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Degree zero and negative homology of the cycle complex

PDF p. 13: Proof of Proposition 6.1: ordinary CH3 identification and degree-seven vanishing

Statement

Let $k$ be a field, $X$ a smooth finite-type $k$ -scheme and $r \geq 0$ . With ordinary Chow group ${C H}^{r} (X)$ defined as codimension- $r$ cycles modulo principal-divisor rational equivalence as specified by the classical relation comparison, the identity on cycles induces $H_{0} z^{r} (X, ∙) ≅ {C H}^{r} (X)$ , and $H_{n} z^{r} (X, ∙) = 0$ for $n < 0$ . Consequently, if for every integer $p$ there is an isomorphism $H^{p, r} (X) ≅ {C H}^{r} (X, 2 r - p)$ , then $H^{2 r, r} (X) ≅ {C H}^{r} (X)$ and $H^{p, r} (X) = 0$ for $p > 2 r$ . The existence of the comparison isomorphisms is not asserted.

Statement references

Classical Chow relations and Bloch boundaries

Facts & Assumptions

Given: $k$ , $X$ and $r$ as in the statement; for the final conditional assertion, the stated comparison isomorphisms.

Goal: Identify degree-zero homology and derive the negative and reindexed vanishing statements.

[F1]

Degree-zero homology is $Z^{r} (X) / B^{r} (X)$ . The degree-zero homology quotient

[F2]

The local graph-and-norm comparison identifies $B^{r} (X)$ with the subgroup of principal-divisor relations defining ${C H}^{r} (X)$ . Classical Chow relations and Bloch boundaries

[F3]

Negative homology vanishes, and the stated higher-Chow comparison reindexes degree $2 r$ to degree zero and $p > 2 r$ to negative homological degrees. Negative homology of the cycle complex, Reindexing a comparison with higher Chow groups

[F4]

Higher Chow groups are defined as homology of the Bloch complex. Bloch cycle complexes and higher Chow groups

Proof

1.1F1F2F4

The two relation subgroups agree by [F2], so the identity on $Z^{r} (X)$ identifies the quotient in [F1] with the ordinary Chow group. This identification commutes with open restriction: a cycle restricts to its intersection with the open, and the face-length coefficients are unchanged there, so the common boundary/relation subgroup is preserved and the induced quotient maps agree.

2.1givenF3F4step 1.1∎

Negative homology vanishes by [F3]. Under the additional comparison isomorphisms, [F3] identifies the degree- $2 r$ group with degree-zero homology and gives zero for $p > 2 r$ ; composing the former identification with step 1.1 gives the asserted Chow-group isomorphism.

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