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Thomason's completion for K-theory and cyclic homology of quotient stacks

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Pith's one-line read The derived completion of equivariant K-theory at the augmentation ideal of the representation ring is the ordinary K-theory of the bar construction of the action — Thomason's completion problem, proved integrally for special groups and…

desk verdict A serious, ambitious preprint that plausibly settles Thomason's completion problem in several new cases, but the load-bearing A1-homotopy bridge (Proposition 9.3) rests on an unverified imported lemma, so the integral special-group theorem is not yet fully certified. read the letter →

arxiv 2502.09462 v1 pith:HSGZFEUB submitted 2025-02-13 math.AG

Amalendu Krishna , Ritankar Nath This is my paper · ORCID
classification math.AG MSC 14F4319D55
keywords equivariantK-theoryThomasoncompletiontheoremderivedofspectrabarconstructionBorelcyclichomologyquotientstacksaugmentationideal
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

Thomason's completion problem asks whether equivariant algebraic K-theory of a group action, after being completed at the augmentation ideal of the representation ring, becomes the ordinary K-theory of the action's bar construction — the simplicial scheme whose $n$-simplices are $G^n \times X$. This paper proves that statement: for special groups the derived completion of equivariant $K'$-theory is weakly equivalent to $K'(X^\bullet_G)$ integrally, while for arbitrary groups the equivalence holds rationally and for Bott-inverted finite coefficients prime to the characteristic. The proof runs through a second, cheaper statement that the same completion is always — integrally, for every linear algebraic group — the ordinary $K'$-theory of the Borel construction $X_G$, an ind-space approximating the homotopy orbit space. Because equivariant K-theory admits few direct computational tools, the upshot is that completed equivariant K-theory can be computed by ordinary K-theory, where motivic-cohomology machinery applies. For actions with finite stabilizers the paper removes even the completion: equivariant K-theory and all equivariant homology theories are described without completion by bar and Borel constructions of the inertia scheme, and the equivariant Hochschild and cyclic homology groups are computed explicitly.

What carries the argument

The argument is carried by four objects. The bar construction $X^\bullet_G$ is the simplicial scheme with $n$-simplices $G^n \times X$ and face maps built from the action and projections; it is the Čech nerve of the quotient map $X \to [X/G]$, and its K-theory is $\mathrm{holim}_n K'(G^n \times X)$. The Borel construction $X_G$ is the homotopy limit of ordinary K-theories of the quotients $X \times^G U_i$, where $(V_i, U_i)$ is an admissible gadget — a sequence of $G$-representations with free open subsets whose complements have growing codimension — and its $K'$-theory is independent of the gadget. Between them sits the derived completion of spectra at $I_G$, the spectrum-level completion whose homotopy groups are compatible with classical algebraic $I_G$-adic completions. The proof of the harder bar-construction theorem uses the A1-homotopy category: a motivic Borel space $X_G(\rho)$ is shown to be independent of the admissible gadget and, for special $G$, canonically isomorphic in that category to the bar construction, via the universal torsor $EG \to BG$ over the classifying space; this is Proposition 9.3, the step that forces the specialness hypothesis. A twisting operator $t_g$, acting by characters on the decomposition of equivariant K-theory over the character group of a central diagonalizable subgroup, then transports the completion at any maximal ideal $m_g$ to the augmentation ideal $I_G$, reducing completions at other ideals to the same theorem. For the homology theories, the equivariant Hochschild homology is identified with the derived loop space of the quotient stack (via the formal HKR theorem), and the cyclic and periodic cyclic homology statements are obtained by commuting the $S^1$-orbit construction past homotopy limits using a HKR filtration and boundedness of the derived odd tangent bundle.

What would settle it

Test the specialness hypothesis directly: take a non-special group, for instance the orthogonal group $O_2$, acting on a point over an algebraically closed field, and compare the homotopy groups of the derived completion of $K'_{O_2}(\mathrm{pt})$ at the augmentation ideal with the ordinary K-theory of the simplicial scheme $O_2^\bullet$. The rational comparison must be an equivalence by Theorem 1.1(2); the integral comparison is predicted to fail, with the obstruction tracing to the failure of Galois descent for algebraic K-theory of fields. Agreement of all integral homotopy groups would show the specialness hypothesis in Theorem 1.1(1) is unnecessary, while a single non-vanishing integral class would confirm it is exactly the right boundary.

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Extended reading notes

Core claim

The paper's central claim is that the pull-back along the stack quotient map induces a weak equivalence of spectra $\tilde\pi^* : K'_G(X)^{\wedge}_{I_G} \to K'(X^\bullet_G)$, where the left-hand side is the derived completion of $G$-equivariant $K'$-theory at the augmentation ideal $I_G$ of the representation ring $R(G)$ and the right-hand side is the ordinary $K'$-theory of the bar construction $X^\bullet_G$ (the simplicial scheme with $n$-simplices $G^n \times X$, the Čech nerve of $X \to [X/G]$). For a special linear algebraic group $G$ the equivalence is integral; for arbitrary $G$ it holds after rationalization, and for Bott-inverted K-theory with $\mathbb{Z}/m$-coefficients with $m$ prime to the characteristic when the base field contains all roots of unity. The paper also proves the companion Borel-construction statement, $K'_G(X)^{\wedge}_{I_G} \simeq K'(X_G)$, where $X_G$ is built from admissible gadgets, and this one holds integrally for every linear algebraic group. To recover all of equivariant K-theory, not just the augmentation-ideal completion, the paper proves the analogous completion theorem at every maximal ideal $m_\Psi$ of $R_k(G)$ associated to a conjugacy class $\Psi$ of a semisimple element $g$: the completed equivariant $K'$-theory is the ordinary $K'$-theory of the bar (or Borel) construction of the fixed-point locus $X^g$ under the centralizer $Z_g$. Finally, when the action has finite stabilizers, no completion is needed at all: equivariant $KH$-theory and the equivariant Hochschild, negative cyclic, cyclic, and periodic cyclic homology spectra are weakly equivalent to the corresponding ordinary theories of the bar and Borel constructions of the inertia scheme $IX$, yielding explicit formulas for their homotopy groups in terms of étale cohomology of the inertia stack.

Load-bearing premise

The load-bearing premise is that the two geometric models of the orbit space — the Borel approximation built from open subsets of representation spaces, and the simplicial bar construction built from $G^n \times X$ — are the same space in the A1-homotopy category of motivic spaces, an identification the paper only establishes for special groups; if it failed, the integral bar-construction theorem would collapse.

Editorial extensions

If this is right

  • For a special group, equivariant $K'$-theory up to derived completion can be computed as the ordinary $K'$-theory of the Čech nerve $G^\bullet \times X$ of the action, so standard motivic-cohomology machinery applies to equivariant problems.
  • For every linear algebraic group, the Borel-construction form holds integrally: the completion $K'_G(X)^{\wedge}_{I_G}$ is always the ordinary $K'$-theory of the ind-space $X \times^G U_i$ built from admissible gadgets.
  • If a $G$-equivariant morphism between smooth schemes is an A1-weak equivalence after forgetting the action, then the completed equivariant K-theories are weakly equivalent (rationally; integrally for special $G$), as stated in Corollary 1.3.
  • With finite stabilizers, equivariant $KH$-theory and equivariant Hochschild, negative cyclic, cyclic, and periodic cyclic homology are uncompleted ordinary theories of the bar and Borel constructions of the inertia scheme $IX$, and Theorem 1.14 gives explicit formulas for all equivariant Hochschild and cyclic homology groups of a smooth scheme in terms of étale cohomology of the inertia stack.
  • Completions at any maximal ideal $m_\Psi$ of $R_k(G)$ are also computable: they are bar or Borel constructions of the fixed-point locus $X^g$ under the centralizer of $g$, for each semisimple conjugacy class $\Psi$.

Reading between the lines

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

  • The contrast between the integral Borel theorem (all groups) and the integral bar theorem (special groups only) suggests the obstruction is purely the A1-homotopic identification of the two orbit-space models; any future tool that classifies $G$-torsors by maps into $BG$ for non-special $G$ would immediately extend Theorem 1.1(1) to those groups — a direction the paper does not pursue.
  • The twisting operator that carries completions between maximal ideals and the augmentation ideal is a transferable mechanism: any equivariant theory with an action of the representation ring and a character decomposition analogous to Proposition 4.2 should admit the same completion theorems, and equivariant motivic cohomology or hermitian K-theory are natural candidates to test.
  • The finite-stabilizer theorems recast completion as an inertia effect: the uncompleted equality with bar and Borel constructions of $IX$ suggests that for general actions the deficit between $KH_G(X)$ and its completion is governed by the non-quasi-finite part of the inertia scheme, so replacing $IX$ by the derived inertia stack might remove completions for all actions.
  • The characteristic-zero homology theorems parallel the K-theory theorems through the trace maps from K-theory to Hochschild and cyclic homology; since the trace to topological Hochschild homology is $S^1$-equivariant in the same way, the positive-characteristic analogue the authors flag as future work would likely follow once an equivariant cdh-descent statement exists in that setting.
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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 / 4 minor

Summary. The paper proves several completion theorems for equivariant K-theory and cyclic homology of quotient stacks over a field. For a linear algebraic group G acting on an algebraic space X, the authors use Lurie's derived completion at the augmentation ideal I_G and prove (Theorem 1.1) that the pull-back from the stack quotient to the bar construction induces a weak equivalence K'_G(X)^hat_{I_G} ≃ K'(X●_G) when G is special, with rational and Bott-inverted finite-coefficient versions for arbitrary G. Theorem 1.4 gives the analogous statement for the Borel construction for all groups with integral coefficients. The paper also proves completions at other maximal ideals, results for homotopy K-theory, and completion theorems for Hochschild, negative cyclic, cyclic and periodic cyclic homology, with applications to Deligne-Mumford quotient stacks. The proof of Theorem 1.1 is based on an A1-homotopy identification, Proposition 9.3, between the motivic Borel space and the bar construction.

Significance. If the main theorems are correct, the paper solves a fundamental problem posed by Thomason in 1986 in important cases, giving an integral algebraic Atiyah-Segal completion theorem for special groups and a clean formulation of the Borel-construction version for all groups. The strategy of proving Theorem 1.1 by combining the Borel-construction theorem with an A1-homotopy equivalence between the Borel and bar constructions is original and potentially very useful. The paper also contains several results of independent interest, such as the derived nonabelian completion theorem (Theorem 8.8) and decomposition theorems for equivariant K-theory. The proof of Theorem 1.4 in Section 7 is detailed and proceeds by a convincing reduction to split tori using Morita spaces. However, two load-bearing points obstruct certification: Proposition 9.3 relies on an unverified assertion that admissible gadgets are 'nice' in the sense of Morel-Voevodsky, and Lemma 8.3 is stated without proof. These points are directly used in the proofs of Theorems 1.1 and 1.9, respectively, so the central claims cannot be considered fully established without further work.

major comments (2)
  1. [Section 9.3, Proposition 9.3] The proof of Proposition 9.3, in the paragraph after Eq. (9.11), asserts that since the base field is a field and G is special, the admissible gadget ρ of Definition 6.7 is a 'nice admissible gadget' in the sense of [72, Chap. 4, Defn. 2.4], and then applies [72, Chap. 4, Lem. 2.9] to conclude that the diagonal map u_X: X/G × F → X/G × E●_F is an A1-weak equivalence. Neither the verification that ρ is nice nor the precise statement of Lemma 2.9 is provided. The codimension-growth conditions in Definition 6.7(2)-(3) are visibly weaker than contractibility-type conditions on the complements U_i\setminus U_{i-1}, and no independent argument is given that F = colim_i U_i is A1-contractible. This step is load-bearing: the zig-zag in (9.12) is the only bridge from the Borel construction (Theorem 1.4) to the bar construction (Theorem 1.1(1)). If the 'nice' assertion fails, the identification X_G(ρ) ≅ X●_G in H(k) may not hold and the proof of Theorem 1.1 collapses. Please supply a complete proof of the 'nice' property for the gadget constructed in Lemma 6.8(3) (or for arbitrary admissible gadgets) and state the hypotheses of Lemma 2.9 explicitly.
  2. [Section 8.1, Lemma 8.3] Lemma 8.3 states that if g ∈ Z(G) and X^g = ∅, then the derived completion K'_G(X,k)^hat_{m_g} is weakly contractible. The proof is omitted with the remark that it is 'easily proved by a routine modification of the proof of [31, Thm. 5.1]'. This lemma is used directly in Corollary 8.4 and hence in the derived nonabelian completion theorem (Theorem 8.8), which in turn feeds Theorems 1.9 and the homology-theoretic completion results. Since the passage from Edidin-Graham's classical nonabelian completion theorem (which concerns homotopy groups) to a statement about derived completions is precisely the kind of step the paper elsewhere emphasizes as non-formal, this omission is significant. Please either include a full proof of Lemma 8.3 or give a precise statement of [31, Thm. 5.1] together with a detailed explanation of the necessary modifications.
minor comments (4)
  1. [Throughout] The manuscript contains numerous formatting and OCR artifacts (e.g., '/slash.left', '/d47/d47', 'K-theor y') that make it difficult to read. A production-level cleaning is needed before final submission.
  2. [Section 7.2, Proof of Theorem 7.5] The proof asserts that I_B = sqrt(I_G R(B)) and I_T = sqrt(I_G R(T)) by [29, Cor. 6.1]. As stated in Section 2, [29, Cor. 6.1] is about the I_G-adic topology on R(G) itself, not about the images of I_G in R(B) or R(T). Since this equality of radicals is used to pass between the I_G-completion and the I_B- or I_T-completion, a proof or precise citation for these equalities should be supplied.
  3. [Section 6.2, Definition 6.7] In condition (4) of Definition 6.7, the wording 'Ui/G ∈ Smk' presupposes that the quotient exists as a smooth scheme. It would be clearer to state explicitly that Ui/G is required to be a smooth algebraic space (or scheme), and to note that the existence of the quotient in the relevant cases follows from Corollary 6.6.
  4. [Remark 1.2(3)] The remark that K'-theory can be replaced by K-theory when X is regular should be stated as a precise corollary with the necessary hypotheses, since the bar construction target K'(X●_G) is defined via pseudo-coherent complexes while K-theory uses perfect complexes.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: Theorem 1.1 is assembled from the independently proved Borel-construction completion (Theorem 1.4) and an A1-identification of Borel and bar models (Proposition 9.3); flagged unverified imports are gaps, not self-referential reductions.

full rationale

The derivation chain is not circular. Theorem 1.4, which identifies K'_G(X)^hat_{I_G} with the ordinary K'-theory of the Borel construction, is proved in §7 directly from admissible gadgets, the split-torus case, split-reductive reduction, and a Morita embedding into a general linear group; it does not use the bar-construction target of Theorem 1.1. The bridge from Borel to bar is Proposition 9.3, whose proof compares X_G(ρ) with X●_G in the A1-homotopy category using the universal G-torsor and Lemma 2.9 quoted from Morel-Voevodsky [72, Chap. 4]. That cited lemma and the assertion that specialness makes the gadget 'nice' are not proved in the paper, and the Skeptic's concern about [72, Chap. 4, Lem. 2.9] is a legitimate correctness or completeness risk; however, nothing in the quoted argument assumes the conclusion of Theorem 1.1 or an equivalent completion statement. The omitted Lemma 8.3, asserted by 'routine modification' of Edidin-Graham [31, Thm. 5.1], likewise feeds Theorem 8.8 without importing the target theorem. Citations to the authors' prior work provide auxiliary inputs such as cdh descent, Borel K-theory formalism, and representation-ring facts; these do not contain the completion equivalence being proved, so they are real evidence rather than circular premises. No equation in the manuscript identifies the claimed output with an input by construction, and no fitted parameter is relabeled as a prediction. Hence no circular step is exhibited; score 0.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The results are pure mathematics; no numerical parameters are fitted. The axioms are imported theorems and standard domain assumptions. The load-bearing imported machinery includes Lurie's derived completion, A1-homotopy theory, cdh descent, HKR, and equivariant resolution of singularities. No speculative entities are introduced.

assumptions (6)
  • domain assumption Existence of equivariant resolution of singularities for G-schemes in characteristic zero.
    Used in Proposition 5.8, Proposition 6.10, Theorem 7.6, Lemma 8.10 and Corollary 8.11 to reduce singular cases. The authors acknowledge this is a hindrance in positive characteristic.
  • standard math Lurie's derived completion of spectra exists and has the properties listed in Proposition 5.3.
    Imported from Lurie's Higher Algebra, Chapter 7.3. The formulation of all main theorems depends on this derived-completion machinery.
  • standard math A1-homotopy theory of Morel-Voevodsky provides universal G-torsors for special groups and representability of K-theory by KGL.
    Used in Section 9 to construct the motivic Borel space and to identify X_G(ρ) with the bar construction X●_G. Specialness of G is essential here.
  • standard math Thomason's theorem for finite-coefficient Bott-inverted equivariant K-theory over separably closed fields.
    Serves as a prior benchmark and as an ingredient in some comparisons. The new contribution removes separably closed restrictions and adds integral and rational cases.
  • standard math cdh descent for equivariant KH-theory, taken from the authors' earlier work [50].
    Used in Proposition 6.10 and Theorem 7.6 to reduce singular schemes and to extend statements from K'-theory to KH-theory.
  • standard math HKR theorem and Toën's derived loop space description of Hochschild homology.
    Used in the proof of Theorem 1.10, where equivariant Hochschild homology is written as derived global sections of the derived loop space and then computed via the HKR theorem.

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Pith. "Pith review of Thomason's completion for K-theory and cyclic homology of quotient stacks." pith.science (2026). https://pith.science/paper/HSGZFEUB

@misc{pith2026250209462,
  author       = {Pith},
  title        = {Pith review of: Thomason's completion for K-theory and cyclic homology of quotient stacks},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/HSGZFEUB}},
  note         = {Machine review of arXiv:2502.09462}
}
read the original abstract

We prove several completion theorems for equivariant K-theory and cyclic homology of schemes with group action over a field. One of these shows that for an algebraic space over a field acted upon by a linear algebraic group, the derived completion of equivariant K'-theory at the augmentation ideal of the representation ring of the group coincides with the ordinary K'-theory of the bar construction associated to the group action. This provides a solution to Thomason's completion problem. For action with finite stabilizers, we show that the equivariant K-theory and cyclic homology have non-equivariant descriptions even without passing to their completions. As an application, we describe all equivariant Hochschild and other homology groups for such actions.

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