LemmaStatement: AI-generatedProof: AI-generated

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Point-supported cycles give all-weight chain purity and low-weight zero

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

Statement

Assume the Axiom of Choice and the fixed compatible site and universe bounds of the supplied constructions. Let $k$ be a perfect field, let $V$ be a smooth affine $k$ -scheme of finite type and pure dimension $d$ , and let $x \in V$ be a closed point of codimension $d$ . Set $L = κ (x)$ ; this is a finite separable extension of $k$ , so $Spec L$ is smooth of finite type over $k$ . For $r \geq 0$ , write $C_{r} (T)$ for the specified unshifted EM motivic coefficient complex on $T$ , put $C_{r} (L) = C_{r} (Spec L)$ and $C_{r, x} (V) = fib (C_{r} (V) \to C_{r} (V ∖ {x}))$ , and let $z_{x}^{r} (V, ∙)$ be the subcomplex of Bloch's unnormalized cycle complex generated by admissible integral cycles supported on ${x} \times_{k} Δ_{k}^{∙}$ .

For every $n \geq 0$ one has $z_{x}^{j} (V, n) = 0$ if $0 \leq j < d$ . If $j \geq d$ , the closed immersion $x \times Δ^{n} = Δ_{L}^{n} ↪ V \times Δ_{k}^{n}$ identifies integral generators and gives an isomorphism of chain complexes $z^{j - d} (Spec L, ∙) \tilde{\to} z_{x}^{j} (V, ∙) .$ It preserves the actual face-intersection multiplicities and the alternating boundary signs. Consequently $C_{j, x} (V) ≃ C_{j - d} (L) [- 2 d] (j \geq d), C_{j, x} (V) ≃ 0 (0 \leq j < d),$ using the preceding actual smooth comparisons on both sides. At $j = d$ , the positive cycle $[x]$ corresponds to $1 \in z^{0} (L, 0)$ , with no residue-field degree factor.

Facts & Assumptions

[A1]

The assumed The Axiom of Choice and fixed compatible site and universe bounds discharge the inherited premises of the neighbourhood and actual-support comparison in [F2] at the annotated applications below. The chain, dimension and formal fiber computations introduce no additional choice principle.

Given: The Axiom of Choice, the fixed compatible site and universe bounds, a perfect field $k$ , a smooth affine finite-type $k$ -scheme $V$ of pure dimension $d$ , a closed point $x \in V$ of codimension $d$ , its residue field $L = κ (x)$ finite separable over $k$ , the supported-cycle and unshifted coefficient conventions in the Statement, and $j \geq 0$ .

Goal: Prove the point-cycle calculation, including the complete low-weight range and coefficients of the boundary.

[F1]

Cycle complexes with fixed support defines admissible integral supported cycles and their actual alternating face boundary. Bloch cycle complexes and higher Chow groups supplies the same convention over $L$ .

[F2]

A smooth point has the exact finite-type excision pair and External finite-type dimension, rational-point and etale-fiber inputs give dimension $d + n$ of the ambient product and dimension $n$ of $Δ_{L}^{n}$ . The actual smooth restriction arrow gives the point-supported cycle fiber gives its actual motivic fiber comparison. The same supplier applies to the smooth zero-dimensional neighbourhood $Spec L$ with its unique closed point and empty open complement; the source component of its arrow equivalence gives $C_{q} (L) ≃ z^{q} (Spec L, ∙) [- 2 q]$ for every $q \geq 0$ . Finite separability verifies smoothness over the same perfect base $k$ .

Proof

1.1A1F1F2

Every integral cycle supported in $x \times Δ^{n}$ has dimension at most $n$ , so its codimension in $V \times Δ^{n}$ is at least $d$ . Thus there are no generators in weights $j < d$ , in any degree. For $j \geq d$ , an integral closed subscheme of this support has ambient codimension $j$ precisely when it has codimension $j - d$ in $Δ_{L}^{n}$ . These are literally the same integral subschemes with reduced structure.

2.1F1F2step 1.1

For a face of simplex dimension $m$ , the ambient admissibility bound is $\dim (Z \cap (V \times Δ^{m})) \leq d + m - j$ . On the support this is exactly $\dim (Z \cap Δ_{L}^{m}) \leq m - (j - d)$ . Thus admissible generators correspond in every degree. For a codimension-one face the quotient by its coordinate equation is formed on the same integral cycle on both sides. Its generic quotient-local lengths, and hence all coefficients of the face cycle, are identical. The simplex face order and signs are unchanged. The generator bijections therefore commute with the full chain differential, giving the stated chain isomorphism, rather than equality only modulo rational equivalence.

3.1A1F1F2step 1.1step 2.1∎

In cochains the support complex is $z_{x}^{j} [- 2 j]$ . Apply step 2.1 and write $[- 2 j] = [- 2 (j - d)] [- 2 d]$ . The finite residue extension $L / k$ is separable because $k$ is perfect. Thus $Spec L$ is a smooth affine finite-type $k$ -scheme of pure dimension zero, and [F2]'s source comparison applies to its unique closed point and empty open complement with $q = j - d \geq 0$ . It identifies $z^{j - d} (Spec L, ∙) [- 2 (j - d)]$ with $C_{j - d} (L)$ . This proves both derived statements using step 1.1 for the zero case. In degree zero and $j = d$ , the same generator is $Spec L$ itself, so its coefficient is $+ 1$ , not $[L : k]$ ; no pushforward to $Spec k$ has occurred.

Depends on

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