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Chern classes of the alternating exterior class

PDF p. 17: Splitting-principle calculations in the proof of Corollary 6.10

Statement

Assume the Axiom of Choice and the fixed site and universe bounds required by the cited motivic filtration and Chern constructions. Let $k$ be an algebraically closed field, let $X$ be a normal affine threefold over $k$ , and let $E$ be a rank-three vector bundle on $X$ . Put $β (E) = [O_{X}] - [E] + [Λ^{2} E] - [\det E] \in K_{0} (X) .$ Then $β (E)$ has rank zero and its actual motivic Chern classes satisfy $c_{1} (β (E)) = 0, c_{2} (β (E)) = 0, c_{3} (β (E)) = - 2 c_{3} (E) .$ Consequently $β (E) \in F_{m o t}^{3} K_{0} (X)$ . On the flag space supplied by the splitting principle, if $ℓ_{1}, ℓ_{2}, ℓ_{3}$ are the first Chern classes of the successive line quotients of the pulled-back $E$ , the corresponding line roots of $Λ^{2} E$ are $ℓ_{1} + ℓ_{2}$ , $ℓ_{1} + ℓ_{3}$ and $ℓ_{2} + ℓ_{3}$ .

Here a vector bundle is a finite locally free sheaf, its rank is the local rank, $Λ^{2} E$ is its second exterior power, and $\det E = Λ^{3} E$ . The group $K_{0} (X)$ is the zeroth homotopy group of the K-theory spectrum of perfect complexes; a bundle is regarded as a complex in degree zero. The integral motivic Chern class $c_{i}$ lies in $H_{m o t}^{2 i, i} (X, Z)$ . The group $F_{m o t}^{j} K_{0} (X)$ is the image of the $j$ th motivic filtration stage in $K_{0} (X)$ . A normal scheme has integrally closed local domains, and a threefold here has dimension three. A line root means the first Chern class of a line-bundle quotient in the indicated locally split flag filtration; it does not require the pulled-back bundle to split as a direct sum.

Facts & Assumptions

Given: The field, scheme, bundle, Choice and site and universe bounds in the statement.

Goal: Compute the rank and first three actual Chern classes of $β (E)$ , identify the exterior-square roots, and prove membership in the third image filtration.

[F1]

Rank-three alternating exterior class defines $β (E)$ and its four bundle terms. Motivic splitting principle supplies, under Choice, a flag map $q : Y \to X$ with locally split line quotients $L_{i}$ and injective pullback on integral motivic cohomology in every bidegree.

[F2]

Exterior-square line factors on a flag space gives the locally split filtration of $Λ^{2} q^{*} E$ with quotients $L_{i} \otimes L_{j}$ for $i < j$ , and identifies $\det q^{*} E$ with $L_{1} \otimes L_{2} \otimes L_{3}$ . Whitney identities and virtual Chern arithmetic gives the filtered product formula and $c_{1} (L \otimes M) = c_{1} (L) + c_{1} (M)$ .

[F3]

On noetherian finite-dimensional field-schemes, The actual perfect K-space operations satisfy total Cartan gives a natural multiplicative total Chern series for actual perfect $K_{0}$ classes. External formal universal Chern maps and supported construction identifies its restriction to bundle classes with the supplied universal bundle Chern classes. Formal Chern arithmetic through degree three gives the formal inverse through degree three.

[F4]

For the stated normal finite-type scheme, Zeroth projection is rank identifies $p_{0}$ with rank; First projection is first Chern class identifies $p_{1}$ on $F^{1}$ with $c_{1}$ ; and Second projection is minus second Chern class identifies $p_{2}$ on $F^{2}$ with $- c_{2}$ and its kernel with $F^{3}$ . The low-stage projections and stage-to-image isomorphisms are those of Graded filtration projections. Stage projections, kernels and image groups identifies the kernel of a transported stage projection with the next image filtration. The Axiom of Choice states the Choice assumption.

Proof

1.1givenF1F4

A rank-three bundle has exterior-square rank three and determinant rank one, as can be checked on a local basis. Thus the alternating rank of the four terms of $β (E)$ is $1 - 3 + 3 - 1 = 0$ at every point. The scheme is affine of finite type over a field and hence noetherian, finite-dimensional and quasi-compact quasi-separated; normality gives the seminormality required in [F4].

1.2F1F2given

Apply [F1] to $E$ . Write $x_{i} = c_{1} (L_{i})$ and set $s = x_{1} + x_{2} + x_{3}$ , $e_{2} = x_{1} x_{2} + x_{1} x_{3} + x_{2} x_{3}$ , and $e_{3} = x_{1} x_{2} x_{3}$ in the even-degree motivic cohomology ring of $Y$ . By [F2], the exterior-square line roots are $x_{1} + x_{2}$ , $x_{1} + x_{3}$ , and $x_{2} + x_{3}$ , while the determinant line has first class $s$ . Moreover $c_{t} (q^{*} E) = 1 + s t + e_{2} t^{2} + e_{3} t^{3} .$ The equality is the Whitney formula for the flag filtration, not an assertion that the flag splits globally.

2.1F2F3step 1.2algebra

The actual perfect Chern series is natural and multiplicative under addition by [F3]. Its values on the four bundle terms are their bundle Chern series, so [F2] and [F3] give, through degree three, $q^{*} c_{t} (β (E)) = \frac{\prod_{i < j} (1 + (x_{i} + x_{j}) t)}{(1 + s t) (1 + s t + e_{2} t^{2} + e_{3} t^{3})} .$ The numerator is $1 + 2 s t + (s^{2} + e_{2}) t^{2} + (s e_{2} - e_{3}) t^{3}$ , while the denominator is $1 + 2 s t + (s^{2} + e_{2}) t^{2} + (s e_{2} + e_{3}) t^{3}$ modulo $t^{4}$ . Their constant coefficients are one. Formal inversion therefore yields $q^{*} c_{t} (β (E)) = 1 - 2 e_{3} t^{3} (m o d t^{4}) .$ These polynomial identities hold integrally, including in characteristic two; no division by two is used.

3.1F1step 1.2step 2.1

By step 1.2, $e_{3} = q^{*} c_{3} (E)$ . Comparing coefficients in step 2.1 gives zero pullbacks of $c_{1} (β)$ and $c_{2} (β)$ and of $c_{3} (β) + 2 c_{3} (E)$ . The injectivity in [F1], separately in bidegrees $(2, 1)$ , $(4, 2)$ and $(6, 3)$ , proves the three stated identities on $X$ .

4.1F4step 1.1step 3.1∎

By step 1.1 and [F4], $p_{0} (β) = 0$ . Stage exactness and the low-stage identification in [F4] put $β$ in $F^{1}$ . By step 3.1 and the first-projection identity, $p_{1} (β) = 0$ ; the same exactness puts it in $F^{2}$ . Finally $p_{2} (β) = - c_{2} (β) = 0$ , so the kernel statement in [F4] puts $β$ in $F^{3}$ . This is membership in the image filtration and does not assert that the third stage itself maps injectively to $K_{0}$ . Together with steps 1.1 through 3.1, this proves all assertions.

Depends on

Used by · 1

Third Chern vanishing and a trivial summand

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. 17, proof of Corollary 6.10, final paragraph