Full projective Quillen–Waldhausen comparison and the local free-class map

PDF p. 14: Lemma6.4; scheme conventions p3

Assume AC and fixed universe bounds. For every commutative unital ring $A$, let $P(A)$ be finite projective modules with their split exact structure, and let $C(A)$ be bounded cochain complexes in $P(A)$, with degreewise split monomorphisms as cofibrations and quasi-isomorphisms as weak equivalences. The canonical exact-category S/Q comparison followed by degree-zero inclusion gives a natural equivalence of full K-spaces $K^Q(P(A))=\mathrm{\Omega }BQ(P(A))\simeq K^W(C(A))\mathrm{.}$ It sends a projective object to the same object in degree zero and is natural for extension of scalars. No noetherian or regularity hypothesis is required. For every commutative local ring $R$, the stable free-class map $\mathbb{Z}\times BGL(R)\to K^Q(P(R))$ is, component by component, the integral-homology-acyclic plus map. This includes Zariski local rings.

This is an explicitly imported established external input: the comparisons and plus theorems of Weibel, The K-book, Chapter IV, IV.1.1.1–1.1.2, IV.1.4–1.6, IV.4.9–4.11.1, IV.7.1–7.2, IV.8.3.2–8.6 and Exercises 8.5–8.7, and the Gillet–Waldhausen theorem of Weibel, The K-book, Chapter V, V.2.2 and V.2.7.2, with the hypotheses and reading limits recorded in the source metadata. The imported proof foundations are literature-derived; the numbered argument below is a local application check and does not reconstruct those foundations. The comparison with the enhanced perfect category and nonconnective degree zero is supplied separately in the consumer.