{"id":"ff5d8c4e-967f-4a66-a66c-5f12d0bcd383","arxiv_id":"2502.01119","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"New 'almost Auslander regular' rings are used to prove Auslander regularity, with sharp global dimension, for completed enveloping algebras, Tate-Weyl algebras, and distribution algebras over non-discretely valued p-adic fields.","lead":"This paper proves that several p-adic Banach algebras, including completed enveloping algebras and norm-completed distribution algebras of compact p-adic Lie groups, are Auslander regular even when the coefficient field is not discretely valued. The proof introduces a new 'almost' version of Auslander regularity and uses almost mathematics to lift regularity from a quotient by a nonzero element back to characteristic zero.","discovery_kind":"new_method","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Appendix A's proof of gl.dim(A'_{m,n})=m rests on an unproved assertion that completed Weyl algebras are not almost finitely generated over proper sub-Weil algebras; if wrong, the sharp dimension claims in Theorem 1.1 collapse.","rationale":"The reader's weakest_assumption identifies the same unproved assertion in Appendix A as the load-bearing concern. I independently reviewed the main lifting theorem (Theorems 1.2 and 3.10) and the polynomial ring arguments of Section 4, and I did not find a fatal gap there; the almost mathematics framework and the lifting of the Auslander condition appear coherent, and the use of one-sided versus two-sided Auslander conditions is defensible because the definition of the almost Auslander condition is phrased for left modules but applies to the right submodules of Ext^i(left module) that arise in the proof. The appendix's assertion, however, is genuinely load-bearing: it is the step that rules out free rank-1 submodules of N, without which the proof that Ext^j vanishes for j>m (and hence the sharp upper bound on the global dimension of the Tate-Weyl algebras) does not go through. The assertion is likely true by a straightforward degree argument modulo π, but because it is stated without proof, the paper is not fully vetted. Adding the short proof in Appendix A would resolve this concern, so the conditional verdict remains appropriate. I therefore keep the reader's verdict unchanged rather than moving to accept or reject.","tokens_in":35608,"tokens_out":38099,"duration_ms":364497,"concrete_test":"Add to Appendix A the following proof, and verify each step: Let Z⊆W be a proper free direct summand, and set A=R/π[W/π], B=R/π[Z/π]. Suppose A is almost finitely generated as a B-module. For any ǫ∈m, there exist f_1,...,f_r∈A with ǫ·A⊆∑ B f_i. Let d be the maximum total degree in the complementary variables (a basis of W/Z) among the f_i. Then every element of ∑ B f_i has complementary degree at most d, but ǫ·y^{d+1} (for y a complementary variable) lies in ǫ·A and has complementary degree d+1, contradiction. Hence A is not almost finitely generated over B. Carrying this through the π-adic completion gives the asserted non-finite-generation for R_ω⟨W⟩_n over R_ω⟨Z⟩_n. Confirm that this degree argument does not require B to be finitely generated or ǫ to be nilpotent; if it fails, identify the exact step where almost generation differs from ordinary finite generation.","verdict_should_be":"UNCHANGED","load_bearing_attack":"To prove Theorem 1.1(ii), the paper needs Theorem A.11: for a finitely generated π-torsionfree module M over the completed Weyl algebra R_ω[V]_n, Ext^j = 0 for j>m. In the proof, after reducing to the annihilator of N_K, it claims that if Ann_{K⟨W⟩_n}(m_i)=0 for a generator, then N would contain a free rank-1 R_ω⟨W⟩_n-submodule; but this is impossible because N is almost finitely generated over R_ω⟨Z⟩_n for a proper direct summand Z of W, and 'R_ω⟨W⟩_n is not almost finitely generated over R_ω⟨Z⟩_n.' This assertion is used without proof in the proof of Theorem A.11. It is needed to conclude that each Ann(m_i) is nonzero, and hence that the global annihilator Ann_{K⟨V⟩_n}(N_K) is nonzero by the integral domain property. Without this step, the contradiction proving π-torsionness of Ext fails, and the upper bound gl.dim ≤ m for the Tate-Weyl algebras is not established. Since Theorem 1.1(ii) asserts the exact equality gl.dim(A'_{m,n})=m, and the distribution algebra dimension claims in Theorem 1.1(iii) rely on the same upper bound, the sharp dimension statements depend on this unstated lemma. The assertion is plausible and can be proved by a degree argument modulo π (as described in the concrete test), but the omission leaves a genuine gap in the proof as written.","agreement_with_reader":"agree"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":35965,"tokens_out":6375,"duration_ms":64102,"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":[{"comment":"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.","section":"Appendix A (proof of Theorem A.11)"},{"comment":"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.","section":"§5.1, §5.2, and Appendix A (proof of Theorem A.1)"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"The algebra denoted '÷Am,2n' should presumably be '’Am,2n' (the Tate–Weyl algebra); the typesetting is inconsistent.","section":"Appendix A"},{"comment":"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.","section":"§5.2"},{"comment":"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.","section":"Proof of Theorem 2.27"},{"comment":"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.","section":"§5.3"}],"recommendation":"major_revision","confidential_remarks":"The main obstacle is the unproved assertion in Appendix A; the rest of the paper is coherent. The author should also clarify the status of [6], since the sharp dimension claims rely on it. If these points are addressed, the paper would make a solid contribution to non-commutative almost mathematics and p-adic representation theory."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things up front. The main lifting result -- Theorem 1.2, \"almost Auslander regular mod pi implies Auslander regular over K\" -- is new and looks correct. The framework of almost Auslander regularity itself is a sensible adaptation of Zariskian filtration methods, and the polynomial ring Theorem 1.3 is worked out in careful detail. If it stands, Theorem 1.1 extends the discretely valued results of Ardakov--Wadsley and Schmidt to all complete non-discretely valued fields, which matters for p-adic locally analytic representation theory.\n\nSections 3 and 4 are where the paper's real value lives: the almost version of the Auslander condition, the Ext vanishing criteria, and the induction for polynomial rings are coherent, and I did not find a circular step. The external benchmarks (Kiehl, Gabber--Ramero, Ardakov--Wadsley) are used appropriately.\n\nThe soft spot is the appendix. The proof that gl.dim('A_{m,n}) = m depends on an assertion used without proof: that R_omega<W>_n is not almost finitely generated over R_omega<Z>_n when Z is a proper direct summand of W. This is exactly what forces the annihilator of each generator of the relevant Ext module to be nonzero, and without it the lower bound on global dimension collapses. The assertion is plausible -- a degree-counting argument modulo pi should do it -- but the author does not supply it. That is a genuine gap in the written proof, not a minor omission. Since the exact dimension statements in Theorem 1.1(ii) and (iii) rest on this appendix, the gap affects the paper's headline claims.\n\nThere is also a dependency on the author's preprint [6] for flatness of U(g) -> \\bar U(g) and for the Spencer resolution used to compute the global dimension in the Weyl algebra case. This is not circular in a damaging sense, but the referee should check whether [6] proves the stated facts in the needed generality.\n\nBottom line: this deserves a serious referee. Send it to review, with a request that the author either prove the missing almost-finite-generation lemma or cite a proof, and state clearly which results from [6] are being imported. If the gap is filled, the paper should be accepted.","headline":"A genuinely new lifting theorem with a load-bearing gap in the appendix that needs fixing before the sharp dimension claims are accepted.","tokens_in":36475,"tokens_out":3127,"would_cite":true,"duration_ms":32128,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["16E65","16E10","16W70","46S10","22E50"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["Auslander regular","p-adic Banach algebras","almost mathematics","Tate-Weyl algebras","distribution algebras","completed enveloping algebras","global dimension","non-discretely valued fields"],"falsifier":"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.","tokens_in":35379,"feed_emoji":"🧮","tokens_out":9778,"duration_ms":86737,"temperature":0.7,"pith_summary":"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.","feed_headline":"p-adic Banach algebras proved Auslander regular via almost math","feed_subtitle":"A new almost-regularity condition lifts from mod π rings to K, covering Weyl and distribution algebras.","key_machinery":"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 $2m$.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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."],"supporting_citations":[{"why":"Establishes the discretely valued case and supplies the deformed algebra construction and Bernstein's inequality proof adapted in Appendix A.","marker":"[3]"},{"why":"Provides the almost mathematics framework: almost modules, almost Noetherian rings, and the lifting lemmas used throughout.","marker":"[9]"},{"why":"Supplies the classical Zariskian filtration proof of the Auslander condition for polynomial rings that the almost version imitates.","marker":"[13]"},{"why":"Defines the distribution algebras $D(G,K)$ and the Fréchet–Stein framework in which Auslander regularity yields a dimension theory.","marker":"[20]"},{"why":"Proves the discretely valued analogue for norm-completed distribution algebras, including Corollary 7.3 used for finite free extensions.","marker":"[18]"},{"why":"Supplies flatness and Fréchet–Stein facts for completions of enveloping algebras and the Spencer resolution used to compute the global dimension of $\\mathcal{A}_{m,n}$.","marker":"[6]"},{"why":"Provides the almost coherent module theory, including almost Nakayama and almost flatness lemmas, underlying the lifting arguments.","marker":"[23]"},{"why":"Used for the Ext-group identifications that transfer almost vanishing between graded, Rees, and filtered modules.","marker":"[1]"}],"fun_headline_variants":["Almost mathematics proves Auslander regularity for p-adic Banach algebras","Auslander regularity for p-adic Banach algebras via almost math","Almost Auslander regularity bridges characteristic p to zero for Banach algebras","Characteristic p to zero: almost Auslander gives regularity for p-adic algebras","From mod p to characteristic 0: almost Auslander regularity in p-adic algebras"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Almost mathematics proves Auslander regularity for p-adic Banach algebras","Auslander regularity for p-adic Banach algebras via almost math","Almost Auslander regularity bridges characteristic p to zero for Banach algebras","Characteristic p to zero: almost Auslander gives regularity for p-adic algebras","From mod p to characteristic 0: almost Auslander regularity in p-adic algebras"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000568,"raw_usage":{"total_tokens":2658,"prompt_tokens":882,"completion_tokens":1776,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":498,"completion_tokens_details":{"reasoning_tokens":1678}},"tokens_in":498,"tokens_out":1776,"duration_ms":12085,"temperature":1.0,"reasoning_tokens":1678,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-09T16:31:35.856518+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Ardakov, S","cited_arxiv_id":null,"evidence_quote":"Establishes the discretely valued case and supplies the deformed algebra construction and Bernstein's inequality proof adapted in Appendix A."},{"cited_title":"Gabber, L","cited_arxiv_id":null,"evidence_quote":"Provides the almost mathematics framework: almost modules, almost Noetherian rings, and the lifting lemmas used throughout."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the classical Zariskian filtration proof of the Auslander condition for polynomial rings that the almost version imitates."},{"cited_title":"Schneider, J","cited_arxiv_id":null,"evidence_quote":"Defines the distribution algebras $D(G,K)$ and the Fréchet–Stein framework in which Auslander regularity yields a dimension theory."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Proves the discretely valued analogue for norm-completed distribution algebras, including Corollary 7.3 used for finite free extensions."},{"cited_title":"Zavyalov","cited_arxiv_id":null,"evidence_quote":"Provides the almost coherent module theory, including almost Nakayama and almost flatness lemmas, underlying the lifting arguments."},{"cited_title":"Ajitabh, S","cited_arxiv_id":null,"evidence_quote":"Used for the Ext-group identifications that transfer almost vanishing between graded, Rees, and filtered modules."}],"review_version":1}