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Face boundaries preserve closed support

PDF p. 13: Supported Chow group

Statement

Let $k$ be a field, $U$ a smooth finite-type $k$ -scheme, $T \subset U$ closed, and $r \geq 0$ . In Bloch's complex $z^{r} (U, ∙)$ , the free subgroup generated in each degree $n$ by admissible integral cycles contained in $T \times Δ^{n}$ is preserved by the boundary.

Statement references

Bloch cycle complexes and higher Chow groups

Facts & Assumptions

Given: $k, U, T, r$ as stated.

Goal: Show that the boundary of each supported generator remains supported.

[F1]

Bloch's boundary is the alternating sum of its admissible face intersections. Bloch cycle complexes and higher Chow groups

Proof

1.1givenF1

For a supported generator $W$ , every component occurring in a face intersection is contained in $W \cap (U \times {t_{i} = 0})$ , hence in $T \times Δ^{n - 1}$ under the face identification.

2.1F1step 1.1∎

The coefficients in the face intersections are integers and the boundary is their alternating sum, so linearity gives the asserted preservation in every degree; degree zero has zero boundary.

Depends on

Bloch cycle complexes and higher Chow groups

Used by · 2

Dependency tree · 3 levels

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