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Third-stage graded isomorphism

PDF p. 12: Section 6 opening projection; PDF p. 13: Opening sentence

Statement

Assume the Axiom of Choice. Let $X$ be a Noetherian scheme of Krull dimension $3$ over a field $k$ . Then the actual third-stage projection followed by the graded comparison is an isomorphism $p_{3} : π_{0} F_{m o t}^{3} K (X) \overset{≅}{\to} H_{m o t}^{6, 3} (X, Z) .$ In particular, this applies to the normal threefolds in Section 6 under the article's standing conventions.

Here $F_{m o t}^{r} K (X)$ is stage $r$ of the decreasing motivic filtration of the nonconnective $K$ -theory spectrum; its third graded spectrum is the cofiber of the transition $F_{m o t}^{4} K (X) \to F_{m o t}^{3} K (X)$ . The map $p_{3}$ is induced by the canonical map from the third stage to this cofiber and the specified graded equivalence. For the weight-three motivic complex $C = Z (3)_{m o t} (X)$ , the group $H_{m o t}^{6, 3} (X, Z)$ means $H^{6} (C)$ . The symbol $π_{i}$ denotes the $i$ th stable homotopy group. No injectivity of the map from the third stage to $K (X)$ is asserted here.

Facts & Assumptions

Given: The Axiom of Choice, a field $k$ , and a Noetherian $k$ -scheme $X$ of Krull dimension $3$ .

Goal: Prove that the actual third-stage projection $p_{3}$ is an isomorphism.

[F1]

A Noetherian scheme is quasi-compact and quasi-separated. Under the choice assumption, Motivic filtration and image groups supplies the decreasing motivic tower over $k$ , and Motivic complexes and cohomology defines the indicated motivic cohomology group.

[F2]

For a qcqs $k$ -scheme, Motivic graded pieces and spectral sequence gives the actual cofiber triangle $F_{m o t}^{4} K (X) \to F_{m o t}^{3} K (X) \to Z (3)_{m o t} (X) [6]$ . The natural shift-index identification in Complex shifts and spectral homotopy indices identifies the zeroth homotopy group of this graded spectrum with $H_{m o t}^{6, 3} (X, Z)$ .

[F3]

High filtration vanishing states, for a Noetherian scheme of finite dimension $d$ over a field and under the Axiom of Choice, that $π_{0} F_{m o t}^{i} K (X) = 0$ for $i > d$ and $π_{j} F_{m o t}^{i} K (X) = 0$ for $i \geq d$ and $j < 0$ .

[F4]

Stage projections, kernels and image groups gives the homotopy exact sequence of a filtered stage and its cofiber: the image of each map equals the kernel of the next.

Proof

technique · Apply the homotopy exact sequence to the third graded cofiber and use the two stage-four vanishing statements

1.1F1F2given

Since $X$ is Noetherian, it is qcqs; therefore the tower and the actual third graded cofiber in [F1] and [F2] apply. Write $G^{3}$ for that cofiber. The supplied graded equivalence and shift-index identification give a natural isomorphism $π_{0} G^{3} ≅ H_{m o t}^{6, 3} (X, Z)$ .

1.2F4step 1.1

Apply [F4] with stage index $3$ and homotopy degree $0$ . The resulting exact segment is $π_{0} F_{m o t}^{4} K (X) ⟶ π_{0} F_{m o t}^{3} K (X) ⟶ π_{0} G^{3} ⟶ π_{- 1} F_{m o t}^{4} K (X) .$ The middle arrow is the map induced by the canonical third-stage-to-cofiber map.

2.1F3step 1.2algebra

In [F3], take $d = 3$ and $i = 4$ . Since $4 > 3$ , the first group in step 1.2 is zero. Since $4 \geq 3$ and $- 1 < 0$ , the last group is also zero. Exactness therefore makes the middle arrow both injective and surjective.

3.1step 1.1step 2.1∎

Compose this middle arrow with the isomorphism of step 1.1. By definition this composite is $p_{3}$ , so $p_{3}$ is an isomorphism. The article's normal threefold case is included because its schemes satisfy the stated Noetherian dimension-three hypotheses; normality and algebraic closedness are not needed for this deduction.

Depends on

Used by · 5

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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 · pp. 12–13, opening of Section 6; p. 9, Lemma 4.2

For a normal threefold in Section 6 the third-stage projection is an isomorphism, using the vanishing of the degree zero and degree minus one homotopy groups at stage four.

Hypotheses: The article uses its standing field and scheme conventions; Lemma 4.2 states the relevant vanishing for Noetherian finite-dimensional schemes over a field.

Reading limits: The cited opening of Section 6 and the statement of Lemma 4.2 were read in the supplied article text. The proof below uses the completed local supplier contracts.

Sources

Jean Fasel and Maria Yakerson, Vector bundles on singular surfaces and threefolds · pp. 12–13, opening of Section 6; p. 9, Lemma 4.2