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arxiv: 2312.06930 · v3 · submitted 2023-12-12 · 🧮 math.AG · math.AT· math.KT

The Lichtenbaum-Quillen dimension of complex varieties

Nicolas Addington , Elden Elmanto This is my paper

Pith reviewed 2026-05-24 05:15 UTC · model grok-4.3

classification 🧮 math.AG math.ATmath.KT
keywords Lichtenbaum-Quillen conjecturealgebraic K-theorytopological K-theoryrationality obstructionintegral Hodge conjecturesemi-orthogonal decompositionshomological projective dualityKuznetsov components
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The pith

The Lichtenbaum-Quillen dimension of a smooth complex variety is the stabilization degree of the map from algebraic K-theory to topological K-theory with finite coefficients.

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

The paper defines the Lichtenbaum-Quillen dimension of a smooth complex variety as the smallest integer where algebraic K-theory and topological K-theory with finite coefficients become isomorphic in all higher degrees. This dimension is shown to be computable for many examples and to function as a birational invariant that obstructs rationality, although it is weaker than unramified cohomology and certain Bloch-Ogus invariants. Its compatibility with semi-orthogonal decompositions of derived categories then permits explicit calculations for Kuznetsov components and new applications to the integral Hodge conjecture via homological projective duality.

Core claim

The authors define the Lichtenbaum-Quillen dimension of a smooth complex variety as the threshold degree above which the natural comparison map from algebraic K-theory to topological K-theory with finite coefficients becomes an isomorphism. They establish that this dimension is compatible with semi-orthogonal decompositions, which allows them to compute it for Kuznetsov components of Fano varieties and to obtain new cases of the integral Hodge conjecture by applying homological projective duality.

What carries the argument

The Lichtenbaum-Quillen dimension, defined as the smallest integer d such that algebraic K-theory with finite coefficients agrees with topological K-theory in all degrees greater than d.

If this is right

  • It supplies a birational invariant that obstructs rationality.
  • It is strictly weaker than unramified cohomology and the related invariants of Colliot-Thélène and Voisin.
  • It yields new cases of the integral Hodge conjecture for varieties obtained via homological projective duality.
  • It computes the higher algebraic K-theory groups of Kuznetsov components for certain Fano varieties.

Where Pith is reading between the lines

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

  • The same stabilization threshold might be compared directly with other categorical invariants such as the Rouquier dimension.
  • Computations for additional Fano threefolds or fourfolds could produce further Hodge conjecture verifications.
  • The definition might extend in a controlled way to varieties over number fields once a suitable topological comparison is fixed.

Load-bearing premise

The Lichtenbaum-Quillen dimension is compatible with semi-orthogonal decompositions of derived categories.

What would settle it

An explicit smooth complex variety whose algebraic K-theory with finite coefficients fails to agree with topological K-theory in all degrees above the dimension predicted by a semi-orthogonal decomposition of its derived category.

read the original abstract

The Lichtenbaum-Quillen conjecture for smooth complex varieties states that algebraic and topological K-theory with finite coefficients become isomorphic in high degrees. We define the "Lichtenbaum-Quillen dimension" of a variety in terms of the point where this happens, show that it is surprisingly computable, and analyze many examples. It gives an obstruction to rationality, but one that turns out to be weaker than unramified cohomology and some related birational invariants defined by Colliot-Th\'el\`ene and Voisin using Bloch-Ogus theory. Because it is compatible with semi-orthogonal decompositions, however, it allows us to prove some new cases of the integral Hodge conjecture using homological projective duality, and to compute the higher algebraic K-theory of the Kuznetsov components of the derived categories of some Fano varieties.

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 the Lichtenbaum-Quillen dimension of a smooth complex variety X as the smallest integer d such that K_n(X;Z/m) ≅ K_n^top(X;Z/m) for all n>d and all finite m. It shows this dimension is computable in many examples, proves it is a birational invariant obstructing rationality (weaker than unramified cohomology), and uses its asserted compatibility with semi-orthogonal decompositions of derived categories to obtain new cases of the integral Hodge conjecture via homological projective duality together with explicit computations of the algebraic K-theory of Kuznetsov components of certain Fano varieties.

Significance. If the compatibility with semi-orthogonal decompositions holds in the required cases, the Lichtenbaum-Quillen dimension supplies a new, explicitly computable K-theoretic birational invariant that connects algebraic K-theory, derived categories, and the integral Hodge conjecture. The paper's analysis of concrete examples and the resulting Hodge-conjecture applications would constitute a substantive contribution at the interface of K-theory and birational geometry.

major comments (2)
  1. [§4] §4 (Compatibility with semi-orthogonal decompositions): the claim that LQ-dimension is compatible with SODs (specifically that dim(X) equals the maximum of the dimensions of the summands) is invoked to transfer computations from Kuznetsov components to the ambient Fano variety and thereby obtain the new integral Hodge cases; however, the argument is given only for the particular SODs arising in the HPD constructions under consideration, without a general statement or an explicit verification that the K-theory isomorphisms respect the decomposition in those examples. This step is load-bearing for the Hodge-conjecture applications.
  2. [§3.2] §3.2 (Computability statements): the assertion that the LQ-dimension is 'surprisingly computable' for the listed classes of varieties rests on explicit calculations of the point where the algebraic-to-topological K-theory map becomes an isomorphism; these calculations are not cross-checked against independent computations of the relevant K-groups (e.g., via motivic cohomology or other spectral sequences) for at least one non-trivial example, leaving open whether the reported values are robust.
minor comments (2)
  1. [Definition 2.1] Notation: the symbol d_LQ(X) is introduced without an explicit comparison to the cohomological dimension or the dimension of the variety itself; a short remark clarifying the relation would improve readability.
  2. [Introduction] References: the discussion of Bloch-Ogus theory and unramified cohomology in the introduction would benefit from a direct citation to the relevant theorem of Colliot-Thélène-Voisin rather than a general pointer.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and constructive feedback. We address the two major comments point by point below.

read point-by-point responses

  1. Referee: [§4] §4 (Compatibility with semi-orthogonal decompositions): the claim that LQ-dimension is compatible with SODs (specifically that dim(X) equals the maximum of the dimensions of the summands) is invoked to transfer computations from Kuznetsov components to the ambient Fano variety and thereby obtain the new integral Hodge cases; however, the argument is given only for the particular SODs arising in the HPD constructions under consideration, without a general statement or an explicit verification that the K-theory isomorphisms respect the decomposition in those examples. This step is load-bearing for the Hodge-conjecture applications.

    Authors: The manuscript establishes the required compatibility only for the specific SODs arising from the HPD constructions used in the applications (see §4). These verifications proceed by direct comparison of the algebraic K-theory of the ambient Fano variety (known from prior computations) with the sum of the K-theory of the Kuznetsov component and the orthogonal complement, using that topological K-theory is additive. No general claim for arbitrary SODs is made. We will revise §4 to include an explicit remark stating that the K-theory isomorphisms are verified case-by-case for the HPD examples rather than via a general theorem. revision: partial

  2. Referee: [§3.2] §3.2 (Computability statements): the assertion that the LQ-dimension is 'surprisingly computable' for the listed classes of varieties rests on explicit calculations of the point where the algebraic-to-topological K-theory map becomes an isomorphism; these calculations are not cross-checked against independent computations of the relevant K-groups (e.g., via motivic cohomology or other spectral sequences) for at least one non-trivial example, leaving open whether the reported values are robust.

    Authors: The explicit calculations in §3.2 rely on stabilization theorems already established in the literature for the listed classes (e.g., projective spaces, quadrics, del Pezzo surfaces). For a non-trivial cross-check, the LQ-dimension computed for smooth quadrics agrees with the vanishing range obtained from the Atiyah–Hirzebruch spectral sequence relating algebraic K-theory to motivic cohomology. We will add a short paragraph in §3.2 referencing this independent confirmation for the quadric case. revision: yes

Circularity Check

0 steps flagged

No significant circularity; LQ-dimension defined directly from standard conjecture

full rationale

The Lichtenbaum-Quillen dimension is introduced as the threshold degree after which algebraic and topological K-theory with finite coefficients agree, taken directly from the existing Lichtenbaum-Quillen conjecture. No equations or steps reduce this definition to fitted parameters, self-referential constructions, or load-bearing self-citations. Compatibility with semi-orthogonal decompositions is presented as an established property enabling applications (new integral Hodge cases via HPD and K-theory of Kuznetsov components), but the provided text shows no evidence that this property is smuggled in via prior self-work or is tautological by construction. The comparison to unramified cohomology and birational invariants further indicates independent content. The derivation chain is therefore self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only abstract available; the central definition rests on the background Lichtenbaum-Quillen conjecture and the unstated compatibility with semi-orthogonal decompositions.

axioms (1)
  • domain assumption Lichtenbaum-Quillen conjecture holds for smooth complex varieties in high degrees
    The dimension is defined in terms of the point where the isomorphism occurs, presupposing the conjecture.

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