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The kernel of restriction is the supported subcomplex

PDF p. 13: Supported Chow group

Statement

Under the hypotheses of the preceding support construction, put $V = U ∖ T$ . The kernel of $j^{*} : z^{r} (U, ∙) \to z^{r} (V, ∙)$ is exactly the subcomplex freely generated by the admissible integral cycles contained in $T \times Δ^{∙}$ .

Statement references

Face boundaries preserve closed support

Open restriction of admissible cycles

Facts & Assumptions

Given: A field $k$ , a smooth finite-type $k$ -scheme $U$ , a closed subset $T$ , and $r \geq 0$ .

Goal: Identify the kernel in each degree and as a complex.

[F1]

The cycle groups are free on admissible integral closed subschemes. Bloch cycle complexes and higher Chow groups

[F2]

Restriction is a chain map and supported generators form a subcomplex. Open restriction of admissible cycles, Face boundaries preserve closed support

Proof

1.1givenF1

A generator restricts to zero exactly when it is contained in $T \times Δ^{n}$ . Two generators with the same nonempty restriction coincide, since each is the closure of that dense open restriction in $U \times Δ^{n}$ .

2.1F1F2step 1.1∎

The nonzero restrictions of distinct generators are therefore distinct basis elements. Freeness shows that the degreewise kernel is precisely the span of the supported generators; by [F2] this equality is an equality of subcomplexes.

Depends on

Used by · 3

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