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Third Chern vanishing and a trivial summand

PDF p. 3: Splitting criterion after Corollary D; PDF p. 17: Corollary 6.10 proof

Statement

Assume the Axiom of Choice and the site and universe size bounds required by the cited motivic filtration, support and Chern constructions. Let $k$ be an algebraically closed field of arbitrary characteristic, let $X = Spec A$ be a normal affine $k$ -scheme of finite type and Krull dimension three, and let $E$ be a rank-three vector bundle on $X$ . Then $c_{3} (E) = 0 ⟺ E ≅ E^{'} \oplus O_{X}$ for some rank-two vector bundle $E^{'}$ on $X$ .

Here algebraically closed means that every nonconstant polynomial over $k$ has a root in $k$ ; finite type means that $A$ is a finitely generated $k$ -algebra; normal means that every local ring is an integrally closed domain; and Krull dimension is the supremum of lengths of chains of distinct irreducible closed subsets. A vector bundle is a finite locally free $O_{X}$ -module, and rank $r$ means locally isomorphic to $O_{X}^{r}$ . The class $c_{3} (E)$ is the third integral motivic Chern class in $H_{m o t}^{6, 3} (X, Z)$ , the degree-six cohomology of the weight-three motivic complex. The Axiom of Choice says that every family of nonempty sets has a choice function; the size bounds fix the sites and universes used by the cited constructions.

Facts & Assumptions

Given: $k$ , $X$ and $E$ as in the statement, the Axiom of Choice and the stated site and universe bounds.

Goal: Prove both implications of the displayed equivalence.

[F1]

Rank-three alternating exterior class defines the alternating exterior class $β (E) = [O_{X}] - [E] + [Λ^{2} E] - [\det E] \in K_{0} (X),$ where $\det E = Λ^{3} E$ and $K_{0} (X)$ is the degree-zero K-theory group of perfect complexes. A perfect complex is locally quasi-isomorphic to a bounded complex of finite locally free sheaves.

[F2]

Under the given hypotheses, Chern classes of the alternating exterior class proves that $β (E)$ lies in $F_{m o t}^{3} K_{0} (X)$ and that $c_{3} (β (E)) = - 2 c_{3} (E)$ . The group $F_{m o t}^{3} K_{0} (X)$ is the image in $K_{0} (X)$ of the degree-zero homotopy group of the third motivic filtration stage.

[F3]

Under the same hypotheses, Third Chern isomorphism says that the actual third Chern operation on perfect K-classes restricts to an isomorphism $c_{3} : F_{m o t}^{3} K_{0} (X) \to H_{m o t}^{6, 3} (X, Z)$ . In particular this restriction is injective.

[F4]

Murthy splitting input in rank three says that, under the given hypotheses, $β (E) = 0$ in $K_{0} (X)$ implies $E ≅ E^{'} \oplus O_{X}$ for a rank-two bundle $E^{'}$ .

[F5]

Motivic Chern classes defines the total Chern polynomial $c_{t} (V) = \sum_{i} c_{i} (V) t^{i}$ of a bundle $V$ and sets $c_{i} (V) = 0$ for $i$ greater than its rank. Whitney identities and virtual Chern arithmetic gives $c_{t} (V \oplus W) = c_{t} (V) c_{t} (W)$ ; in particular a trivial line has total Chern polynomial $1$ . The Axiom of Choice states the Choice assumption used by these constructions.

Proof

technique · Apply the third-Chern isomorphism to the alternating exterior class, then use the splitting criterion; the converse follows from Whitney and rank vanishing

1.1givenF2

Suppose $c_{3} (E) = 0$ . By [F2], $β (E)$ belongs to the third motivic image piece and $c_{3} (β (E)) = - 2 c_{3} (E) = 0$ .

2.1F3step 1.1

The restricted third Chern operation in [F3] is injective. Since $β (E)$ lies in its domain and maps to zero, $β (E) = 0$ in $F_{m o t}^{3} K_{0} (X)$ and hence in its ambient group $K_{0} (X)$ .

3.1givenF4step 2.1

Apply [F4] to this vanishing. All its field, normality, finite-type, dimension, rank, Choice and size hypotheses are those given here. It yields a rank-two bundle $E^{'}$ and an isomorphism $E ≅ E^{'} \oplus O_{X}$ .

4.1givenF5step 3.1

Conversely, suppose $E ≅ E^{'} \oplus O_{X}$ with $E^{'}$ of rank two. By [F5], $c_{t} (E) = c_{t} (E^{'}) c_{t} (O_{X}) = c_{t} (E^{'})$ . The coefficient of $t^{3}$ on the right is zero because $E^{'}$ has rank two. Thus $c_{3} (E) = 0$ .

5.1step 3.1step 4.1∎

Steps 1.1–4.1 prove both implications and hence the stated equivalence.

Depends on

Used by · 1

Rank-two bundles with prescribed first two classes

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. 3, explanation after Corollary D; p. 17, proof of Corollary 6.10