LemmaStatement: AI-generatedProof: AI-generated

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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.

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Affine characteristic-p syntomic amplitude without regularity

PDF p. 5: Proposition 2.6: torsion freeness

Statement

For every $F_{p}$ -algebra $A$ and $j \geq 0$ , $H^{i} (F_{p} (j)_{s y n} (A)) = 0$ for $i > j + 1$ . In particular its degree-five group vanishes when $j = 3$ .

Facts & Assumptions

[F1]

External EM affine syntomic filtration and actual obstruction map supplies EM Lemma 4.16 and its notation $L_{A / F_{p}}^{a} = L Λ^{a} L_{A / F_{p}}$ . These derived exterior-power complexes lie in $D^{\leq 0}$ ; they are computed by a simplicial polynomial resolution with ordinary differential forms in degree zero.

Proof

Given: Any $F_{p}$ -algebra $A$ , with no smoothness, reducedness or finite-type hypothesis.

Goal: Establish the stated upper bound.

1.1F1

In [F1] the first block of graded pieces is $L^{a} [- a - 1]$ for $0 \leq a \leq j$ and the second block is $L^{a} [- a]$ for $0 \leq a \leq j$ . A complex in $D^{\leq 0}$ shifted by $- b$ lies in $D^{\leq b}$ . Every listed piece therefore lies in $D^{\leq j + 1}$ .

2.1F1step 1.1∎

The filtration is finite, starts at zero, and ends at the syntomic complex. The cohomology exact sequence for each successive extension shows inductively that the entire complex lies in $D^{\leq j + 1}$ . For $j = 3$ , $5 > 4$ , yielding the assertion.

Depends on

External EM affine syntomic filtration and actual obstruction map

Used by · 1

Characteristic-prime degree-five vanishing for singular affine threefolds

Dependency tree · 3 levels

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