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.

Coordinate faces of the algebraic simplex

PDF p. 4: Smooth comparison with Bloch cycles

Statement

Let $k$ be a field, $n \geq 0$ , $Δ^{n} = Spec k [t_{0}, \dots, t_{n}] / (\sum_{a} t_{a} - 1)$ , and $X$ a $k$ -scheme. For a proper subset $I ⊊ {0, \dots, n}$ , the coordinate face $F_{I} = V (t_{i} : i \in I)$ is canonically $Δ^{n - ∣ I ∣}$ after ordering the remaining coordinates. On each nonempty affine open of $X$ , the coordinates indexed by $I$ form a regular sequence in any order: successive multiplication is injective and the final quotient is nonzero. Imposing all coordinates gives the empty scheme. If $δ_{i}^{n}$ inserts zero at position $i$ , then for $0 \leq i < j \leq n$ and $n \geq 2$ , $δ_{j}^{n} δ_{i}^{n - 1} = δ_{i}^{n} δ_{j - 1}^{n - 1}$ .

Facts & Assumptions

Given: The field, scheme, coordinates and index set in the Statement.

Goal: Establish the face descriptions, regular sequences and coordinate identities.

[F1]

On an affine open $Spec B \subset X$ , the product ring is $B [t_{0}, \dots, t_{n}] / (\sum t_{a} - 1)$ ; quotienting by the indicated coordinates gives the face.

Proof

1.1F1construct

Choose $b \notin I$ . Eliminate $t_{b}$ using $t_{b} = 1 - \sum_{a \neq b} t_{a}$ . The product ring becomes a polynomial ring in the remaining coordinates over $B$ , and quotienting by the coordinates in $I$ leaves a polynomial ring in $n - ∣ I ∣$ variables. For $B = k$ this gives the indicated simplex with its ordered barycentric coordinates.

2.1step 1.1algebra

Multiplication by an indeterminate in a polynomial ring is injective by coefficient comparison. After quotienting by earlier distinct indeterminates, the remaining ring is again polynomial over $B$ . Thus the sequence is regular in every order on a nonempty affine open. If all coordinates vanish, $\sum t_{a} = 1$ becomes $0 = 1$ , so the resulting scheme is empty.

3.1step 1.1algebra∎

Both displayed composites insert zero in positions $i$ and $j$ and leave the other coordinates in their original order. Their coordinate maps agree, proving the identity and the lemma.

Used by · 1

Admissible face restrictions and their cycle identities