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arxiv: 2604.11235 · v3 · submitted 2026-04-13 · 🧮 math.RT · math.NT

Pro-p Iwahori-Hecke modules in semisimple rank one and singularity categories

Nicolas Dupr\'e This is my paper

Pith reviewed 2026-05-10 15:56 UTC · model grok-4.3

classification 🧮 math.RT math.NT
keywords pro-p Iwahori-Hecke algebrahomotopy categorysingularity categoryGorenstein projective model structureGL2SL2PGL2
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The pith

For GL2 over local fields the homotopy category of pro-p Iwahori-Hecke modules equals the singularity category of an explicit scheme.

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

The paper shows that the homotopy category obtained from a Gorenstein projective model structure on modules over the pro-p Iwahori-Hecke algebra of GL2 is equivalent to the singularity category of a certain scheme. The same construction yields complete explicit descriptions of the homotopy category when the group is SL2 or PGL2. This link lets the algebraic data of Hecke modules be read off from geometric objects attached to the scheme. A reader would care because the equivalence supplies a concrete bridge between module categories in representation theory and singularity categories in algebraic geometry.

Core claim

When G equals GL2 of the local field the homotopy category Ho of the Gorenstein projective model structure on modules over the pro-p Iwahori-Hecke algebra is equivalent to the singularity category Sing of the explicit scheme X_q,G. The same methods produce fully explicit descriptions of Ho when G is SL2 or PGL2 and supply further calculations in the GL2 case. The GL2 equivalence recovers a known correspondence between Hecke modules and mod-p representations.

What carries the argument

The Gorenstein projective model structure on the category of modules over the pro-p Iwahori-Hecke algebra, which produces a homotopy category that matches the singularity category of the associated scheme for GL2.

If this is right

  • The equivalence Ho(H_GL2) ≃ Sing(X_q,GL2) holds.
  • This equivalence recovers the mod-p correspondence for Hecke modules.
  • Ho(H_G) admits a complete explicit description when G is SL2 or PGL2.
  • Further explicit computations of the homotopy category are available in the GL2 case.

Where Pith is reading between the lines

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

  • The explicit descriptions for SL2 and PGL2 may make it feasible to compute the same invariants directly inside the singularity category.
  • The GL2 equivalence suggests that the scheme X_q,G encodes the full derived structure of the Hecke-module category.
  • Similar model-structure comparisons could be tested for other semisimple groups of rank one.

Load-bearing premise

The paper's explicit constructions succeed in identifying the homotopy category of the model structure with the singularity category of the scheme.

What would settle it

A concrete H_G-module whose class in the homotopy category fails to correspond to any object in the singularity category, or whose Ext groups computed in one category differ from those in the other.

read the original abstract

Let $\mathfrak{F}$ be a non-archimedean local field of residue characteristic $p$ and $G$ be one of the groups $\mathrm{GL}_2(\mathfrak{F})$, $\mathrm{SL}_2(\mathfrak{F})$ or $\mathrm{PGL}_2(\mathfrak{F})$. Let $\mathcal{H}_G$ denote the pro-$p$ Iwahori-Hecke algebra of $G$ over $\overline{\mathbb{F}}_p$. We study the homotopy category $\mathrm{Ho}(\mathcal{H}_G)$ of Hovey's Gorenstein projective model structure on the category of $\mathcal{H}_G$-modules and relate it to the singularity category $\mathrm{Sing}(X_{q,\mathbf{G}})$ of an explicit scheme. When $G=\mathrm{GL}_2(\mathfrak{F})$, this scheme was first introduced by Dotto-Emerton-Gee \cite{DEG22}. We obtain in that case an equivalence $\mathrm{Ho}(\mathcal{H}_{\mathrm{GL}_2})\simeq \mathrm{Sing}(X_{q,\mathrm{GL}_2})$ and recover from this Grosse-Kl\"onne's mod-$p$ Langlands correspondence for Hecke modules \cite{GK20}, building on work of P\'epin-Schmidt \cite{PeSch25_2}. We furthermore describe $\mathrm{Ho}(\mathcal{H}_G)$ completely explicitly when $G=\mathrm{SL}_2(\mathfrak{F})$ or $\mathrm{PGL}_2(\mathfrak{F})$, and make additional computations in the $\mathrm{GL}_2$ case.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 3 minor

Summary. The paper studies the homotopy category Ho(H_G) of Hovey's Gorenstein projective model structure on modules over the pro-p Iwahori-Hecke algebra H_G for G = GL_2(F), SL_2(F) or PGL_2(F), where F is a non-archimedean local field of residue characteristic p. It relates this to the singularity category Sing(X_{q,G}) of an explicit scheme (the Dotto-Emerton-Gee scheme in the GL_2 case). The central result is an equivalence Ho(H_{GL_2}) ≃ Sing(X_{q,GL_2}) from which Grosse-Klönne's mod-p Langlands correspondence for Hecke modules is recovered; explicit descriptions of Ho(H_G) are given for SL_2 and PGL_2, together with further computations in the GL_2 case.

Significance. If the claimed equivalence holds, the work supplies a geometric model for the homotopy category of pro-p Iwahori-Hecke modules in semisimple rank one and recovers a known mod-p Langlands correspondence via singularity categories. The explicit descriptions for SL_2 and PGL_2 make the homotopy categories computable and constitute a concrete contribution. The approach combines model-categorical techniques with geometric constructions in a novel way for this area of representation theory.

major comments (2)
  1. [GL_2 equivalence theorem and preceding functor constructions] The proof that the equivalence Ho(H_{GL_2}) ≃ Sing(X_{q,GL_2}) is an equivalence of triangulated categories (the main theorem in the GL_2 section): the argument must verify that the functors induced by the H_{GL_2}-action send Gorenstein projective modules to objects whose cones lie in the thick subcategory of perfect complexes on X_{q,GL_2}, so that the quotient by weak equivalences coincides with the quotient by perfect complexes. Explicit checks that these functors are fully faithful and essentially surjective after passage to the quotient are required; without them the triangulated equivalence does not follow from the underlying categorical relationship.
  2. [Recovery paragraph after the main GL_2 theorem] Recovery of Grosse-Klönne's correspondence (the paragraph following the equivalence statement): the precise mechanism by which the equivalence produces the mod-p Langlands correspondence for Hecke modules should be spelled out, including the identification of the relevant objects on the singularity-category side with the Hecke modules studied by Grosse-Klönne.
minor comments (3)
  1. [Introduction] The introduction should briefly recall the definition of the scheme X_{q,G} (or at least its key properties) rather than referring exclusively to DEG22, to improve readability for readers outside the immediate circle of references.
  2. [Section introducing the model structure] Notation for the homotopy category Ho(H_G) is used consistently, but a short reminder of Hovey's model structure axioms (weak equivalences, cofibrations) in the first section where it appears would help readers who are not experts in model categories.
  3. [Notation and preliminaries] A few typographical inconsistencies appear in the notation for the residue field and the parameter q; these should be standardized throughout.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for the constructive major comments. We have revised the paper to address both points by adding the requested explicit verifications and expanded explanations.

read point-by-point responses

  1. Referee: [GL_2 equivalence theorem and preceding functor constructions] The proof that the equivalence Ho(H_{GL_2}) ≃ Sing(X_{q,GL_2}) is an equivalence of triangulated categories (the main theorem in the GL_2 section): the argument must verify that the functors induced by the H_{GL_2}-action send Gorenstein projective modules to objects whose cones lie in the thick subcategory of perfect complexes on X_{q,GL_2}, so that the quotient by weak equivalences coincides with the quotient by perfect complexes. Explicit checks that these functors are fully faithful and essentially surjective after passage to the quotient are required; without them the triangulated equivalence does not follow from the underlying categorical relationship.

    Authors: We agree that the triangulated structure of the equivalence requires these explicit verifications, which were only sketched in the original submission. In the revised manuscript we have inserted a new subsection (immediately preceding the statement of the main theorem) that constructs the H_{GL_2}-action on the singularity category, verifies that Gorenstein projective modules are sent to objects whose cones lie in the thick subcategory of perfect complexes, and supplies direct arguments for full faithfulness and essential surjectivity of the induced functors on the quotient categories. These checks rely on the explicit descriptions of both sides already developed in the paper. revision: yes

  2. Referee: [Recovery paragraph after the main GL_2 theorem] Recovery of Grosse-Klönne's correspondence (the paragraph following the equivalence statement): the precise mechanism by which the equivalence produces the mod-p Langlands correspondence for Hecke modules should be spelled out, including the identification of the relevant objects on the singularity-category side with the Hecke modules studied by Grosse-Klönne.

    Authors: We thank the referee for highlighting the need for greater precision here. The revised paragraph now explicitly traces the mechanism: the equivalence Ho(H_{GL_2}) ≃ Sing(X_{q,GL_2}) identifies homotopy classes of Gorenstein projective H_{GL_2}-modules with objects of the singularity category; the Hecke modules appearing in Grosse-Klönne's correspondence are recovered as the images, under the equivalence, of the indecomposable Gorenstein projective modules classified in the preceding sections, which correspond to the structure sheaves of the irreducible components of the special fiber of X_{q,GL_2} (as identified via the work of Pépin-Schmidt). revision: yes

Circularity Check

0 steps flagged

No significant circularity; equivalence derived from explicit constructions and external citations.

full rationale

The paper claims to establish the equivalence Ho(H_GL2) ≃ Sing(X_q,GL2) via Hovey's Gorenstein projective model structure and compatibility with the DEG scheme, building explicitly on DEG22 and PeSch25_2. No step reduces the central claim to a self-definition, fitted input renamed as prediction, or load-bearing self-citation chain. The derivation relies on constructing functors and verifying triangulated category properties independently of the target result, making the chain self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

The paper rests on standard constructions in homological algebra and algebraic geometry; no free parameters or new entities are introduced.

axioms (2)
  • domain assumption Hovey's Gorenstein projective model structure exists on the category of H_G-modules
    Invoked to define the homotopy category Ho(H_G)
  • domain assumption The scheme X_q,G is well-defined and its singularity category is as constructed in DEG22
    Required for the equivalence statement

pith-pipeline@v0.9.0 · 5594 in / 1263 out tokens · 49932 ms · 2026-05-10T15:56:45.471966+00:00 · methodology

discussion (0)

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Reference graph

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