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Perfect-field cancellation removes exactly d coefficient twists

PDF p. 7: Lemma2.9: smooth-point supported purity

Statement

Assume the Axiom of Choice and the fixed site and universe bounds of the supplied classical derived-category model. Let $k$ be perfect, let $A$ be an object of $D M_{N i s}^{e f f, -} (k, Z)$ , and let $d \geq 0$ , $j \geq d$ and $i \in Z$ . The canonical Tate coefficient convention and repeated cancellation give $Hom (A (d) [2 d], Z (j) [i]) ≅ Hom (A, Z (j - d) [i - 2 d]) .$ For $A = M (Spec L)$ with $L / k$ finite separable, the right side is $H_{m o t}^{i - 2 d, j - d} (L, Z)$ under the actual smooth/classical comparison. This argument supplies no effective negative-weight coefficient when $j < d$ ; that range is computed by the preceding cycle lemma.

Facts & Assumptions

Given: The effective category, perfect field and exact integer range just stated, together with the ambient Choice and site/universe assumptions.

Goal: Prove the previously missing coefficient-twist comparison with its correct bounds.

[A1]

The Axiom of Choice and the fixed small-site/universe bounds are inherited from the source contract of the classical derived-category input in [F2]. They are used for its coefficient models and representability, including the field-group identification in step 2.1. The smooth-comparison use in [F3] is only its complex comparison, not its Choice-qualified filtered-spectrum assertion. The Axiom of Choice

[F1]

External perfect-field Tate cancellation imports Voevodsky's perfect-field full faithfulness of tensoring by the Tate motive.

[F2]

External classical coefficient, derived Yoneda and effective-localization interface fixes the actual Tate convention $Z (r) (d) ≅ Z (r + d)$ for $r, d \geq 0$ , and classical representability on smooth schemes. Purity twists and cohomological shifts with explicit premises supplies the source-shift sign.

[F3]

The smooth comparison identifies the classical motivic complex with the EM complex on a smooth finite-type scheme over a field in nonnegative weight. Elmanto–Morrow smooth comparison input

Proof

1.1A1F1F2given

Put $r = j - d \geq 0$ . Apply [F1] repeatedly $d$ times to $A$ and $Z (r) [i - 2 d]$ . At each step both objects remain effective and the coefficient twist remains nonnegative, so every application has its stated hypotheses. This gives the inverse of the canonical bijection $Hom (A, Z (r) [i - 2 d]) \tilde{\to} Hom (A (d), Z (r) (d) [i - 2 d]) .$ Use [F2] to identify the last coefficient with $Z (j)$ .

2.1A1F2F3step 1.1∎

Shifting the source by $[2 d]$ subtracts $2 d$ from the target cohomological shift, by [F2]. This gives exactly the displayed formula. For $A = M (Spec L)$ , representability identifies its right side with the classical weight- $r$ field group; $Spec L$ is smooth separated finite type over the perfect field $k$ because $L / k$ is finite separable, and $r = j - d \geq 0$ . Thus [F3] identifies that classical group with the specified EM field group under its exact hypotheses. If $j < d$ , step 1.1 would require $r < 0$ and is not invoked. Thus there is no concealed cancellation into a negative effective coefficient.

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