TheoremStatement: Literature-sourcedProof: Literature-sourced

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

External Levine–Weibel quotient cycle map with exact generators

PDF p. 3: Levine–Weibel image after TheoremC; PDF p. 14: Definition6.5 pure-dimensional smooth-point subgroup

Statement

For a normal affine scheme $X$ of finite type over a perfect field and of dimension $d \geq 1$ , use Krishna's Section2.1 dimension- $d$ convention: the free generators are smooth closed points on its dimension- $d$ components, the forbidden locus includes all lower-dimensional components, and the relations are divisors of semilocal units on the specified relative Cartier curves. There is an established quotient cycle map ${cyc}_{X}^{K r} : {CH}_{K r}^{d} (X) ⟶ K_{0} (X), [x] ⟼ [κ (x)] .$ In particular every specified Cartier-curve principal-divisor relation maps to zero. The target is the classical projective-module $K_{0}$ in the affine case, equivalently the quasi-projective perfect/Thomason–Trobaugh convention recorded immediately after equation2.2. Comparisons with the local perfect-spectrum model and the local pure-dimensional Levine–Weibel presentation are separate application arguments.

Source contract

Given: Exactly the restricted perfect-field, normal affine finite-type and dimension- $d$ point/Cartier-curve convention above.

Goal: Import the outstanding established relation theorem with its actual map normalization.

[F1] Krishna §2.1 equation(2.2) recalls the quotient map from Levine–Weibel Proposition2.1 and gives its residue-class values. Its p6 conventions specify which components contribute generators.

External input

Import [F1], with the full exact source record. The factorization through the Cartier-curve relations is substantial external input; it is not inferred locally merely from the existence of point classes. The five local drafts establish component conventions, actual K-class transport, filtered zero extension and the resulting image equality, not a new reconstruction of the Levine–Weibel relation proof.

Adjudication reading limits

For this maintenance, read the primary PDF text of Krishna, arXiv:1511.04221v3 (18 March 2019), Section 1.9 and Section 2.1 through equation (2.2) and the following K-theory convention, printed/PDF pp. 6–8; also checked the introductory point-class and image statements on pp. 1–2. These passages supply the finite-type scope and the precise imported map. The original Levine–Weibel Proposition 2.1 proof was not read or reconstructed. PDF screenshot retrieval failed; no visual inspection is claimed.

Depends on

Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv1511.04221v3,18March2019 · Section1.9 and opening Section2.1, printed/PDF p6; Cartier curves and quotient equation(2.1),p7; cycle map equation(2.2) and K-theory convention,p8. Intro p1 height-d regular maximal ideals and p2 §1.3 cycle-map/image statement.

Used by · 3

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv1511.04221v3,18March2019 · Section1.9 and opening Section2.1, printed/PDF p6; Cartier curves and quotient equation(2.1),p7; cycle map equation(2.2) and K-theory convention,p8. Intro p1 height-d regular maximal ideals and p2 §1.3 cycle-map/image statement.

The Cartier-curve principal-divisor quotient admits cyc_X:CH^d(X,Y)→K_0(X), sending a permitted regular closed point x to [k(x)]; it recalls Levine–Weibel Proposition2.1 for its existence. Its image is the subgroup classically denoted F^dK_0(X). In §2.1 the permitted X_reg uses smooth loci of d-dimensional components, excluding lower-dimensional components in a nonpure scheme.

Hypotheses: Section2 assumes a perfect field unless stated otherwise; X is reduced quasi-projective finite type of dimension d≥1; forbidden Y contains the source X_sing and no d-dimensional component. This import is used only for normal affine X, where components are disjoint, or its pure-dimensional normal pieces; then the source generator/Cartier-curve definitions match the completed local definition exactly. Algebraically closed k in the assigned article is permitted in every characteristic.

Reading limits: Read actual v3 PDF pp6–8 in full for notation, generator convention, Cartier curves, semilocal units, quotient and equation2.2, including the immediately following K-theory convention. Read relevant introductory p1/p2 statements. The original Levine–Weibel Proposition2.1 proof recalled by Krishna was not obtained or reconstructed; existence and relation-vanishing of the precise quotient map are explicit established external inputs. No torsion, injectivity, isomorphism, rational-equivalence or later K-theory theorem from Krishna is imported here.

Sources

Amalendu Krishna, Murthy’s conjecture on 0-cycles, arXiv1511.04221v3,18March2019 · Section1.9 and opening Section2.1, printed/PDF p6; Cartier curves and quotient equation(2.1),p7; cycle map equation(2.2) and K-theory convention,p8. Intro p1 height-d regular maximal ideals and p2 §1.3 cycle-map/image statement.