LemmaStatement: AI-generatedProof: AI-generated

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.

The origin of the affine line has zero perfect K class

Statement

Assume the Axiom of Choice and suitable universes for perfect complexes and their K-theory. Let $k$ be a field, $A = k [t]$ , and $X = Spec A$ , the affine line over $k$ . Let $x$ be the point corresponding to the ideal $(t)$ , and let $i : Spec k ↪ X$ be induced by evaluation at $t = 0$ . Then $X$ is quasi-compact and quasi-separated and is smooth over $k$ , $x$ is closed with residue field $κ (x) = k$ , and the residue sheaf $i_{*} k$ is perfect. Its degree-zero perfect K-theory class vanishes:

$[i_{*} k] = 0 in K_{0} (X) = π_{0} K (X) .$

Here $K$ is nonconnective K-theory of perfect complexes; $i_{*} k$ denotes the sheaf with sections $k$ on opens containing $x$ and zero on opens avoiding $x$ , with scalar action through evaluation at $x$ . A perfect complex is locally represented by a bounded complex of finite locally free modules; a sheaf in a perfect-complex class is placed in degree zero.

Facts & Assumptions

Given: $k, A, X, x, i$ as in the statement, AC and compatible suitable universes. The polynomial ring consists of finite coefficient sequences, with coefficientwise addition and convolution multiplication. The constants embed $k$ into $A$ and $t$ is the sequence with coefficient $1$ at degree one and zero elsewhere.

Goal: Check the geometric hypotheses and prove the vanishing by an explicit two-term resolution and the affine Euler-class comparison.

[A1]

The Axiom of Choice supplies the assumed AC, retained for the affine compactness, smoothness, associated-sheaf and perfect K-theory suppliers below. Suitable universes are assumed with the closure convention of Grothendieck universe closure convention and inherited compatibly from the perfect-complex and K-theory interfaces, not inferred from AC.

[F1]

Article scheme conventions supplies scheme-over-field, qcqs and residue-field conventions, with the base field here explicitly arbitrary. The polynomial ring over a commutative ring as finitely supported coefficient sequences with convolution supplies the coefficient model of $k [t]$ used in the resolution. Smooth morphisms and closed immersions use Smooth morphism of schemes and Closed immersions of schemes.

[F2]

Every affine scheme is quasi-compact and Affine schemes are quasi-separated give the two affine compactness properties. Polynomial rings are flat and smooth with base $k$ and one variable gives smoothness over $k$ , under AC.

[F3]

Finite-variable polynomial algebras over fields are Noetherian by finite generators makes $A$ Noetherian. For a maximal ideal of such a ring, A coherent closed-point skyscraper identifies the associated sheaf of the quotient module with the closed-point residue sheaf and identifies its residue field with that quotient. A surjective ring map induces a closed immersion of affine spectra gives the closed immersion induced by the quotient map.

[F4]

Exact sequences and short exact sequences of modules defines short exactness by equality of successive images and kernels, including injectivity and surjectivity at the ends. The stalk of an associated sheaf is the localisation identifies stalks of associated sheaves, naturally in module maps, with module localizations. Localisation of modules is exact preserves a short exact module sequence under localization. A sequence of abelian sheaves is exact exactly when it is exact on every stalk then establishes exactness of its associated sheaf sequence.

[F5]

Perfect complexes on schemes and their support defines perfect complexes, quasi-isomorphisms and degreewise associated-sheaf representatives on an affine scheme. Nonconnective K theory of perfect complexes defines $K (X)$ and $K_{0} (X)$ , using the degree-zero stable homotopy group of Stable spectra and exact triangles. Under the assumed AC and universes, Affine perfect K zero and bundle K zero identifies $K_{0} (X)$ with vector-bundle K zero and sends a bounded finite-projective complex to the alternating sum of its associated bundle terms.

Proof

1.1A1F1F2F3given

Evaluation $ϵ : A \to k$ , $\sum a_{j} t^{j} \mapsto a_{0}$ , is a surjective ring homomorphism: convolution gives constant coefficient $a_{0} b_{0}$ for a product, and constants give all values. Its kernel is $(t)$ , since a polynomial has zero constant term exactly when it is $t$ times the polynomial obtained by shifting its coefficients down one place. Every ideal properly containing $(t)$ contains an element with nonzero constant term; subtracting its multiple of $t$ and then multiplying by that nonzero constant's inverse puts $1$ in the ideal. Thus $(t)$ is maximal and $A / (t) ≅ k$ . By [F3], $x$ is closed, its residue field is $k$ , and $i$ is the quotient closed immersion with residue sheaf $\tilde{A / (t)} = i_{*} k$ . By [F2], $X$ is qcqs and smooth over $k$ . The polynomial ring is generated over $k$ by $t$ , so it also satisfies the article's finite-type affine convention.

2.1A1F1F4F5step 1.1∎

Multiplication by $t$ on $A$ is injective, since it shifts a finite coefficient sequence one place up without losing a coefficient. Together with step 1.1 this proves exactness of $0 \to A \overset{t}{\to} A \overset{ϵ}{\to} k \to 0$ . Apply [F4] at every prime and then its stalkwise criterion: the associated sheaf sequence is exact, with last term $i_{*} k$ by step 1.1. Hence the associated complex of $P^{- 1} = A$ , $P^{0} = A$ , $d^{- 1} = t$ and all other terms zero has an augmentation to $i_{*} k [0]$ inducing all cohomology isomorphisms. It is a quasi-isomorphism by [F5]. The two terms are free of rank one, hence finite projective (a map from $A$ lifts through a surjection by lifting the image of $1$ ), and their associated bundles are both $O_{X}$ . Thus $i_{*} k$ is perfect and its class maps by [F5] to $- [O_{X}] + [O_{X}] = 0$ . That comparison is an isomorphism, so $[i_{*} k] = 0$ in $K_{0} (X)$ itself. The assumed Choice and universe conditions discharge the hypotheses of all invoked constructions.

Depends on

Used by · 1

Smooth-point K theory class map

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.