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External Quillen acyclicity and the actual local free-class model

PDF p. 11: Lemma5.3, second Chern additivity

Statement

Assume the Axiom of Choice and the fixed compatible universe and site-size bounds of the supplied constructions.

At a Nisnevich local ring $R$ , the actual free-module map into the sheaf K-space model is $Z \times B G L (R) ⟶ Z \times B G L (R)^{+} ≃ Ω^{\infty} K_{P e r f} (R),$ with $K_{0} (R) = Z$ and the first arrow integral-homology acyclic on each component. It is the canonical plus/completion arrow used by the completed universal Chern construction. The categorical comparison with perfect K-theory is natural and matches free object/core maps. Block sum corresponds to the direct-sum K-space operation. On arbitrary affine rings with nonfree projectives the displayed rank component model is not asserted for all $K_{0}$ ; it is used only locally in the sheaf sense.

External input

Import the exact plus acyclicity and explicit free-module comparison, retaining the completed enhancement/model inputs. The passage from this local comparison to a global identity is the separately authored normalized-chain adjunction argument; it is not justified by affine group agreement.

Facts & Assumptions

Given: The stated scheme and model hypotheses, Choice and the fixed universe and site-size bounds.

[A1]

Assume the Axiom of Choice as formulated in The Axiom of Choice and the stated universe and site-size bounds. They are inherited from the completed universal Chern construction and enhanced perfect K-space comparison in the dependency list, including small presentations and local descent. The external local plus comparison is applied under those same assumptions.

Depends on

Used by · 3

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, Chapter IV · Definition1.1.1/Functoriality1.1.2 p3; Definitions1.4/1.4.1, Theorem1.5, Lemma1.6 pp4–5; Constructions1.9(ii) p6; §4 group-completion and plus comparisons4.9/4.11.1 pp38–45; Theorem IV.7.1, printed p61 (PDF page61), for group-completion/Q; Corollary IV.7.2, printed p62 (PDF page62), for the resulting plus/Q comparison

The canonical plus map is acyclic and hence an integral homology isomorphism; Bousfield–Kan integral completion is a functorial model for BGL⁺. Actual K-space addition is induced by direct sum/block sum through the explicit group-completion/Q comparison. This yields the stated acyclic free-class test on local stalks, not all global affine K-spaces.

Hypotheses: The plus construction is on connected CW spaces/componentwise; for the local free-module K-space use local rings, whose finite projectives are free and K0=Z. The actual sheaf/model comparison is that of the completed universal-Chern and connective/nonconnective suppliers. Theorem IV.7.1 requires a split exact category. Finite projective modules satisfy this hypothesis because a surjection onto a projective module splits; over the local ring these modules are finite free.

Reading limits: The inherited source-authoring record reports reading pp3–6 completely, including Theorem1.5 statement and Lemma1.6 proof, and the stated §4 actual comparisons in earlier K1 source authoring. This adjudication independently read the author-hosted Chapter IV PDF text accessed 2026-11-10, §7 opening through Theorem7.1 and its following explanation on printed p61, and Corollary7.2 on printed p62. Theorem7.1 and Corollary7.2 are imported as established inputs; the full proof of Theorem7.1 and its prerequisites were not reconstructed or checked here. The split-exact hypothesis was checked in the present finite-projective application. Plus-existence obstruction theory is imported, not reconstructed. No universal all-affine K0=Z claim is made.

Sources

Charles Weibel, The K-book, Chapter IV · Definition1.1.1/Functoriality1.1.2 p3; Definitions1.4/1.4.1, Theorem1.5, Lemma1.6 pp4–5; Constructions1.9(ii) p6; §4 group-completion and plus comparisons4.9/4.11.1 pp38–45; Theorem IV.7.1, printed p61 (PDF page61), for group-completion/Q; Corollary IV.7.2, printed p62 (PDF page62), for the resulting plus/Q comparison