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.

Admissible face restrictions and their cycle identities

PDF p. 4: Smooth comparison with Bloch cycles

Statement

Let $k$ be a field, $X$ smooth of finite type over $k$ , and $r, n \geq 0$ . Write $z^{r} (X, n)$ for the free abelian group on integral closed subschemes $W \subset X \times Δ^{n}$ of codimension $r$ meeting every nonempty coordinate face properly: each intersection component has codimension $r$ in the face, and an empty intersection is allowed. For $n \geq 1$ , define $d_{i}^{n} [W] = \sum_{T} ℓ_{O_{W, η_{T}}} (O_{W, η_{T}} / (t_{i})) [T],$ where $T$ runs over the components of $W \cap {t_{i} = 0}$ and the target face is identified with $X \times Δ^{n - 1}$ in increasing coordinate order. The sum is zero if the intersection is empty. These formulas give homomorphisms $d_{i}^{n} : z^{r} (X, n) \to z^{r} (X, n - 1)$ and, for $n \geq 2$ and $0 \leq i < j \leq n$ , satisfy $d_{i}^{n - 1} d_{j}^{n} = d_{j - 1}^{n - 1} d_{i}^{n}$ as equalities of integral cycles. Neither separatedness nor pure dimension is assumed for $X$ .

Facts & Assumptions

Given: $k, X, r, n$ and the groups in the Statement.

Goal: Establish finite coefficients, admissibility and the cycle-level face identities.

[F1]

Coordinate faces are regular embeddings with the stated order identifications. Coordinate faces of the algebraic simplex

[F2]

A proper single-coordinate intersection has the specified local tensor-resolution length. Length multiplicity for one proper Cartier face

[F3]

Cycle equality, the relevant codimension conditions and the displayed lengths are open local. Cycle coefficients and face intersections are open local

[F4]

The separated equidimensional smooth case, including the simplicial identity with intersection multiplicities, is an external input. External input for simplicial Bloch cycle groups

[F5]

A smooth finite-type scheme admits a cover by integral smooth affine opens. External geometric inputs for affine reduction

[F6]

Generator images extend uniquely to homomorphisms. Free abelian group on a set

Proof

1.1F3F4F5construct

By [F5], cover $X$ by integral smooth affine opens $U$ . Each is separated and equidimensional, so [F4] applies. Restrictions of admissible generators remain admissible by [F3].

1.2F1F2F4given

At a component $T$ of a single face intersection, put $A = O_{X \times Δ^{n}, η_{T}}$ and let $I$ define $W$ there. By [F1], $t_{i}$ is a nonzerodivisor on $A$ . Properness prevents $W$ from lying in the face, so $t_{i} \notin I$ ; since $η_{T}$ is generic in the intersection, $\sqrt{I + (t_{i})}$ is the maximal ideal of $A$ . The Tor coefficient in [F4] is therefore the finite quotient length in [F2]. This is also the length over $A / I = O_{W, η_{T}}$ , since the submodules coincide.

2.1F3F4F6step 1.2construct

The intersection has finitely many components because it is noetherian. The global sum consequently has finite coefficients and restricts to the local face cycles by [F3] and step 1.2. Those local cycles are admissible by [F4]; [F3] detects the codimension conditions globally. Extend to a homomorphism by [F6].

3.1F1F3F4step 1.1step 1.2step 2.1∎

Restrict the two sides of the claimed identity to each open in step 1.1. Both successive face operations commute with restriction by [F3] and step 1.2. On each open they agree by [F4], with the same target face identified by [F1]. Local detection in [F3] gives equality of global cycles with all coefficients. This proves the lemma; compatibility of multiplicities is imported in [F4].

Depends on

Used by · 2

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