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Rational third-edge comparison

PDF p. 14: Proof of Proposition 6.2, rational degeneration; PDF p. 14: Remark 6.3

Statement

Assume the Axiom of Choice. Let $X$ be a Noetherian scheme of Krull dimension $3$ over a field $k$ . Let $F_{m o t}^{q} K (X)$ denote stage $q$ of the Elmanto–Morrow motivic filtration of nonconnective $K$ -theory and put $F_{m o t}^{q} K_{0} (X) = im (π_{0} F_{m o t}^{q} K (X) \to K_{0} (X)) .$ The associated motivic spectral sequence degenerates after tensoring its terms and differentials with $Q$ . Moreover its actual third edge homomorphism induces an isomorphism $ϵ_{3} \otimes Q : H_{m o t}^{6, 3} (X, Z) \otimes_{Z} Q \overset{≅}{\to} F_{m o t}^{3} K_{0} (X) \otimes_{Z} Q .$ Here $H_{m o t}^{6, 3} (X, Z)$ is degree-six cohomology of the weight-three motivic complex; $ϵ_{3}$ is the inverse of the actual third-stage projection followed by the stage-to- $K_{0}$ map restricted to its image. Rational degeneration means that every differential on every page from $E_{2}$ onward becomes zero after tensoring with $Q$ .

In particular, this applies to the normal affine threefolds over algebraically closed fields in Proposition 6.2 of the article. It also applies to $X ∖ T$ under Proposition 6.1's hypotheses: $X$ is an affine finite-type threefold over an algebraically closed field and $T$ is a closed integral curve contained in its smooth locus. No normality assumption is needed for this rational comparison on the complement.

Facts & Assumptions

Given: Choice, a Noetherian dimension-three scheme $X$ over a field $k$ , and the actual motivic tower and third edge map described above.

Goal: Prove rational degeneration and that the rationalization of the actual edge map is an isomorphism, including the two article applications.

[F1]

Finite valuative dimension for noetherian and finite type schemes gives finite valuative dimension for a Noetherian finite-dimensional scheme under Choice. Such a scheme is quasi-compact and quasi-separated. The structure map over $k$ factors through the prime field, which is $Q$ in characteristic zero and $F_{p}$ in characteristic $p$ .

[F2]

Under the qcqs, prime-field and finite-valuative-dimension hypotheses, Rational splitting of the motivic tower supplies an equivalence of the actual rationalized augmented tower with a direct-sum tail tower and supplies degeneration of its spectral sequence. This is an imported splitting; its construction is not reconstructed here. Image filtration for a split tower states that every tail inclusion has a retraction and identifies the successive image quotients with the corresponding graded homotopy groups.

[F3]

Third-stage graded isomorphism identifies the actual third-stage-to-cofiber map, followed by the specified graded comparison, with an isomorphism $p_{3} : π_{0} F_{m o t}^{3} K (X) \to H_{m o t}^{6, 3} (X, Z)$ . Third edge homomorphism defines $ϵ_{3}$ as $p_{3}^{- 1}$ followed by the actual map to the image subgroup. Motivic graded pieces and spectral sequence supplies the actual graded and spectral-sequence identifications.

[F4]

Article scheme conventions makes affine schemes in the article finite type over the base field. Over an algebraically closed field, Nonempty opens preserve irreducible dimension says that a nonempty open subset of an irreducible classical variety has the same dimension. Passing to the reduced structure does not change the underlying topological space or its chain dimension.

Proof

1.1F1F2F3given

The given Noetherian scheme is qcqs and has finite valuative dimension by [F1]. Its map to $Spec k$ also makes it a scheme over the appropriate prime field. Thus the rational filtered splitting in [F2] applies. Its stated spectral-sequence consequence, with the actual page identification of [F3], proves rational degeneration.

2.1F2step 1.1

Let $A = π_{0} F_{m o t}^{3} K (X)$ , $B = K_{0} (X)$ , and let $u : A \to B$ be the stage-to- $K$ homomorphism. The augmented tower equivalence in [F2] identifies the rationalized stage-three map with a tail inclusion. That inclusion has a retraction, so $u \otimes Q$ is injective: rationalization of spectra induces tensoring of homotopy groups with $Q$ .

3.1F3step 2.1algebra

By definition, $F_{m o t}^{3} K_{0} (X) = im u$ . The rational numbers are flat over $Z$ , so tensoring the exact sequence $0 \to \ker u \to A \to im u \to 0$ with $Q$ remains exact. Step 2.1 makes $(\ker u) \otimes Q = 0$ ; hence $A \otimes Q \to F_{m o t}^{3} K_{0} (X) \otimes Q$ is an isomorphism. This uses the actual image subgroup, not merely an abstract group isomorphic to it.

4.1F3step 3.1

By [F3], $p_{3}$ is an isomorphism and $ϵ_{3}$ is its inverse followed by $u$ with codomain restricted to $im u$ . Tensoring this composite with $Q$ and applying step 3.1 proves the displayed edge isomorphism.

5.1F4givenstep 4.1∎

An affine finite-type scheme over a field is Noetherian. For the first article application its dimension is three by hypothesis, so steps 1.1 through 4.1 apply. For the complement, $U = X ∖ T$ is an open subscheme of the Noetherian threefold and is Noetherian of dimension at most three. Choose an irreducible component of $X_{r e d}$ of dimension three. The closed integral curve $T$ has dimension one and cannot contain that component, so its intersection with $U$ is a nonempty open subset. By [F4], this intersection has dimension three; its chains are also chains in $U$ . Consequently $\dim U = 3$ . It is a scheme over the same field, and steps 1.1 through 4.1 apply to it without normality.

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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

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 14, proof of Proposition 6.2 and Remark 6.3; p. 13, Proposition 6.1

The proof uses rational degeneration to identify the rational third edge with the third image-filtration group; the subsequent remark applies the comparison to a curve complement.

Hypotheses: The proposition concerns a normal affine threefold over an algebraically closed field; the remark uses the hypotheses of Proposition 6.1, an affine threefold and an integral curve in its smooth locus.

Reading limits: The cited statements and proof paragraph were read in the supplied article text. The rational splitting is used through the completed external-input supplier, not reconstructed here.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · p. 14, proof of Proposition 6.2 and Remark 6.3; p. 13, Proposition 6.1