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External matrix, exact and bounded-projective K-theory comparisons

PDF p. 9: Surjectivity of the determinant on K1

Statement

Assume AC. For every commutative unital ring A, let P(A) be the exact category of finite projective modules and let C(A)=Chᵇ(P(A)) be bounded cochain complexes, with degreewise split monomorphisms as cofibrations and quasi-isomorphisms as weak equivalences. The canonical matrix-automorphism maps and degree-zero inclusion give natural isomorphisms

GL(A)ab→∼π1KQ(P(A))→∼π1KW(C(A)).

The first is the classical matrix/Quillen comparison; the second includes the canonical exact-category S/Q comparison and Gillet–Waldhausen inclusion. The image of M∈GLn(A) is the standard automorphism class of M acting on Aⁿ in degree zero. In particular the one-by-one unit u maps to the class of multiplication by u on A. These comparisons are natural for extension of scalars along unital ring maps.

Source contract

Given: A commutative unital ring, AC, and the exact/Waldhausen structures specified above.

Goal: Import the actual classical comparison maps with their matrix-loop normalization.

[F1] Weibel III.1.1 identifies matrix K₁ with GL(A)_ab. IV.1.1, IV.4.9/4.11.1 and IV.7.1–7.2 identify its canonical automorphism maps with exact-category Quillen K-theory.

[F2] Weibel Exercise IV.7.9(c) matches the Quillen automorphism class with its group-completion class. IV.8.3.2/8.5 and Exercises IV.8.5–8.7 supply the canonical S/Q comparison and its automorphism-class normalization. V.2.2 supplies the K-theory equivalence induced by P(A)→C(A).

External input

Import [F1] and [F2] with the precise reading limits in the frontmatter. These are comparisons induced by actual categorical constructions, not an unqualified existence of abstract π₁ isomorphisms. The elementary checks that P(A) is closed under kernels of its surjections and that base change is weakly exact on C(A) are given in the target’s application. The local helpers independently prove determinant and its unit calculation; no determinant compatibility is inferred from a π₀ comparison.

Source correction: Exercise IV.8.7 cites IV.7.8 for the automorphism class; the intended reference is IV.7.9. This source cross-reference error does not change the imported comparison.

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, Chapter III · Definition III.1.1 and Example III.1.1.1, PDF pp. 1–2

Classical K₁(A)=GL(A)/[GL(A),GL(A)] is natural in unital ring maps. For a commutative ring determinant gives a surjection to A× split by the one-by-one matrices.

Hypotheses: Unital rings for matrix K₁; commutative unital rings for determinant and the unit section. No regularity or noetherian hypothesis.

Reading limits: Read the full definition and adjacent determinant example on pp. 1–2. The matrix determinant and its naturality are independently authored in the first two local helpers; no general Whitehead theorem is reconstructed.

Charles Weibel, The K-book, Chapter IV · Definition IV.1.1; Theorem IV.4.9 and explicit maps η_n(g)=(g,1), pp. 42–43; Theorem IV.4.11/Corollary IV.4.11.1, pp. 44–45; Theorem IV.7.1/Corollary IV.7.2, pp. 61–64; Exercises IV.7.8–7.9, printed and PDF p. 65; IV.8.3.2, IV.8.5 and Exercises IV.8.5–8.7, pp. 68,74–75

The canonical free-module automorphism maps identify matrix K₁(A)=π₁BGL(A)^+ with π₁ of exact-category K-theory of finite projectives. The split-exact group-completion and Quillen Q constructions agree. Waldhausen S and Quillen Q agree for exact categories, with the indicated automorphism classes matched by the explicit comparison.

Hypotheses: P(A), finite projective modules, is split exact. Stabilization sends M to diag(M,1). The exact-category S/Q comparison applies to any small exact category. Positive generator orientation is the usual automorphism class of M, as fixed by the source η_n map, Exercise IV.7.9(c), and Exercise IV.8.7 (with its cross-reference corrected to IV.7.9).

Reading limits: Prior recorded reading covered the stated plus/group-completion/Q comparisons and their displayed proofs/reductions in the listed passages, including the explicit η_n map and cofinality hypotheses. Read the complete canonical S/Q comparison construction and automorphism statement in Exercises 8.5–8.7; those are a published proof outline, not a locally completed reconstruction of Waldhausen’s original proof. Did not read Waldhausen’s original paper or reconstruct all group-completion and Theorem-B prerequisites. This adjudication read Exercises IV.7.8–7.9 and IV.8.5–8.7 only, including IV.7.9(c), in the Rutgers web text accessed 2026-11-10; visually checked printed/PDF pp. 65 and 75 in the primary cached PDF /tmp/k1-authoring/Kbook.IV.pdf (93 pages; PDF creation date 2012-08-17; SHA256 9f1c1b8cccfe19d547c27dd04c61f198fd7a0cddd0018a0b84442b00fa575248). The web text and the two cached pages agree on the cited passages; no byte identity with the live PDF is claimed. No new reading of the other comparison proofs is claimed.

Charles Weibel, The K-book, Chapter V · Theorem V.2.2 (Gillet–Waldhausen) and proof, pp. 13–14; §V.2.7.2, p. 18

For an exact category closed under kernels of surjections in an ambient abelian category, inclusion as complexes concentrated in degree zero induces a K-theory equivalence with its bounded chain-complex Waldhausen category. In particular K(P(A))≃K(Chᵇ(P(A))).

Hypotheses: P(A) is the category of finite projective A-modules, with split exact sequences. Chᵇ(P(A)) has degreewise split monomorphisms as cofibrations and quasi-isomorphisms as weak equivalences; the ambient abelian category is all A-modules. Kernels of surjections between finite projectives are finite projective because the sequence splits. Cochain reindexing does not change the statement.

Reading limits: Read Theorem 2.2 and its full displayed proof and §2.7.2, including the actual inclusion functors. The proof imports Waldhausen localization and preceding exact-complex calculations; those substantial foundational results are not reconstructed. Only the bounded-projective bridge is needed here; the larger perfect-complex approximation in §2.7.2 is context, not a required new model.

Sources