W⟩_n}(m_i) is nonzero, hence that the global annihilator Ann_{K⟨W⟩_n}(N_K) is nonzero by the integral domain property. Without it, the contradiction proving π-torsionness of Ext^j fails and the upper bound gl.dim(A'_{m,n}) ≤ m is not established. Since Theorem 1.1(ii) asserts equality and Theorem 1.1(iii) uses the same upper bound, this gap must be repaired. A degree argument modulo π appears plausible and should be supplied.
  2. [§5.1, §5.2, and Appendix A (proof of Theorem A.1)] The sharp dimension claims in Theorem 1.1 depend on results cited from the author's preprint [6]: flatness of U(g)→Ū(g) ([6, Theorem 2.13]), the Fréchet–Stein property ([6, Theorem 2.8]), and the Spencer resolution for Tate–Weyl algebras ([6, subsection 6.3]). These results are used without proof and are not restated as lemmas here. The author should either prove these inputs or state them precisely and, if [6] is not yet published, indicate its status; as written, the lower bounds gl.dim = m and gl.dim(U_n) = dim_K g rest on this external dependence.
minor comments (5)
  1. [Throughout] The same symbol A is used for the R-algebra and for A ⊗_R K (e.g., in §1 and Theorem 3.10); this is a frequent source of confusion. Consider using A_K or another notation for the generic fibre.
  2. [Appendix A] The algebra denoted '÷Am,2n' should presumably be '’Am,2n' (the Tate–Weyl algebra); the typesetting is inconsistent.
  3. [§5.2] The notation U(π^nL) for the enveloping algebra of the Lie–Rinehart algebra L is introduced only indirectly via [4]; please define it explicitly or give the construction.
  4. [Proof of Theorem 2.27] The final annihilation exponent 'ǫ6j+13' is stated without derivation; this is harmless because only almost-zero-ness matters, but the derivation is hard to follow and would benefit from a brief explanation.
  5. [§5.3] After establishing regularity for Dr_n(H,K) and for finite free extensions, the step 'for any 1/p ≤ s < 1 there exists r < 1 with s ≤ r' is not spelled out; one should choose n with r_n ≥ s and set r = r_n.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the main almost-mathematics lifting theorem is self-contained, and the cited dependencies (including the author's earlier preprint [6]) are not equivalent to the conclusions they support.

full rationale

Walking the derivation chain: Section 3 defines almost Auslander regularity for R/π-algebras in terms of Ext vanishing over A/πA, and Theorem 3.10 proves that if A/πA is almost Auslander regular of almost global dimension d, then A⊗_R K is Auslander regular of global dimension at most d. The proof proceeds through Lemma 3.5 and Lemma 3.9(iv), using almost Nakayama and derived completeness; the conclusion is not assumed in the definition, and no equation identifies the two conditions. Section 4 proves Theorem 1.3 by induction, with the almost Noetherianity input supplied by Kiehl's Satz 5.1 and the base case computed in Lemma 4.13; the polynomial ring result is an independent almost-module statement, not a restatement of Theorem 1.1. Section 5 applies these results to completed enveloping algebras, Tate-Weyl algebras, and distribution algebras; the upper bounds follow from Corollary 4.15 after identifying A/πA with R/π[x_1,...,x_d]. Exact global dimensions use the Koszul resolution in Section 5.1 and, for Tate-Weyl algebras, the Spencer resolution cited from [6, subsection 6.3] together with [6, Theorems 2.8 and 2.13]. These are self-citations to the author's earlier work, and they are load-bearing for the sharp dimension claims, but they are not circular: the cited results are stated with assumptions that do not include Auslander regularity, and no equation in this paper makes the target theorem an input to itself. Under the review rules, this counts as independent support or dependency, not circularity. The one notable weakness is in the proof of Theorem A.11: the statement that 'R_ω⟨W⟩_n is not almost finitely generated over R_ω⟨Z⟩_n' is used without proof to ensure each Ann(m_i) is nonzero; if false, the lower bound gl.dim(A'_{m,n}) = m would collapse. That is a proof gap and a correctness risk, not a circular step, because the assertion is not equivalent to Auslander regularity and is not derived from the theorem being proved. No fitted parameter is renamed as a prediction; no known result is merely relabelled; no uniqueness claim is imported to forbid alternatives. Overall score 0.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The central lifting theorem rests on standard almost mathematics background plus a handful of domain assumptions about the coefficient field and the examples. The most fragile items are the unproved non-finite-generation assertion in Appendix A and the reliance on the author's prior preprint [6] for sharp dimension bounds.

assumptions (5)
  • domain assumption The maximal ideal m of R satisfies m^2 = m for densely valued K.
    Used throughout Section 2 and repeatedly via 'epsilon arbitrary and m^2 = m' to promote almost vanishing to m-annihilation, e.g., in Theorem 2.27.
  • standard math R<x_1,...,x_m> is almost Noetherian (Kiehl, [11, Satz 5.1]).
    Provides the almost Noetherianity of the polynomial reductions R/pi[x_1,...,x_m] used in Proposition 2.7 and Section 4.
  • domain assumption A[X,Y] is almost Noetherian, an explicit hypothesis in Theorems 4.7 and 4.12.
    Replaces the missing almost Hilbert basis theorem ([23, Warning 2.7.9]); for the examples this follows from Kiehl's theorem as in Proposition 2.7.
  • ad hoc to paper R_omega<W>_n is not almost finitely generated over R_omega<Z>_n for rank Z < rank W.
    Asserted without proof in Appendix A, Proposition A.11 proof, to force nonzero annihilators; load-bearing for gl.dim('A_{m,n}) = m.
  • domain assumption Flatness of U(g) to ar U(g) ([6, Theorem 2.13]) and of ar U(g) to U_n ([20, Remark 3.2]).
    Used in Section 5.1 to show gl.dim(U_n) = d; [6] is the author's prior preprint.

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Pith. "Pith review of Auslander regularity of $p$-adic Banach algebras via almost mathematics." pith.science (2026). https://pith.science/paper/XEEGRCOJ

@misc{pith2026250201119,
  author       = {Pith},
  title        = {Pith review of: Auslander regularity of $p$-adic Banach algebras via almost mathematics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/XEEGRCOJ}},
  note         = {Machine review of arXiv:2502.01119}
}
abstract

We discuss an "almost" version of Auslander regularity and use it to prove the Auslander regularity of various Banach algebras over non-discretely valued fields appearing naturally in $p$-adic locally analytic representation theory: completed Weyl algebras, the completed enveloping algebra of a Lie algebra, and the Banach completion of the distribution algebra $D(G, K)$ for a compact $p$-adic Lie group $G$.

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Reviewed papers in the Pith corpus that reference this work. Sorted by Pith novelty score. Full citation record

  1. A survey on Auslander-Gorenstein algebras

    math.RT 2025-08 conditional novelty 3.0 of 10

    A survey of finite-dimensional Auslander-Gorenstein algebras, including classifications for monomial and incidence algebras and the identification of the Auslander-Reiten permutation with rowmotion, Ringel's homologic...

Reference graph

Works this paper leans on

24 extracted references · 23 canonical work pages · cited by 1 Pith paper

  1. [6]

    A. Bode. Six operations for ÙD-modules on rigid analytic spaces. arXiv 2110.09398v3

    arXiv

  2. [1]

    Ajitabh, S

    K. Ajitabh, S. Smith, J. Zhang. Injective resolutions of some regular rings

    work page

  3. [2]

    K. Ardakov. Equivariant D-modules on rigid analytic spaces. Astérisque 423 (2021), 1–161

    work page 2021

  4. [3]

    Ardakov, S

    K. Ardakov, S. W adsley. On irreducible representations of compact p-adic groups. Annals of Math. 178 (2013), 453–557

    work page 2013

  5. [4]

    Ardakov, S

    K. Ardakov, S. W adsley. ÙD-modules on rigid analytic spaces I. J. Reine Angew. Math. 747 (2019), 221–276

    work page 2019

  6. [5]

    Ardakov, K.A

    K. Ardakov, K.A. Brown. Primeness, semiprimeness and lo calisation in Iwasawa algebras. Trans. Amer. Math. Soc. 359 (2007), 1499–1515

    work page 2007

  7. [7]

    Fujiwara, O

    K. Fujiwara, O. Gabber, F. Kato. On Hausdorff completions of commutative rings in rigid geometry. J. Algebra 332 (2011), 293–321

    work page 2011

  8. [8]

    Gabber, L

    O. Gabber, L. Ramero. Foundations for almost ring theory

    work page

Show all 24 references
  1. [9]

    Gabber, L

    O. Gabber, L. Ramero. Almost ring theory . Lecture Notes in Math. 1800. Springer, 2003

    2003

  2. [10]

    Greuel, G

    G.-M. Greuel, G. Pfister. A singular introduction to commutative algebra. Springer, Berlin, extended edition, 2008. With contributions by Olaf Bachman n, Christoph Lossen and Hans Schönemann. With 1 CD-ROM (Windows, Macintosh and UNIX)

    2008

  3. [11]

    R. Kiehl. Der Endlichkeitssatz für eigentliche Abbild ungen in der nichtarchimedischen Funk- tionentheorie. Invent. Math. 2 (1967), 191–214

    1967

  4. [12]

    T. Y. Lam. A first course in noncommutative rings. Volume 131 of Graduate Texts in Math- ematics, second edition, Springer, New York, 2001

    2001

  5. [13]

    H. Li, F. van Oystaeyen. Zariskian filtrations. K-Monogr. Math., 2. Kluwer Academic Pub- lishers, Dordrecht, 1996

    1996

  6. [14]

    J. C. McConnell, J. C. Robson. Noncommutative Noetherian rings. Volume 30 of Graduate Studies in Mathematics , American Mathematical Society, Providence, RI, revised e dition,

  7. [15]

    Orlik, M

    S. Orlik, M. Strauch. On the irreducibility of locally a nalytic principal series representations. Represent. Theory 14 (2010), 713–746

    2010

  8. [16]

    Perez-Garcia, W.H

    C. Perez-Garcia, W.H. Schikhof. Locally convex spaces over non-Archimedean fields. Volume 119 of Cambridge Studies in Advanced Mathematics . CUP, Cambridge, 2010

    2010

  9. [17]

    platification

    M. Raynaud, L. Gruson. Critères de platitude et de proje ctivité. Techniques de "platification" d’un module. Invent. Math. 13 (1971), 1–89

    1971

  10. [18]

    T. Schmidt. Auslander regularity of p-adic distribution algebras. Represent. Theory 12 (2008), 37–57

    2008

  11. [19]

    T. Schmidt. Verma modules over p-adic Arens–Michael envelopes of reductive Lie algebras. Journal of Algebra 390 (2013), 160–180

    2013

  12. [20]

    Schneider, J

    P. Schneider, J. Teitelbaum. Algebras of p-distributions and admissible representations. In- vent. Math. 153 (2003), no. 1, 145–196

    2003

  13. [21]

    Stacks project

    The Stacks Project Authors. Stacks project. https://stacks.math.columbia.edu, 2024

    2024

  14. [22]

    An introduction to homological algebra

    W eibel. An introduction to homological algebra

  15. [23]

    Zavyalov

    B. Zavyalov. Almost coherent modules and almost cohere nt sheaves

  16. [2001]

    With the cooperation of L. W. Small

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