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Affine perfect K zero and bundle K zero
PDF p. 12: Bundle classes in spectral K zero
Statement
Assume the Axiom of Choice and suitable universes. For a commutative unital ring A and X=Spec A, there is a canonical isomorphism π₀K(X)≅K⁰(X), where K(X) is the nonconnective spectrum of Nonconnective K theory of perfect complexes and K⁰(X) is the vector-bundle Grothendieck group. Under this isomorphism the class of the associated sheaf complex of a bounded complex P of finite projective A-modules maps to Σₙ(−1)ⁿ[P̃ⁿ], and the spectral class of a vector bundle E placed in degree zero maps to [E]. Equivalently, the inverse sends [E] to the class of E[0]. No noetherian or regularity hypothesis is imposed.
A vector bundle is a locally finite free -module. Its Grothendieck group is generated by isomorphism classes of vector bundles with [E]=[E′]+[E″] for every short exact sequence 0→E′→E→E″→0 of vector bundles. A finite projective A-module is a finitely generated module having the lifting property through surjective module maps; a bounded complex has finitely many nonzero terms. This is a comparison with vector-bundle K zero, not an identification with coherent-sheaf G-theory.
Facts & Assumptions
Given: AC, suitable universes, a commutative unital ring A, and X=Spec A.
Goal: Identify the degree-zero group of the specified nonconnective spectrum with the vector-bundle group and establish the stated formulas.
Choice is assumed for the essentially small perfect-category and K-theory constructions invoked in [F1]; it is not inferred from affineness. The Axiom of Choice
An affine scheme is qcqs. The supplied interface defines K(X)=𝕂(Perf(X)), with an essentially small stable idempotent-complete perfect category and invariance under an equivalent small presentation. Nonconnective K theory of perfect complexes
The cofiber Grothendieck group of a small stable infinity-category is canonically the triangle Grothendieck group of its homotopy category. Cofiber and triangle presentations of K zero
For a small idempotent-complete stable infinity-category, the object-class map identifies its cofiber Grothendieck group with π₀ of the specified nonconnective K-theory. This is an external input with the source and reading limits recorded there. Nonconnective K theory in degree zero
The triangle group of Perf(X) identifies with the vector-bundle group; it takes a bounded finite-projective representative to the alternating sum of its associated bundle terms and E[0] to [E]. The exact affine equivalence used there is degreewise associated sheaf. Affine perfect triangle classes and bundle Euler classes, Perfect complexes on schemes and their support
Proof
Choose a small stable idempotent-complete presentation C equivalent to Perf(X), available by [F1]. The equivalence identifies the corresponding K-theory spectra and hence their π₀ groups. Apply [F3] to C and invert its object-class isomorphism; then apply [F2] to obtain an isomorphism from π₀K(X) to the triangle Grothendieck group of hPerf(X). The construction is independent of the presentation by the naturality and equivalence invariance in [F1] and [F3].
Compose the isomorphism of step 1.1 with the affine Euler isomorphism of [F4]. This gives π₀K(X)≅K⁰(X). By [F3], the spectral class of an object is sent in step 1.1 to its triangle class. Therefore [F4] sends the class represented by P to Σₙ(−1)ⁿ[P̃ⁿ], and sends the class of E[0] to [E].
The classes [E] generate K⁰(X), so the final formula also describes the inverse on its generators. The hypotheses of all invoked suppliers are satisfied by the given commutative unital ring, AC and affine qcqs scheme; none requires noetherianity or regularity. This proves the bridge and both formulas.
Depends on
Used by · 13
- Isomorphism classes of vector bundles
- Critical-rank representatives of rank-zero classes
- Prescribing three Chern classes successively
- Murthy splitting input in rank three
- Weight zero is the locally constant rank function
- External determinant and pointed one-type inputs
- Normal dimension strata and the actual residue class model
- The actual first EM stage is the rank-zero K space on all qcqs schemes
- Actual supported Chern operation and its smooth comparison
- Krishna's affine zero-cycle divisibility and kernel input
- Local free chains detect natural perfect operations on every qcqs base
- Euler-group comparison and the article's expectation
- Full affine K-space comparison and its sheafification
Dependency tree · 3 levels
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