{"id":"af727774-2ac3-46fc-a994-63563b894ba3","arxiv_id":"2502.09462","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":8.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"The paper proves completion theorems identifying completed equivariant K-theory and cyclic homology of quotient stacks with ordinary K-theory and homology of bar or Borel constructions, and gives explicit formulas for finite-stabilizer actions.","lead":"This paper proves long-conjectured completion theorems: for group actions on algebraic spaces, the completed equivariant K-theory equals the ordinary K-theory of the bar or Borel construction, with integral form for special groups and rational or finite-coefficient form in general. It also computes equivariant Hochschild and cyclic homology for quotient stacks with finite stabilizers.","discovery_kind":"unification","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Proposition 9.3's A1-identification of the Borel and bar constructions rests on an unverified imported lemma ([72, Chap. 4, Lem. 2.9]) and an unproved claim that every admissible gadget for a special group is 'nice'; if that application fails, the bridge to Theorem 1.1(1) collapses.","rationale":"The reader's weakest assumption, Proposition 9.3, is indeed the load-bearing point for the headline integral theorem. My concern sharpens this: even granting the framework of A1-homotopy theory, the proof of Proposition 9.3 depends on an imported lemma whose hypotheses are not verified in the text. This is not an external-consensus objection; it is a gap in the written argument. If the cited lemma applies, the main construction likely goes through; if not, there is no visible substitute. I do not see a contradiction or a reason to reject outright, so the reader's conditional verdict remains appropriate. The unproved Lemma 8.3 is a real but secondary gap, since it supports the maximal-ideal and homology completions rather than the central Theorem 1.1. My concrete test is deliberately targeted: it asks for a verifiable translation of the cited lemma into the paper's setting rather than a broad re-derivation of the whole theory.","tokens_in":70661,"tokens_out":11312,"duration_ms":115241,"concrete_test":"Write out the proof of [72, Chap. 4, Lem. 2.9] and check its hypotheses for the gadget ρ = (V_i,U_i) constructed in Lemma 6.8(3) with G = GL_n: (a) verify that the recursive open U_{i+1} = (U_i ⊕ V) ∪ (V ⊕ U_i) satisfies the codimension and free-action conditions of [72, Chap. 4, Defn. 2.4]; (b) prove that F = colim_i U_i is A1-contractible, or exhibit an explicit A1-homotopy from u_X to the projection X/G × F → X/G; (c) confirm that the quotients U_i/G are schemes with the required properties. If all three checks hold for G_m and GL_n, the bridge stands; if any one fails, Proposition 9.3 and Theorem 1.1(1) are unsupported at that step.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The integral statement of Theorem 1.1(1) is obtained by combining Theorem 1.4, which identifies K'_G(X)^hat with the K'-theory of the Borel construction, with Proposition 9.3, which identifies the motivic Borel space X_G(ρ) with the bar construction X●_G in H(k). Proposition 9.3 is the least secure point. Its proof reduces to showing that the diagonal map u_X: X/G × F → X/G × E●_F is an A1-weak equivalence, and the sole justification is the assertion that G special guarantees that ρ is a nice admissible gadget for G in the sense of [72, Chap. 4, Defn. 2.4], followed by an application of the cited Lemma 2.9. Neither the verification that the gadget constructed in Lemma 6.8(3), or an arbitrary gadget of Definition 6.7, satisfies the nice definition, nor the precise statement and hypotheses of Lemma 2.9 are given. The codimension-growth conditions of Definition 6.7 are visibly weaker than the contractibility-type conditions that would make F = colim_i U_i A1-contractible, which is what the conclusion of Lemma 2.9 needs. This is exactly the place where specialness does essential work; if the implication from special to nice fails, the zig-zag in (9.12) does not produce an A1-equivalence and the Borel and bar models need not agree. A secondary gap is Lemma 8.3, asserted by routine modification of [31, Thm. 5.1] and omitted; it feeds Theorem 8.8 and hence the maximal-ideal and homology completions, though it is not the central bridge for Theorem 1.1.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":71045,"tokens_out":8445,"duration_ms":77133,"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":[{"comment":"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.","section":"Section 9.3, Proposition 9.3"},{"comment":"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.","section":"Section 8.1, Lemma 8.3"}],"minor_comments":[{"comment":"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.","section":"Throughout"},{"comment":"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.","section":"Section 7.2, Proof of Theorem 7.5"},{"comment":"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.","section":"Section 6.2, Definition 6.7"},{"comment":"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.","section":"Remark 1.2(3)"}],"recommendation":"major_revision","confidential_remarks":"The paper is extremely long and dense, and the two gaps identified above are concentrated in exactly the places where the most delicate comparisons are made. The A1-homotopy comparison in Proposition 9.3 is the hinge of the main theorem, and the 'nice gadget' assertion cannot be taken as a black box because Definition 6.7 does not obviously imply the properties needed for [72, Chap. 4, Lem. 2.9]. Similarly, Lemma 8.3 is a derived analogue of a known result and should not be left as an exercise. I would recommend asking the authors to write out these two proofs in full in the revision. If they can be supplied without introducing new hypotheses, the paper is likely to be acceptable in a top journal; the overall strategy and the breadth of applications are compelling."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Here's the short version: this is a serious, ambitious preprint that plausibly settles Thomason's completion problem in several new cases. The genuinely new material includes an integral solution for special groups, rational and finite-coefficient solutions for all linear algebraic groups, an integral Borel-construction theorem for all groups, and substantial results for cyclic homology and Deligne-Mumford stacks. The strategy is coherent: Theorem 1.1 is reduced to Theorem 1.4 plus an A1-homotopy identification of Borel and bar constructions, and the homology theorems go through HKR and derived loop spaces. I did not find circular reasoning, and the paper is honest about what is imported from Lurie and Morel-Voevodsky.\n\nThe soft spots are real and localized. Proposition 9.3 is the bridge that gets from the easier Borel construction to the bar construction, and its proof depends on two things that are not actually shown: that every admissible gadget for a special group is 'nice' in the sense of [72, Chap. 4, Defn. 2.4], and that Lemma 2.9 there has the precise hypotheses needed. The codimension-growth conditions in Definition 6.7 look weaker than the contractibility-type condition Lemma 2.9 would require. If the special-to-nice implication fails, the zig-zag in (9.12) does not produce an A1-equivalence, and Theorem 1.1(1) has a real gap. This is exactly where specialness does essential work, so it is not a cosmetic issue. Lemma 8.3 is also asserted without proof ('routine modification' of [31, Thm. 5.1]); it feeds Theorem 8.8 and the maximal-ideal and homology completions, so it should be filled in, though it is not the central bridge for Theorem 1.1. The abstract slightly overstates the integral bar-construction result, since the special-group restriction is structural.\n\nWho is this for? People working in equivariant K-theory, motivic homotopy, and Hochschild/cyclic homology of stacks. It deserves a serious referee: the claims are important, the proof structure is clear, and the suspect points are localized enough to check. I would not desk-reject it. Send it to peer review with referees who can actually verify Proposition 9.3 and the imported Lemma 2.9. I would want to see those details before relying on Theorem 1.1(1), but the paper is well worth referee time.","headline":"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.","tokens_in":71561,"tokens_out":3781,"would_cite":true,"duration_ms":34464,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["14F43","19D55"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["equivariant K-theory","Thomason completion theorem","derived completion of spectra","bar construction","Borel construction","cyclic homology","quotient stacks","augmentation ideal"],"falsifier":"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.","tokens_in":70463,"feed_emoji":"🔄","tokens_out":16277,"duration_ms":126732,"temperature":0.7,"pith_summary":"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.","feed_headline":"Equivariant K-theory, completed, matches bar-construction K-theory","feed_subtitle":"Derived completion equals ordinary K-theory of the bar construction; finite-stabilizer actions match without completion.","key_machinery":"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.","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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 $\\Psi$."],"supporting_citations":[{"why":"Thomason's 1986 paper that posed the completion problem and proved the Bott-inverted finite-coefficient case over separably closed fields; the statement Theorem 1.1 extends.","marker":"[86]"},{"why":"Thomason's construction of equivariant K-theory and of the bar construction as the Čech nerve of the stack quotient map.","marker":"[87]"},{"why":"The topological Atiyah-Segal completion theorem, whose algebraic analogue the paper proves.","marker":"[2]"},{"why":"Lurie's derived completion of module spectra, which supplies the spectrum-level completion $K'_G(X)^{\\wedge}_{I_G}$ that Thomason predicted but did not construct.","marker":"[70]"},{"why":"Morel-Voevodsky's A1-homotopy theory, which provides the motivic Borel space, the classifying space $BG$, and the universal torsor $EG \\to BG$ used to compare Borel and bar constructions.","marker":"[72]"},{"why":"Edidin-Graham's quotient approximations (admissible gadgets) used to define the Borel construction $X_G$.","marker":"[28]"},{"why":"Edidin-Graham's nonabelian completion theorem, whose derived version (Theorem 8.8) the paper proves and uses for completions at arbitrary maximal ideals.","marker":"[31]"},{"why":"Ben-Zvi-Nadler's formal HKR theorem, used to identify equivariant Hochschild homology with the derived loop space of the quotient stack.","marker":"[7]"},{"why":"Bhatt-Morrow-Scholze's filtration arguments, used to commute the $S^1$-homotopy orbit computation with homotopy limits in the proof for cyclic and periodic cyclic homology.","marker":"[9]"},{"why":"The earlier work showing finite coefficients and Bott inversion are both necessary in Thomason's theorem, which marks the boundary of the integral statement for non-special groups.","marker":"[60]"}],"fun_headline_variants":["Thomason's completion problem solved for equivariant K-theory","Derived completion of equivariant K-theory equals bar construction K-theory","For finite stabilizers, equivariant K-theory and cyclic homology need no completion","Equivariant K-theory completion matches ordinary K-theory of bar construction","Thomason's problem solved: equivariant K-theory from bar construction"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Thomason's completion problem solved for equivariant K-theory","Derived completion of equivariant K-theory equals bar construction K-theory","For finite stabilizers, equivariant K-theory and cyclic homology need no completion","Equivariant K-theory completion matches ordinary K-theory of bar construction","Thomason's problem solved: equivariant K-theory from bar construction"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000882,"raw_usage":{"total_tokens":3896,"prompt_tokens":1115,"completion_tokens":2781,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":731,"completion_tokens_details":{"reasoning_tokens":2683}},"tokens_in":731,"tokens_out":2781,"duration_ms":43668,"temperature":1.0,"reasoning_tokens":2683,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T21:23:06.640458+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Thomason, Comparison of equivariant algebraic and topological K-theory, Duke Math","cited_arxiv_id":null,"evidence_quote":"Thomason's 1986 paper that posed the completion problem and proved the Bott-inverted finite-coefficient case over separably closed fields; the statement Theorem 1.1 extends."},{"cited_title":"Thomason, Algebraic K-theory of group scheme actions , In: ‘Algebraic Topology and Algebraic K-theory’, Ann","cited_arxiv_id":null,"evidence_quote":"Thomason's construction of equivariant K-theory and of the bar construction as the Čech nerve of the stack quotient map."},{"cited_title":"Lurie, Spectral Algebraic Geometry , (2018), available on author’s homepage","cited_arxiv_id":null,"evidence_quote":"Lurie's derived completion of module spectra, which supplies the spectrum-level completion $K'_G(X)^{\\wedge}_{I_G}$ that Thomason predicted but did not construct."},{"cited_title":"Morel, V","cited_arxiv_id":null,"evidence_quote":"Morel-Voevodsky's A1-homotopy theory, which provides the motivic Borel space, the classifying space $BG$, and the universal torsor $EG \\to BG$ used to compare Borel and bar constructions."},{"cited_title":"Edidin, W","cited_arxiv_id":null,"evidence_quote":"Edidin-Graham's quotient approximations (admissible gadgets) used to define the Borel construction $X_G$."},{"cited_title":"Edidin, W","cited_arxiv_id":null,"evidence_quote":"Edidin-Graham's nonabelian completion theorem, whose derived version (Theorem 8.8) the paper proves and uses for completions at arbitrary maximal ideals."},{"cited_title":"Krishna, The completion problem for equivariant K-theory, J","cited_arxiv_id":null,"evidence_quote":"The earlier work showing finite coefficients and Bott inversion are both necessary in Thomason's theorem, which marks the boundary of the integral statement for non-special groups."}],"review_version":1}