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The supported second stage has the actual augmentation inverse

PDF p. 16: Supported leading weight two

Statement

Assume the Axiom of Choice. Let $U$ be a smooth finite-type scheme of pure dimension three over a field $k$ , and let $C \subset U$ be an integral closed curve of codimension two. Let $F^{∙} K$ be the specified motivic filtration. Assume its actual smooth comparison supplies natural graded identifications with Bloch cycle spectra on both $U$ and $U ∖ C$ , compatible with restriction. Then the supported graded spectra in weights zero and one are zero, and the augmentations $F_{C}^{2} K (U) ⟶ F_{C}^{1} K (U) ⟶ F_{C}^{0} K (U) = K_{C} (U)$ are equivalences. Thus its supported leading weight-two homomorphism is unambiguously $π_{0} K_{C} (U) \overset{(π_{0} ϵ_{2})^{- 1}}{\to} π_{0} F_{C}^{2} K (U) ⟶ π_{0} {gr}_{C}^{2} K (U),$ where $ϵ_{2}$ is the actual stage-to-zero augmentation. Under the same comparison the target is $H_{0} z_{C}^{2} (U, ∙) = Z [C]$ . For any specified natural augmented tower comparison, the leading maps commute with the induced maps on support fibers and supported graded spectra provided the target supported stage-two augmentation is also an equivalence. In particular, this condition holds for comparisons which are equivalences on all stages and on the underlying spectra on both $U$ and $U ∖ C$ . Its generator value remains a separate normalization input; it is not implied by the equivalence alone.

Facts & Assumptions

Given: AC, $U, C, k$ as stated, the specified filtration and the explicit natural smooth/Bloch graded comparison, including the supported Bloch fiber comparison. For inverse-augmentation transport the target supported stage-two augmentation is also an equivalence; equivalences on all stages and underlying spectra on both schemes are a sufficient condition.

Goal: Justify the domain, target and exact meaning of the leading supported map before its generator is computed.

[F1]

Under AC and the stated smooth finite-type pure-three-dimensional and integral codimension-two curve hypotheses, the two lower supported cycle complexes vanish and the bottom weight-two group is $Z [C]$ . Bottom supported Bloch groups on a codimension two curve

[F2]

The support construction preserves the graded cofiber sequences. Support fibers commute with graded triangles

[F3]

The stage-zero filtration is naturally $K$ , and the supported spectra are formed by taking fibers of its restriction maps. The assumed AC discharges the Choice hypotheses of these K-theory/filtration interfaces and of the supported Bloch supplier in [F1]. Motivic filtration and image groups, Motivic and filtered K theory with support, The Axiom of Choice

[F4]

A specified natural augmented filtered comparison which is an equivalence on all stages and underlying spectra on a scheme and its open complement induces supported stage, underlying and graded equivalences, preserving augmentations and graded projections. Augmented tower comparisons preserve supported graded projections

Proof

1.1F1F2F3given

By the assumed compatible graded comparison, forming its support fiber identifies each supported layer with the supported cycle spectrum in that weight. By [F1] the layers at weights zero and one are zero. In the stable category a map with zero cofiber is an equivalence; applying this to the two supported transition maps furnished by [F2] gives both displayed equivalences. Their composite is the specified augmentation because the original tower is augmented coherently.

2.1F1F3step 1.1

Inverting that equivalence on $π_{0}$ and then taking the weight-two graded projection gives the displayed homomorphism. Its target, under the same comparison, is the zeroth homology of the weight-two supported cycle complex and equals $Z [C]$ by [F1]. The cohomological convention is $π_{0} {gr}_{C}^{2} = H_{C}^{4, 2}$ , since the actual motivic layer has its $[4]$ shift.

3.1F2F3step 1.1step 2.1givenalgebra

Let a specified natural augmented comparison have target supported stage-two augmentation $ϵ_{2}^{'}$ which is an equivalence. Write $ϕ_{K}$ , $ϕ_{2}$ and $ϕ_{{g r}^{2}}$ for its maps on underlying support fibers, supported second stages and supported second layers, and write $q_{2}, q_{2}^{'}$ for the stage-to-layer projections. Functoriality of support fibers and graded cofibers gives $ϵ_{2}^{'} ϕ_{2} = ϕ_{K} ϵ_{2}$ and $q_{2}^{'} ϕ_{2} = ϕ_{{g r}^{2}} q_{2}$ in the homotopy category. Since both augmentations are invertible, these identities imply $q_{2}^{'} (ϵ_{2}^{'})^{- 1} ϕ_{K} = ϕ_{{g r}^{2}} q_{2} ϵ_{2}^{- 1}$ . Taking $π_{0}$ proves commutation of the leading homomorphisms, without requiring the comparison maps themselves to be equivalences.

4.1F4step 1.1step 3.1∎

Under the stronger equivalence hypotheses of [F4] on $U$ and $U ∖ C$ , the induced maps $ϕ_{K}$ and $ϕ_{2}$ are equivalences. Their commuting augmentation square and step 1.1 imply that $ϵ_{2}^{'}$ is an equivalence. Step 3.1 therefore applies, in particular to any specified classical slice/coniveau comparison satisfying those hypotheses. This preserves the actual leading map; its generator value still requires the separate normalization input.

Depends on

Used by · 1

Smooth rank, first Chern, determinant and supported positive leading cycle

Dependency tree · 3 levels

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