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Local free chains detect natural perfect operations on every qcqs base

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

Statement

Assume the Axiom of Choice and the fixed compatible universe and site-size bounds of the supplied perfect-K-space and derived-sheaf constructions. Let $X$ be any qcqs scheme over a field. On its small Zariski site let $K = Ω^{\infty} K_{P e r f}$ and $j : \underline{Z} \times B G L \to K$ be the actual stable free-class map, with locally constant ranks. The maps $j$ , $j \times j$ , and the component-zero map $j_{0} : B G L \to S$ into the actual rank-zero fiber induce local quasi-isomorphisms on normalized free integral chains. Therefore precomposition detects homotopy classes of derived sheaf maps into each actual abelian EM coefficient target $A_{i} = Ω^{\infty} (Z (i)_{m o t} [2 i])$ . No noetherianness, dimension bound, or continuity of motivic complexes is assumed.

Facts & Assumptions

[A1]

AC and fixed universe bounds are assumed for the small presentations and enhanced realizations in the suppliers. The Axiom of Choice

[F1]

Full projective Quillen–Waldhausen comparison and the local free-class map supplies the full natural Quillen-to-bounded-projective Waldhausen K-space comparison, including its degree-zero object map, and the canonical acyclic plus map for every commutative local ring. The latter clause covers Zariski local rings. External Quillen acyclicity and the actual local free-class model fixes the actual free-object/addition comparison used by the completed sheaf model; its stated Nisnevich-local scope alone is not used to infer the all-local-ring clause.

[F2]

External formal universal Chern maps and supported construction supplies the actual sheaf model, coefficient targets and completion construction. On the small site of open subschemes of $X$ , write $D (Sh (X_{Z a r}, Z))$ for the unbounded derived category of abelian sheaves, localized at cohomology-sheaf isomorphisms, with its derived mapping spaces and cochain convention $H^{a} (C [s]) = H^{a + s} (C)$ . Define $R Γ_{Z a r} (U, C) = R Hom (Z [U], C)$ , where $Z [U]$ is the free abelian sheaf on the represented open. For a space-valued sheaf $Y$ , its derived free integral chains are sheafified normalized free chains of a simplicial resolution, placed in nonpositive cochain degrees. The right adjoint is the underlying derived abelian EM-space functor, using the nonpositive truncation for Dold–Kan. Thus the actual target associated to $C_{i} [2 i]$ , with $C_{i} = Z (i)_{m o t}$ , is $A_{i} (U) = Ω^{\infty} H (R Γ_{Z a r} (U, C_{i} [2 i]))$ . These are local Zariski conventions, not an enlargement of the separately scoped derived-sections definition. The actual complexes satisfy Zariski descent, so restriction is evaluated by these derived sections.

[F3]

Affine perfect K zero and bundle K zero identifies the actual nonconnective perfect $K_{0}$ on every affine ring with its finite-projective Grothendieck group, by the object-class map; it uses the idempotent completeness of $P e r f$ . The Quillen and bounded-projective degree-zero object maps in [F1] send each generator to that same class.

[F4]

Bounded finite-projective complexes and affine chain models supplies the actual bounded finite-projective enhanced model and K-projective mapping property. Enhanced affine perfect equivalence supplies the actual associated-sheaf equivalence with the affine perfect category, including preservation of shifts and cones.

[F5]

External connective, nonconnective and automorphism comparisons supplies the natural Waldhausen-to-connective equivalence under factorization and DHKS-saturation, its actual core/object maps, and the canonical connective-to-nonconnective comparison in positive degrees. Its statement does not by itself supply the missing degree-zero comparison; [F3] does.

Given: $X$ as stated, AC and the fixed universe bounds.

Goal: Supply the locality and derived-map argument beyond the earlier finite-dimensional scope.

Proof

1.1givenconstruct

Finite projective modules over a filtered colimit of rings descend: represent one by an idempotent finite matrix, and descend its finitely many entries and idempotent equations. Morphisms, their equalities, inverses, and split exact sequences descend by the same finite-matrix argument. A Quillen Q morphism is a finite admissible diagram modulo an isomorphism of such diagrams; the diagram, its splitting data, and any witnessing isomorphism descend. Consequently every finite simplex of the Q nerve descends, with equality after a further index. Realization preserves filtered colimits, and finite homotopy data and the finite limits defining based loops commute with filtered colimits of spaces. Thus $K^{Q} (P (-))$ commutes with these ring colimits, including its components.

1.2A1F1F3F4F5

The full Q/Waldhausen comparison of [F1] applies because surjections of finite projectives split. Bounded projective complexes admit mapping-cylinder factorization within this category, and their acyclic cones are contractible by [F4]. A quasi-isomorphism is therefore a homotopy equivalence; conversely a map invertible in the localization induces an isomorphism on cohomology, so DHKS-saturation holds. Their actual localization is the affine enhanced perfect category by [F4], and extension of scalars preserves split cofibrations and contractible cones. Hence [F5] applies naturally. The resulting comparison to the actual nonconnective perfect K-space is an isomorphism in every positive degree. On degree zero, the Q and Waldhausen object classes generate their Grothendieck groups, and [F3] identifies their image with the same finite-projective class in actual perfect $K_{0}$ . Thus this canonical comparison is also an isomorphism on components. Translation in these grouplike K-spaces identifies all components, so it is an equivalence of the full K-spaces. Choice is used in the supplied small presentations and enhanced realizations, not as a substitute for the component argument.

2.1step 1.1step 1.2

Combine the natural full-space comparison in step 1.2 with the Q continuity in step 1.1. This identifies the connective space of actual perfect K-theory of a filtered colimit ring with the filtered colimit of the corresponding spaces, compatibly with free-object maps. Only this K-theoretic continuity is used. No assertion about continuity of motivic complexes enters.

3.1F1F2step 2.1

At a Zariski point $x$ , the affine-neighborhood rings have colimit $O_{X, x}$ . Step 2.1 identifies the stalk of $K$ with its actual K-space. Finite matrices and invertibility descend in the same way, so the source stalk is $Z \times B G L (O_{X, x})$ . Finite projectives over this local ring are free, and the actual free-class map is componentwise the acyclic plus map of [F1]. The rank map on its $K_{0}$ is the isomorphism to $Z$ , so taking the fiber over zero selects precisely the zero component. This proves the integral chain quasi-isomorphism for $j$ and for $j_{0}$ .

4.1step 3.1algebra

Eilenberg–Zilber identifies normalized chains of a product with the tensor product of the two chain complexes. These complexes are degreewise free and bounded below in homological grading, hence K-flat; tensoring the chain quasi-isomorphisms for $j$ therefore preserves them. This proves the assertion for $j \times j$ . Stalks commute with these finite products and chain constructions, and Zariski points detect isomorphisms of abelian sheaves on every scheme, without a dimension hypothesis.

5.1F2step 3.1step 4.1∎

Sheafify normalized free integral chains. Abelian sheafification is exact, so steps 3.1 and 4.1 give quasi-isomorphisms of complexes of sheaves for all three maps. The free-abelian and normalized-chain adjunctions, applied to simplicial resolutions and derived EM targets as in [F2], identify maps of space-valued sheaves into $A_{i}$ with derived maps from these chains into $C_{i} [2 i]$ . Precomposition by a chain quasi-isomorphism is an equivalence of these derived mapping spaces and hence detects their homotopy classes. This uses no stalk calculation or filtered-colimit assertion for the motivic coefficients. Applying derived sections on $X$ to an equality of derived sheaf-map classes retains its descent data.

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