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The divisor presentation and actual Picard surjection

PDF p. 13: Proposition 6.1 support argument

Statement

Assume the Axiom of Choice and the fixed site and universe bounds of the motivic and support constructions. Let $k$ be an arbitrary algebraically closed field, let $X$ be an affine finite-type $k$ -scheme of Krull dimension three, and put $U = X_{s m o o t h}$ . Let $T \subset X$ be a closed integral curve contained in $U$ , with normalization $ν : \tilde{T} \to T$ . Let $U_{d} \subset U$ be the open-and-closed relative-dimension stratum containing $T$ . If $d = 3$ , the fixed-support group is the actual divisor quotient ${C H}_{T}^{3} (U) = coker (k (T)^{\times} \overset{{div}_{T}}{\to} ⨁_{x \in T c l o s e d} Z [x]),$ and normalization gives the surjection $q : Pic (\tilde{T}) ↠ {C H}_{T}^{3} (U)$ induced by $[y] \mapsto [ν (y)]$ . If $d < 3$ , this supported group is zero. Thus the support-fiber identification gives an actual surjection from a divisible group onto $H_{m o t, T}^{6, 3} (X, Z)$ .

Facts & Assumptions

Given: The algebraically closed field $k$ , affine finite-type threefold $X$ , smooth locus $U$ , closed integral curve $T \subset U$ , normalization and stratum specified in the statement, the Axiom of Choice, and the fixed site and universe bounds.

Goal: Construct the stated actual maps and prove their consequence.

[F1]

The actual smooth open pair and its choice scope and Excision and the exact fixed-support cycle fiber.

[F2]

Normalization of an affine integral curve, Top cycles supported on an integral curve and Divisibility of the Picard group of an affine curve apply with the stated Choice and size assumptions. Their normalization, divisor and Picard constructions use the Axiom of Choice, recorded by The Axiom of Choice. The Picard divisibility includes all nonzero integers, with no characteristic restriction.

Proof

1.1F1F2

The curve is affine by the geometry lemma. The completed normalization supplier therefore gives finite smooth affine $\tilde{T}$ over the perfect field, with every closed residue field equal to k. On the smooth pure-three-dimensional stratum containing T, the completed divisor supplier gives its displayed quotient and the actual pushforward $q$ . It proves that pushforward sends principal divisors to the length-order divisor downstairs, with residue degrees one; hence the map descends and is surjective, even if T is singular. The other open-and-closed strata do not meet T and contribute zero support.

2.1F1F2step 1.1∎

If T instead lies on a smooth stratum of dimension $d < 3$ , there are no codimension-three cycles in degree zero on that stratum; hence $z_{T}^{3} (U, 0) = 0$ and $H_{0} z_{T}^{3} (U) = 0$ . This branch is zero and already divisible. In the dimension-three case the completed affine Picard supplier makes $Pic (\tilde{T})$ divisible by every nonzero integer, including the characteristic when the characteristic is positive. Compose $q$ with the actual supported-comparison isomorphism of the support-fiber lemma, obtaining the claimed surjection to supported motivic cohomology.

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