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The reduced universal bundle calculation identifies the second projection

PDF p. 11: Proposition 5.5; pp. 11–12

Statement

Assume the Axiom of Choice and the compatible site-size and universe bounds required by the supplied perfect K-theory, motivic and derived-section constructions. For a vector bundle $V$ of locally constant rank $n$ on a noetherian finite-dimensional field-scheme $Y$ , let $v = [V] - n \in π_{0} S (Y)$ and $\tilde{δ} (V) = r (v) \in π_{0} T (Y)$ . Its underlying perfect class is $δ (V) = [V] - [\det V] - (n - 1)$ . For the actual second stage projection $Q : T \to A = Ω^{\infty} (Z (2)_{m o t} [4])$ and actual second Chern map $C_{2} : Ω^{\infty} K \to A$ , $Q (\tilde{δ} (V)) = - c_{2} (V) = - c_{2} (δ (V)) .$ The assertion holds for actual universal bundles on finite Grassmannians over $Y$ , naturally under their transition maps.

Facts & Assumptions

[A1]

Assume the Axiom of Choice as formulated in The Axiom of Choice and the compatible site-size and universe bounds in the Statement. They are inherited through the determinant-fiber, first-Chern and flag constructions in [F1]–[F2] at steps 1.1–2.1 and the actual perfect Cartan comparison in [F3] at step 3.1.

[F1]

The actual determinant fiber is a natural retract of the rank-zero K-space supplies the determinant retraction $r : S \to T$ for $S = Ω^{\infty} {Fil}^{1} K$ and $T = Ω^{\infty} {Fil}^{2} K$ , and injectivity of $π_{0} T \to K_{0}$ . Filtration projections and multiplication supplies the actual stage product and its projection square on every qcqs field-scheme; only this stage-level assertion is used, without the additional hypotheses of its image-group square. The line lift $κ (L)$ has first projection $c_{1} (L)$ by that section and First Chern class factors through determinant.

[F2]

Flag towers with line-bundle quotients gives a flag pullback with a filtration by line quotients; Detection on a simultaneous flag tower detects equality in every motivic bidegree on this tower. It does not assert that the filtration splits globally. Whitney identities and virtual Chern arithmetic identifies its Chern roots.

[F3]

The actual perfect K-space operations satisfy total Cartan supplies actual perfect Cartan in this noetherian finite-dimensional scope. Reduced bundle class identities supplies the reduced-class ring identities on every qcqs scheme over any field under Choice and its size bounds. On each constant-rank component of the flag tower, apply those identities to partial direct sums of the line quotients. The exact flag sequences identify the final direct-sum class with the pulled-back bundle class. A noetherian scheme is qcqs, and these finite projective-bundle towers and finite Grassmannians remain noetherian and finite-dimensional. The calculation below spells out both the ring expansion and its actual stage lifts.

Proof

Given: $Y, V$ as stated, with the actual maps and finite clopen rank decomposition.

Goal: Compute both actual operations on the reduced determinant-trivial class.

1.1F1F2F3A1algebra

The expression for $δ (V)$ follows from the formula for $r$ . It has rank zero and its determinant is canonically trivial. Under [A1], construct the detecting flag tower in [F2] on each constant-rank clopen stratum. On one such tower put $l_{s} = [L_{s}]$ , $D_{s} = ⨂_{i \leq s} L_{i}$ and $R_{s} = \sum_{i \leq s} l_{i} - [D_{s}] - (s - 1)$ . Exact flag sequences give these perfect $K_{0}$ class equalities without globally split sequences. For $s \geq 2$ , the line-product identity in [F3], or direct distributivity, gives $([L_{s}] - 1) ([D_{s - 1}] - 1) = [D_{s}] - [L_{s}] - [D_{s - 1}] + 1$ , hence $R_{s} = R_{s - 1} - ([L_{s}] - 1) ([D_{s - 1}] - 1)$ and $R_{1} = 0$ . Define ${\tilde{R}}_{s} = r (\sum_{i \leq s} κ (L_{i}))$ . Its argument has underlying class $\sum_{i \leq s} l_{i} - s$ and determinant $D_{s}$ , so ${\tilde{R}}_{s}$ has underlying class $R_{s}$ . Let $P_{s}$ be the actual second-stage product of $κ (L_{s})$ and $κ (D_{s - 1})$ . Its underlying class is the product in the recursion. Injectivity of the second-stage augmentation in [F1] gives ${\tilde{R}}_{1} = 0$ and ${\tilde{R}}_{s} = {\tilde{R}}_{s - 1} - P_{s}$ . At $s = n$ the same injectivity identifies ${\tilde{R}}_{n}$ with the pullback of $\tilde{δ} (V)$ . This comparison uses no additivity of $r$ .

2.1F1F2step 1.1A1

Write $x_{i} = c_{1} (L_{i})$ . The multiplicative projection square and additivity of the actual second-stage map give by induction $Q (\tilde{δ} (V)) = - \sum_{s = 2}^{n} x_{s} (\sum_{i < s} x_{i}) = - \sum_{i < j} x_{i} x_{j} = - c_{2} (V) .$ For rank two this is exactly $([L_{1}] - 1) ([L_{2}] - 1) = - δ (V)$ ; the displayed recursion is the full rank induction. Additivity of line $c_{1}$ identifies $c_{1} (D_{s - 1})$ with the indicated sum. Rank zero and rank one give zero. Detection in [F2] descends the equality; finite clopen reduction handles varying rank.

3.1F3step 1.1step 2.1A1∎

Put $a = c_{1} (V) = c_{1} (\det V)$ . Actual perfect Cartan gives $c_{t} (δ (V)) = c_{t} (V) (1 + a t)^{- 1},$ since a trivial virtual rank has Chern series one. Its $t^{2}$ coefficient is $c_{2} (V) - a^{2} + a^{2} = c_{2} (V)$ . This uses actual perfect Cartan, not a global bundle presentation of arbitrary classes. All constructions commute with pullback, so they apply to the actual universal bundles on each finite Grassmannian.

Depends on

Used by · 1

The second-stage projection is minus the actual second Chern map on its homotopy fiber

Dependency tree · 3 levels

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