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Full affine K-space comparison and its sheafification

PDF p. 17: Corollary 6.9: actual supported Chern operation; affine perfect K-space prerequisite

Statement

Assume the Axiom of Choice and the fixed site and universe bounds. For any commutative unital ring $A$ , let $P (A)$ be its exact category of finite projective modules and let $C (A) = {C h}^{b} (P (A))$ have degreewise split cofibrations and quasi-isomorphisms. The canonical comparisons give natural equivalences of entire pointed spaces $Ω B Q P (A) ≃ Ω^{\infty} K^{W} (C (A)) ≃ Ω^{\infty} K_{P e r f} (Spec A) .$ Here $K^{W}$ is connective Waldhausen K-theory and $K_{P e r f}$ is the specified nonconnective perfect-complex theory. The equivalences include components, preserve direct sum and carry the actual degree-zero object and automorphism maps to those of the same perfect objects.

For every qcqs scheme $X$ , these comparisons identify the local sheaf models of the vector-bundle Q-space presheaf and the perfect K-space presheaf on the small Zariski or Nisnevich site, and on the smooth big site with either topology. Quasi-compact objects and quasi-compact open covering bases suffice. Here the local model uses derived sheafification of spaces; when using the source's stalkwise local homotopy convention, take its hypercomplete replacement consistently on both sides. The perfect K-space has its canonical natural map into derived sections of the chosen common model. No global equality of vector-bundle and perfect $K_{0}$ is asserted.

External input

Import the full S/Q, Gillet–Waldhausen and BGT comparisons at the exact frontmatter locators. In particular, import BGT Remark 9.33's degree-zero clause with its idempotent-completeness hypothesis, in addition to the positive-degree comparison already supplied locally. The following proof checks their application and the local sheaf consequence; it does not reconstruct the imported foundations.

Facts & Assumptions

[A1]

Choice and the fixed universes are used for the supplied essentially small enhanced models and small presentations. The Axiom of Choice

[F1]

The cited Weibel IV.8.5–8.7 and V.2.2 give the natural entire S/Q and bounded-projective K-space comparisons. The object/automorphism maps are the core maps in IV.8.3.2. The current full Quillen–Waldhausen supplier states this entire comparison, with its degree-zero object map; it is used without modification. Full projective Quillen–Waldhausen comparison and the local free-class map

[F2]

The enhanced supplier gives the natural full Waldhausen-to-connective comparison and the canonical positive-degree comparison with the specified nonconnective functor. The additional degree-zero clause of BGT Remark 9.33 applies when the underlying stable category is idempotent complete. External connective, nonconnective and automorphism comparisons

[F3]

The actual bounded-projective and associated-sheaf enhanced equivalences are supplied here. The specified perfect category is idempotent complete, and its degree-zero object-class comparison is the affine Euler bridge. Bounded finite-projective complexes and affine chain models Enhanced affine perfect equivalence Nonconnective K theory of perfect complexes Affine perfect K zero and bundle K zero

[F4]

The exact HTT input above supplies derived space-valued sheafification and its left exactness, and separately the further left exact hypercompletion when a stalkwise local model is chosen. These are not inferred from set-valued sheafification. Asakura–Sato Definition 5.1 and Proposition 5.3 specify the formal local Q-space model. The site and size conventions are supplied here. Derived sections of a space-valued local model mean evaluation after the chosen derived local replacement; this is not an extension of the separate definition scoped to Nisnevich, cdh and étale abelian sections. Scheme sites and sheafification

Proof

Given: The ring, sites, category models, Choice and size bounds in the Statement.

Goal: Establish the full affine comparison and its precise local use.

1.1A1F1F2F3given

Every surjection between finite projective modules splits, and its kernel is finite projective, so the application of Gillet–Waldhausen in [F1] is valid in the ambient module category. The K-projective property in [F3] makes a bounded acyclic projective complex contractible. The mapping cylinder $Q \oplus Cone ({i d}_{P})$ factors a map $P \to Q$ as a degreewise split cofibration followed by a quasi-isomorphism, within bounded finite projectives. A map invertible in the localization is a quasi-isomorphism because its image in the derived category is invertible. This verifies factorization and DHKS-saturation for [F2]. The actual localization is the full affine perfect category of [F3], which is stable and idempotent complete.

2.1F1F2F3step 1.1

Compose the S/Q equivalence, degree-zero bounded-complex inclusion, enhanced comparison and canonical connective-to-nonconnective map from [F1]–[F3]. The comparison is an isomorphism in all positive homotopy degrees. BGT's additional degree-zero clause applies by step 1.1; the affine Euler bridge also identifies this same degree-zero object-class map. Thus it is an equivalence of full spaces, including components. Naturality for extension of scalars follows from the actual functors: it preserves projectives and split cofibrations, and it preserves quasi-isomorphisms here because their bounded projective cones are contractible. The core maps carry each object and automorphism to itself, and direct sums are preserved.

3.1F1F2F3F4step 2.1∎

On each indicated site every object has an affine open covering, whose affine pieces remain objects of that site. On the small Nisnevich site these are affine étale objects; on the smooth big site these are affine smooth objects. Intersections need not be affine: their affine open covers suffice again. The exact degree-zero bundle-to-perfect functor, with the natural S/Q comparison, gives the Q-space-to-perfect-space arrow on every object, compatibly with pullback; on affines it is the arrow of step 2.1. It is therefore an equivalence on a covering basis, and hence an equivalence after derived space-valued sheafification by [F4]: restriction to the affine basis identifies the presheaves, and covering descent reconstructs their sheafifications on all objects. Left exactness also identifies the looped Q model. Applying the further left exact hypercompletion gives the same comparison in the source's stalkwise local convention, without identifying the two conventions on arbitrary presheaves. For the presheaf $K$ of perfect K-spaces, write $L_{τ}$ for the chosen derived local replacement. The natural unit $K \to L_{τ} K$ , followed by derived evaluation on $X$ , supplies its map into sections of that common model. This only uses local comparison and the unit; it neither assumes hyperdescent of the original presheaf nor identifies global vector-bundle $K_{0}$ with perfect $K_{0}$ . Restrictions and the units commute by naturality.

Source versions and reading limits

The cached author-hosted Chapter IV PDF has SHA256 9f1c1b8cccfe19d547c27dd04c61f198fd7a0cddd0018a0b84442b00fa575248; BGT arXiv:1001.2282v4 has SHA256 08a8ae7fb5715d8269fd728c8a20d72403527aaefbeaa974b7040ec21d667a2d; Asakura–Sato's 2019 journal PDF has SHA256 6736b9710dd4d8d5fd89dfa2863a72cf90fc274ba47a45651fc7024209b4dcbf. These cached bytes were hashed and the passages specified above were read in this assignment. The cached Chapter V PDF has SHA256 52dcc8ee3a1764e5ea309c59f093ac8e2a1ea64f3b94bacc05e6a2b6125b1da8; its stated passages were also read directly. HTT was read through the live primary PDF extraction, accessed 9 November 2026, within the stated limits; no local-byte hash is claimed for HTT. The corresponding live URLs were opened. Long comparison, completion and localization foundations remain precise external inputs.

Depends on

Used by · 2

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Charles Weibel, The K-book, Chapters IV and V · V.2.2 and proof, pp. 13–14; V.2.7.2, p. 18; canonical S/Q comparison IV.8.5–8.7, pp. 74–75 in Kbook.IV.pdf

The degree-zero exact inclusion of finite projectives into bounded projective complexes induces a natural equivalence of entire K-spaces. The canonical exact-category S and Q constructions agree, preserving object maps.

Hypotheses: Finite projective modules over an arbitrary commutative unital ring; split exact structure; degreewise split cofibrations and quasi-isomorphisms on bounded complexes. Kernels of surjections of projectives are projective.

Reading limits: This adjudication read the hash-checked cached author-hosted Chapter V PDF, Theorem V.2.2 and its displayed proof, printed pp. 13–14, and V.2.7.2–2.7.3 on p. 18; the live primary URL and relevant extracted passages were also opened. The long foundational localization/approximation results are imported, not reconstructed. The Chapter IV comparison is recorded separately below.

Charles Weibel, The K-book, author-hosted Chapter IV · IV.8.3.2 and IV.8.5, p. 68; Exercises IV.8.5–8.7, pp. 74–75

The canonical exact-category S/Q comparison identifies the entire K-spaces and their object and automorphism maps.

Hypotheses: A small exact category, here finite projective modules in the fixed universes.

Reading limits: This adjudication read the cached author-hosted Chapter IV PDF, p. 68 in full and the complete published S/Q comparison outline in Exercises 8.5–8.7, pp. 74–75. The exercise outline and its cited Waldhausen foundations are imported as a precise external input; no completed independent reconstruction of those foundations is claimed.

Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory, arXiv:1001.2282v4 · Lemma 7.11/Corollary 7.12, pp. 48–49; Proposition 9.32 and Remark 9.33, p. 64; Definition 9.6/Theorem 9.8, p. 58

For a saturated Waldhausen category with factorization the canonical entire connective K-space agrees naturally with the infinity-localization K-space. The canonical connective-to-nonconnective map is an isomorphism in positive degrees and also degree zero for an idempotent-complete underlying category.

Hypotheses: Smallness in the fixed universes, DHKS-saturation and factorization; bounded projective complexes satisfy these. The target perfect category is stable and idempotent complete.

Reading limits: This adjudication read primary arXiv:1001.2282v4, the full text of Lemma 7.11 and Corollary 7.12 with their displayed proofs, pp. 48–49, and Proposition 9.32/Remark 9.33 on p. 64. The canonical identification of the specified nonconnective functor with IK is reused from the inspected enhanced-comparison supplier, not newly reconstructed. No scan inspection or new reading of the long Theorem 9.8 proof is claimed.

Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, 2019 · Definition 5.1 and §5 Steps 1–3, printed pp. 269–271

The K-space sheaf is the local sheafification of the exact-category Q presheaf; the universal class is completed on this sheaf model.

Hypotheses: Chosen local site and simplicial sheaf homotopy category; here only the formal sheaf-model step is used, with actual coefficient classes supplied separately.

Reading limits: This adjudication read the cached 2019 journal PDF text of Definition 5.1, all three construction steps, Proposition 5.3 and formula (5.3.1), printed pp. 269–271, and the supported construction and Proposition 6.1(1), pp. 272–273. Only the formal local construction is imported.

Jacob Lurie, Higher Topos Theory, author-hosted PDF (accessed 9 November 2026) · Proposition 6.2.2.7, printed p. 579, proof pp. 582–583; paragraph following Proposition 6.5.2.8, printed p. 669 (PDF p. 687)

Sheafification of space-valued presheaves on a small Grothendieck site is a left exact localization. Hypercompletion is a further left exact localization. Each has its own natural unit; no assertion that these two localizations coincide is made.

Hypotheses: A small site in the specified universes, or its nerve; for hypercompletion, the associated infinity-topos.

Reading limits: Read the live author PDF extraction: Proposition 6.2.2.7, the construction and adjunction argument in its proof on pp. 582–583, and the defining paragraph on p. 669 immediately after Proposition 6.5.2.8. The stated localization theorem and its foundational inputs are imported; no complete reconstruction of the transfinite argument, full-book reading, or equality of ordinary and hypercomplete descent is claimed.

Sources

Charles Weibel, The K-book, Chapters IV and V · V.2.2 and proof, pp. 13–14; V.2.7.2, p. 18; canonical S/Q comparison IV.8.5–8.7, pp. 74–75 in Kbook.IV.pdf
Charles Weibel, The K-book, author-hosted Chapter IV · IV.8.3.2 and IV.8.5, p. 68; Exercises IV.8.5–8.7, pp. 74–75
Blumberg–Gepner–Tabuada, A universal characterization of higher algebraic K-theory, arXiv:1001.2282v4 · Lemma 7.11/Corollary 7.12, pp. 48–49; Proposition 9.32 and Remark 9.33, p. 64; Definition 9.6/Theorem 9.8, p. 58
Asakura–Sato, Chern Class and Riemann–Roch Theorem for Cohomology Theory without Homotopy Invariance, 2019 · Definition 5.1 and §5 Steps 1–3, printed pp. 269–271
Jacob Lurie, Higher Topos Theory, author-hosted PDF (accessed 9 November 2026) · Proposition 6.2.2.7, printed p. 579, proof pp. 582–583; paragraph following Proposition 6.5.2.8, printed p. 669 (PDF p. 687)