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Spectral K-zero equality gives an actual projective stabilization

Statement

Assume AC and the suitable universes of 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 $X$ , meaning finite locally free $O_{X}$ -modules. Equality $[E] = [F]$ in the article's perfect spectral $K_{0} (X)$ is equivalent to an isomorphism of finite projective $A$ -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 stated universes, a commutative unital noetherian ring $A$ , the affine scheme $X = Spec A$ , and vector bundles $E, F$ on $X$ .

Goal: Establish the stated conclusion.

[A1]

AC is assumed through the supplied affine perfect-K-zero comparison and affine bundle/module interfaces; their universe bounds are also assumed. No choice principle stronger than those supplier hypotheses is introduced. The Axiom of Choice

[F1]

The spectral group identifies with vector-bundle $K^{0}$ and preserves bundle classes. 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 affine comparison identifies these classes with the classes of the bundles $E, F$ in $K^{0} (X)$ . Since $X$ is noetherian affine and $E, F$ are vector bundles, [F2] gives $E \oplus O_{X}^{N} ≅ F \oplus O_{X}^{N}$ for an integer $N \geq 0$ . By [F3], global sections give the asserted isomorphism of finite projective modules. The choice and size hypotheses required by these suppliers are included in the given assumptions.

2.1A1F1F2F3step 1.1∎

Conversely, suppose the displayed module isomorphism holds for some integer $N \geq 0$ . By [F3] it gives a bundle isomorphism $E \oplus O_{X}^{N} ≅ F \oplus O_{X}^{N}$ . By [F2], $E$ and $F$ have equal classes in $K^{0} (X)$ , and [F1] identifies these with their spectral classes. Together with step 1.1 this proves the equivalence.

Depends on

Used by · 1

Local application interface for a critical-rank cancellation theorem

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.