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Lifting the corrected class to the curve-supported third stage

PDF p. 16: Proof of Theorem 6.7, application of Proposition 6.2 and Remark 6.3

Statement

Assume the Axiom of Choice and the site and universe bounds of the cited filtration and support constructions. Let $k$ be algebraically closed, let $X$ be a normal affine finite-type $k$ -scheme of dimension three, and let $C$ be the reduced closed curve constructed in Curve-and-point support reduction. Thus $C$ lies in the smooth locus $W = X_{s m o o t h}$ and, on each three-dimensional clopen integral component of $X$ , its intersection is a closed integral curve; it is empty on the lower-dimensional components. Put $U = X ∖ C$ and $V = W ∖ C$ .

If $β \in F_{m o t}^{3} K_{0} (X)$ restricts to zero in $K_{0} (U)$ , then its unique representative $b \in π_{0} F_{m o t}^{3} K (X)$ lifts to some $\tilde{b} \in π_{0} F_{m o t}^{3} K (X / U) .$ Restriction induces an isomorphism $π_{0} F_{m o t}^{3} K (X / U) \overset{≅}{\to} π_{0} F_{m o t}^{3} K (W / V),$ compatible with support-to-global maps. In particular this applies to the corrected class of Removing isolated points from the support. The support lift need not be unique.

Here an integral curve is a reduced irreducible scheme of dimension one, and the smooth locus is the open set where $X \to Spec k$ is smooth. The spectrum $K$ is nonconnective perfect-complex $K$ -theory; $K_{0} = π_{0} K$ . The image-filtration group $F_{m o t}^{3} K_{0}$ is the image of $π_{0} F_{m o t}^{3} K \to K_{0}$ . For an open complement $A ∖ B$ , the supported third stage $F_{m o t}^{3} K (A / (A ∖ B))$ means the homotopy fiber of the actual restriction of third stages, with its map to the stage on $A$ . A homotopy-cartesian square is a square whose upper-left object is equivalent to the homotopy pullback of the other three objects.

Facts & Assumptions

Given: The field, scheme, curve, class, Choice and size assumptions in the statement.

Goal: Lift the actual third-stage representative of $β$ to the curve-support fiber and identify that fiber on the smooth locus.

[F1]

Curve-and-point support reduction supplies the componentwise curve and its containment in $W$ ; Removing isolated points from the support supplies a corrected class satisfying the stated kernel condition. The components form a finite clopen partition because $X$ is normal and Noetherian, as used in the former supplier.

[F2]

Third edge isomorphism gives injectivity of $π_{0} F^{3} K (X) \to K_{0} (X)$ . The definition of the image filtration in Motivic filtration and image groups gives a representative for every element of $F^{3} K_{0} (X)$ .

[F3]

Third edge on a curve complement gives injectivity of $π_{0} F^{3} K (X_{e} ∖ C_{e}) \to K_{0} (X_{e} ∖ C_{e})$ for each three-dimensional affine component $X_{e}$ and its closed integral curve $C_{e} \subset (X_{e})_{s m o o t h}$ . High filtration vanishing gives $π_{0} F^{3} K (X_{e}) = 0$ on each component of dimension less than three. Finite clopen restrictions split the actual motivic image filtration identifies the actual stage and $K$ maps on a finite clopen union with their componentwise products.

[F4]

Support fibers commute with graded triangles gives the long exact homotopy sequence of the actual stagewise support fiber. Motivic and filtered K theory with support identifies the displayed quotient notation with that fiber. Open excision for a descent theory gives support excision, compatible with support-to-global maps, provided the relevant restriction square is homotopy cartesian.

[F5]

Perfect K theory localization and open excision gives Zariski Mayer–Vietoris for $K$ on quasi-compact opens. Motivic graded pieces and spectral sequence gives the natural fiber triangles $F^{r + 1} K \to F^{r} K \to Z (r)_{m o t} [2 r]$ , and Elmanto–Morrow motivic complex input gives Nisnevich descent, hence Zariski descent, for the graded terms.

Proof

1.1F1given

By [F1], write $X$ as the finite disjoint union of its clopen integral components $X_{e}$ . On every three-dimensional component, $C_{e} = C \cap X_{e}$ is the reduced closure of an integral curve, hence is a closed integral curve; it lies in the smooth locus. On every lower-dimensional component $C_{e}$ is empty. The opens $U$ and $W$ , and their intersections with the components, are quasi-compact: $X$ is finite type over a field and therefore Noetherian.

1.2F3step 1.1

For a three-dimensional component, [F3] makes the actual stage-to- $K_{0}$ map on $X_{e} ∖ C_{e}$ injective. For a lower-dimensional component, its complement is all of $X_{e}$ and its third-stage zeroth group is zero by [F3], so the same map is injective. The finite clopen product equivalences in [F3], applied to $U = ∐_{e} (X_{e} ∖ C_{e})$ and compatible with the stage-to- $K$ maps, therefore show that $π_{0} F^{3} K (U) ⟶ K_{0} (U)$ is injective.

2.1F2F4givenstep 1.2

By [F2], $β$ has a representative $b \in π_{0} F^{3} K (X)$ , and that representative is unique. Naturality of the stage-to- $K$ maps sends the restriction of $b$ to the restriction of $β$ , which is zero in $K_{0} (U)$ . Step 1.2 therefore makes $b ∣_{U} = 0$ in $π_{0} F^{3} K (U)$ . The exact sequence in [F4] supplies $\tilde{b} \in π_{0} F^{3} K (X / U)$ mapping to $b$ . Exactness does not assert that this lift is unique.

2.2F5step 1.1

The opens $U$ and $W$ cover $X$ because $C \subset W$ , and their intersection is $V$ . The restriction square for stage zero $K$ is homotopy cartesian by [F5]. The corresponding square for each graded term is homotopy cartesian by its Zariski descent. If the stage- $r$ square is homotopy cartesian, the natural triangle in [F5] identifies the stage- $(r + 1)$ square as its fiber over the graded-term square; fibers preserve homotopy-cartesian squares. Induction for $r = 0, 1, 2$ proves that the stage-three square for this cover is homotopy cartesian. All opens involved are qcqs by step 1.1.

3.1F1F4step 2.1step 2.2∎

Apply the spectrum form of open excision in [F4] to $C \subset W \subset X$ and the stage-three square of step 2.2. It gives the actual restriction equivalence $F^{3} K (X / U) ≃ F^{3} K (W / V)$ and its compatibility with support-to-global maps. Taking $π_{0}$ gives the displayed isomorphism. Applying step 2.1 to the corrected class supplied by [F1] proves the final assertion.

Depends on

Used by · 1

Smooth points generate the third motivic piece

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.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · PDF p. 16, proof of Theorem 6.7, application of Proposition 6.2 and Remark 6.3