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Reduced bundle class identities

PDF p. 11: Reduced rank-two identity; PDF p. 12: Induction identity

Statement

Assume the Axiom of Choice and the site and universe bounds required by the rank-fiber, determinant-fiber and Chern-class comparisons. Let $k$ be any field and $X$ a quasi-compact quasi-separated $k$ -scheme. For a vector bundle $V$ of constant rank $n \geq 0$ , put $δ (V) = [V] - [\det V] - (n - 1) 1 \in K_{0} (X) .$ Then $δ (V) \in F_{m o t}^{2} K_{0} (X)$ . If $V$ and $W$ have the same constant rank and isomorphic determinant line bundles, then $[V] - [W] = δ (V) - δ (W) .$ If $V = L_{1} \oplus L_{2}$ for line bundles $L_{1}, L_{2}$ , then $([L_{1}] - 1) ([L_{2}] - 1) = [\det V] + 1 - [V] .$ For every $V$ as above, its virtual second Chern class satisfies $c_{2} (δ (V)) = c_{2} (V)$ . Finally, if $L$ is a line bundle, then $δ (V \oplus L) = δ (V) - ([L] - 1) ([\det V] - 1) .$

A vector bundle is a finite locally free $O_{X}$ -module; a line bundle has rank one. The determinant $\det V$ is $⋀^{n} V$ , with $\det 0 = O_{X}$ . Quasi-compact means that every open cover has a finite subcover; quasi-separated means that intersections of quasi-compact opens are quasi-compact. In particular, these assertions apply to all normal qcqs schemes over an algebraically closed field, including the finite-type case. Normal means that every local ring is an integrally closed domain. Here $K_{0} (X)$ is the zeroth group of nonconnective K-theory of perfect complexes; a vector bundle gives a class in it, $1 = [O_{X}]$ , and its product is induced by tensor product. A perfect complex is locally quasi-isomorphic to a bounded complex of vector bundles. The group $F_{m o t}^{j} K_{0} (X)$ is the image of the $j$ th motivic filtration stage in $K_{0} (X)$ . The virtual Chern class in the displayed formula is the second coefficient of the representative-independent total Chern series on the vector-bundle Grothendieck group, evaluated on the indicated virtual bundle expression; no identification of that group with all of perfect-complex $K_{0} (X)$ is assumed. Choice asserts that every family of nonempty sets admits a choice function; the size bounds fix the sites and ambient universes of the cited comparisons.

Facts & Assumptions

Given: $k$ , $X$ , Choice, the stated size bounds, and the vector bundles and line bundles occurring in the assertions.

Goal: Prove the second-filtration membership, both ring identities, the equal-rank comparison, and the virtual second-Chern identity.

[F1]

The zeroth filtration stage is $K (X)$ and the image groups are the images of the stage augmentations. On qcqs field-schemes, put $S = Ω^{\infty} F_{m o t}^{1} K$ and $T = Ω^{\infty} F_{m o t}^{2} K$ . The actual first projection $d : S \to B G_{m}$ is determinant, and $T$ is its homotopy fiber over the trivial line. Thus a first-stage point together with a trivialization of its determinant gives a second-stage lift. Motivic filtration and image groups The actual determinant fiber is a natural retract of the rank-zero K-space

[F2]

For every qcqs scheme over a field, the actual first-stage augmentation identifies $π_{0} F_{m o t}^{1} K (X)$ injectively with the kernel of Euler rank on perfect $K_{0} (X)$ . Hence every rank-zero perfect class has a unique first-stage class lifting it. The actual first EM stage is the rank-zero K space on all qcqs schemes

[F3]

The first perfect Chern class of a virtual bundle is the image of its determinant under the weight-one orientation; this agrees with the first coefficient of its virtual bundle Chern series. First Chern class factors through determinant

[F4]

The total Chern series is multiplicative for direct sums and virtual differences. For a line bundle $M$ , $c_{t} (M) = 1 + c_{1} (M) t$ , and $c_{1} (V) = c_{1} (\det V)$ . The latter follows from [F3], while the line and virtual-series formulas are supplied by Whitney identities and virtual Chern arithmetic.

[F5]

The Axiom of Choice is among the hypotheses of the cited filtration and Chern comparisons. The Axiom of Choice

Proof

technique · Lift the rank-zero class through the actual determinant fiber, then compute in the bundle Grothendieck ring and its virtual Chern series

1.1givenF1F2F5

The rank of $δ (V)$ is $n - 1 - (n - 1) = 0$ , including $n = 0$ . By [F2], it has a unique first-stage class $s \in π_{0} S (X)$ lifting it. Its virtual determinant is $\det V \otimes (\det V)^{- 1} \otimes (\det O_{X})^{- (n - 1)} ≅ O_{X}$ , since the determinant of a virtual trivial bundle is trivial. By [F1], the image of $s$ under the actual determinant map is therefore the trivial line class. Represent $s$ by a point of $S (X)$ and choose a trivialization of its determinant; this is a path of its image to the basepoint of $B G_{m} (X)$ . The point and path define a point of the actual fiber $T (X)$ whose underlying perfect class is $δ (V)$ . By the image-group definition in [F1], this proves $δ (V) \in F_{m o t}^{2} K_{0} (X)$ . These suppliers apply throughout the stated qcqs scope under the assumed Choice and size bounds.

1.2givenconstruct

If $V$ and $W$ have the same rank and isomorphic determinants, subtracting their definitions of $δ$ cancels the determinant and trivial-bundle terms. This gives $δ (V) - δ (W) = [V] - [W]$ .

1.3givenconstruct

For $V = L_{1} \oplus L_{2}$ , its determinant is $L_{1} \otimes L_{2}$ . Distributivity in $K_{0} (X)$ and the direct-sum relation give $([L_{1}] - 1) ([L_{2}] - 1) = [L_{1} \otimes L_{2}] - [L_{1}] - [L_{2}] + 1 = [\det V] + 1 - [V] .$ This calculation is performed on actual bundle classes and their images in perfect-complex $K_{0}$ .

1.4F3F4construct

Write $a = c_{1} (V) = c_{1} (\det V)$ . By [F4], the total series of $δ (V)$ is $c_{t} (V) (1 + a t)^{- 1}$ : the trivial-bundle factor has total class $1$ , even when its coefficient in the virtual expression is negative. Through degree two, $c_{t} (V) (1 + a t)^{- 1} = (1 + a t + c_{2} (V) t^{2} + \dots) (1 - a t + a^{2} t^{2} + \dots) .$ The coefficient of $t^{2}$ is $c_{2} (V) - a^{2} + a^{2} = c_{2} (V)$ . This proves the stated virtual identity without requiring a Chern-series operation on arbitrary perfect classes.

2.1givenconstructstep 1.1step 1.2step 1.3step 1.4∎

For $D = \det V$ , the determinant of $V \oplus L$ is $D \otimes L$ . Therefore $\begin{aligned} δ (V \oplus L) - δ (V) & = [L] - [D \otimes L] + [D] - 1 \\ = - ([L] - 1) ([D] - 1) . \end{aligned}$ The equality uses distributivity and the tensor-product interpretation of multiplication. Together with steps 1.1 through 1.4 this proves all assertions.

Depends on

Used by · 1

The reduced universal bundle calculation identifies the second projection

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 · v1, PDF p. 3 conventions; p. 8, Section 3; pp. 11–12, proof of Proposition 5.5