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Spectral K-zero equality gives an actual projective stabilization

Statement

Assume AC and the suitable universes used for the article's perfect K-theory construction. Let $A$ be a commutative unital noetherian ring, put $X = Spec A$ , and let $E, F$ be vector bundles on $X$ . Equality $[E] = [F]$ in the article's spectral $K_{0} (X)$ is equivalent to an isomorphism of $A$ -modules $Γ (X, E) \oplus A^{N} ≅ Γ (X, F) \oplus A^{N}$ for some integer $N \geq 0$ . This is the bundle/perfect K-zero interface, without a passage through coherent-sheaf G-theory.

Facts & Assumptions

Given: AC, the suitable universes of the supplied perfect K-theory construction, a commutative unital noetherian ring $A$ , $X = Spec A$ , and vector bundles $E, F$ on $X$ .

Goal: Establish the stated conclusion.

[A1]

AC is used through the supplied affine spectral comparison, stable-equivalence result and bundle–module equivalence. The Axiom of Choice

[F1]

In the stated universes, for an affine scheme the spectral group identifies with vector-bundle $K^{0}$ and preserves the classes of bundles concentrated in degree zero. Isomorphism classes of vector bundles

[F2]

Equality in $K^{0}$ is equivalent to stabilization by the same finite free bundle. Equality in affine K zero and stable isomorphism

[F3]

Stabilization is carried to and from modules. Free stabilization passes through global sections

Proof

1.1A1F1F2F3given

Suppose $[E] = [F]$ in spectral $K_{0} (X)$ . The ring hypotheses make $X$ noetherian affine, and $E, F$ are vector bundles, so all three supplied interfaces apply under AC and the stated universes. By [F1] their classes agree in $K^{0} (X)$ . By [F2] there is an integer $N \geq 0$ with $E \oplus O_{X}^{N} ≅ F \oplus O_{X}^{N}$ . Applying [F3] gives $Γ (X, E) \oplus A^{N} ≅ Γ (X, F) \oplus A^{N}$ .

2.1A1F1F2F3step 1.1∎

Conversely, suppose the displayed module isomorphism holds for an integer $N \geq 0$ . By [F3] it gives $E \oplus O_{X}^{N} ≅ F \oplus O_{X}^{N}$ . By [F2] the bundle classes agree in $K^{0} (X)$ , and [F1] identifies them with the given spectral classes. Thus $[E] = [F]$ in spectral $K_{0} (X)$ , proving the equivalence together with step 1.1.

Depends on

Used by · 1

Local application interface for a critical-rank cancellation theorem

Dependency tree · 3 levels

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