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arxiv: 2507.09392 · v2 · submitted 2025-07-12 · 🧮 math.AG · math.KT

Equivariant localizing invariants of simple varieties

Jakub L\"owit This is my paper

Pith reviewed 2026-05-19 04:16 UTC · model grok-4.3

classification 🧮 math.AG math.KT
keywords simple varietiesequivariant localizing invariantstruncating invariantscyclotomic traceGoodwillie-Jones tracehomotopy invariant K-theorySchubert varietiesalgebraic geometry
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The pith

A constructive definition of simple varieties allows control of their equivariant truncating invariants, proving trace equivalences and K-theory formality on them including Schubert varieties.

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

The paper defines a class of simple varieties over a field using a constructive recipe. It shows that this class lets one control their equivariant truncating invariants. This control implies that the p-adic cyclotomic trace becomes an equivalence when the base field is algebraically closed of characteristic p. It also implies that the Goodwillie-Jones trace is an isomorphism in degree zero over the rationals and that homotopy invariant K-theory is equivariantly formal and determined by topological versions. The results apply to finite and affine Schubert varieties for GL_n, which the paper shows belong to the class.

Core claim

We define a certain class of simple varieties over a field k by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if k is algebraically closed and of characteristic p, the p-adic cyclotomic trace is an equivalence; (ii) if k is the rationals, the Goodwillie-Jones trace is an isomorphism in degree zero; (iii) homotopy invariant K-theory is equivariantly formal and determined by its topological counterparts. Both finite and affine Schubert varieties for GL_n lie in this class.

What carries the argument

The constructive recipe defining simple varieties, which enables control over their equivariant truncating invariants.

If this is right

  • The p-adic cyclotomic trace becomes an equivalence on simple varieties over algebraically closed fields of positive characteristic.
  • The Goodwillie-Jones trace is an isomorphism in degree zero on simple varieties over the rationals.
  • Homotopy invariant K-theory on simple varieties is equivariantly formal and determined by its topological counterparts.
  • All the above statements hold for finite and affine Schubert varieties for GL_n.

Where Pith is reading between the lines

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

  • The same constructive approach might simplify invariant computations for other singular varieties that arise in geometric representation theory.
  • Equivariant formality could make explicit calculations of K-theory groups feasible for these varieties by reducing them to topological data.
  • Analogous recipes might extend control to other localizing invariants beyond the truncating ones treated here.

Load-bearing premise

That the constructive recipe for simple varieties sufficiently constrains their equivariant truncating invariants and that both finite and affine Schubert varieties for GL_n satisfy the definition.

What would settle it

A specific simple variety where the p-adic cyclotomic trace fails to be an equivalence, or a direct verification that some finite or affine Schubert variety for GL_n does not meet the constructive definition of simple varieties.

read the original abstract

We define a certain class of simple varieties over a field $k$ by a constructive recipe and show how to control their (equivariant) truncating invariants. Consequently, we prove that on simple varieties: (i) if $k=\overline{k}$ and $\mathrm{char} \ k = p$, the $p$-adic cyclotomic trace is an equivalence; (ii) if $k = \mathbb{Q}$, the Goodwillie-Jones trace is an isomorphism in degree zero; (iii) we can control homotopy invariant $K$-theory $KH$, which is equivariantly formal and determined by its topological counterparts. Simple varieties are quite special, but encompass important singular examples appearing in geometric representation theory. We in particular show that both finite and affine Schubert varieties for $GL_n$ lie in this class, so all the above results hold for them.

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 / 2 minor

Summary. The paper defines a class of simple varieties over a field k by a constructive recipe and shows how to control their (equivariant) truncating invariants. Consequently, it proves that on simple varieties: (i) if k=algebraically closed of characteristic p, the p-adic cyclotomic trace is an equivalence; (ii) if k=Q, the Goodwillie-Jones trace is an isomorphism in degree zero; (iii) homotopy invariant K-theory KH is equivariantly formal and determined by its topological counterparts. The paper shows that both finite and affine Schubert varieties for GL_n lie in this class.

Significance. If the results hold, this provides a constructive framework for controlling equivariant truncating invariants on a class of varieties that includes key singular examples from geometric representation theory. The explicit recipe and the inclusion of Schubert varieties are strengths that could enable new computations in K-theory and related invariants.

major comments (2)
  1. [§5 (Definition of simple varieties)] §5 (Definition of simple varieties): The control over equivariant truncating invariants is derived directly from the constructive recipe; the manuscript must verify that finite and affine Schubert varieties satisfy every clause of this recipe, including any equivariant closure conditions, since partial verification would prevent transfer of the trace results.
  2. [§6.2 (Schubert varieties)] §6.2 (Schubert varieties): The claim that both finite and affine Schubert varieties for GL_n are simple varieties is load-bearing for the three main statements; the proof should explicitly address whether the equivariant truncating invariants are controlled without additional implicit assumptions that may fail in the equivariant setting.
minor comments (2)
  1. [Introduction] The introduction could clarify the precise relationship between truncating invariants and the three main trace/formality statements with a brief diagram or list.
  2. [§2] Notation for the equivariant versions of the invariants is introduced late; moving a summary table to §2 would improve readability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading of the manuscript and for highlighting the importance of a complete verification for Schubert varieties. We address each major comment below and indicate the changes we will make in the revised version.

read point-by-point responses

  1. Referee: §5 (Definition of simple varieties): The control over equivariant truncating invariants is derived directly from the constructive recipe; the manuscript must verify that finite and affine Schubert varieties satisfy every clause of this recipe, including any equivariant closure conditions, since partial verification would prevent transfer of the trace results.

    Authors: We agree that the transfer of the trace results requires a full verification that finite and affine Schubert varieties satisfy every clause of the definition, including the equivariant closure conditions. In the revised manuscript we will expand §6.2 with an explicit, clause-by-clause check for both classes of varieties, paying particular attention to the equivariant aspects of the closure conditions. revision: yes

  2. Referee: §6.2 (Schubert varieties): The claim that both finite and affine Schubert varieties for GL_n are simple varieties is load-bearing for the three main statements; the proof should explicitly address whether the equivariant truncating invariants are controlled without additional implicit assumptions that may fail in the equivariant setting.

    Authors: The control of the equivariant truncating invariants is obtained directly from the constructive recipe without additional implicit assumptions. To make this fully explicit, we will add a short clarifying paragraph in the revised §6.2 stating that the equivariant control follows from the general mechanism of §5 and does not rely on any further assumptions that could fail in the equivariant setting. revision: yes

Circularity Check

0 steps flagged

No significant circularity; central claims follow from new definition plus independent inclusion verification

full rationale

The paper introduces a constructive definition of 'simple varieties' and derives control of equivariant truncating invariants directly from that recipe. It then proves the three main statements (p-adic cyclotomic trace equivalence, Goodwillie-Jones isomorphism in degree zero, and equivariant formality of KH) for any variety satisfying the definition. Finally it verifies that finite and affine Schubert varieties for GL_n meet every clause of the recipe. This is a standard definitional-plus-verification structure with no reduction of a claimed prediction or theorem to a fitted parameter, self-citation chain, or tautological renaming. No equations or steps in the provided text exhibit the specific self-referential collapse required by the circularity criteria.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The paper relies on standard background from algebraic geometry and algebraic K-theory. The only new object introduced is the class of simple varieties itself, defined constructively rather than by additional axioms or free parameters.

axioms (1)
  • standard math Standard axioms and constructions of algebraic geometry and algebraic K-theory over a field k
    The results presuppose the usual framework of schemes, K-theory spectra, and trace maps as developed in prior literature.
invented entities (1)
  • simple varieties no independent evidence
    purpose: A class of varieties on which equivariant truncating invariants can be controlled by a constructive recipe
    Defined in the paper to include important singular examples such as Schubert varieties; no independent evidence outside the definition is supplied.

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Works this paper leans on

2 extracted references · 2 canonical work pages · 1 internal anchor

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    Topological K-theory of complex noncommutative spaces

    doi: 10.1007/s00029-022-00796-w . [Bla16] Anthony Blanc. “Topological K-theory of complex noncommutative spaces”. Compo- sitio Mathematica 152.3 (2016), pp. 489–555. doi: 10.1112/S0010437X15007617. [Bri05] Michel Brion. “Lectures on the geometry of flag varieties”. Topics in Cohomological Studies of Algebraic Varieties . Trends in Mathematics. Birkh¨ ause...

    work page doi:10.1007/s00029-022-00796-w 2016

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    Equivariant $K$-theory, affine Grassmannian and perfection

    url: https://arxiv.org/abs/2409.18925. [LS25] Ishan Levy and Vladimir Sosnilo. Almost connective categories. in preparation. 2025. [LT19] Markus Land and Georg Tamme. “On the K-theory of pullbacks”. Annals of Mathe- matics 190 (2019), pp. 877–930. doi: 10.4007/annals.2019.190.3.4. [Oet21] David Oetjen. “Factorial Schur and Grothendieck polynomials from Bo...

    work page internal anchor Pith review Pith/arXiv arXiv doi:10.4007/annals.2019.190.3.4 2025