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Positive supported coordinates at a smooth threefold point

PDF p. 17: Corollary6.9 supported factor calculation

Statement

Assume AC. Let $k$ be algebraically closed of any characteristic, $X$ a finite-type pure threefold, and $x$ a smooth closed point. There is a smooth affine neighbourhood $V$ of $x$ , and actual open-excision identifications give $K_{0, x} (X) ≅ Z$ with positive generator $u_{x} = [i_{*} k (x)]$ , and $H_{x}^{6, 3} (X, Z) ≅ Z$ with positive generator $δ_{x}$ , the fundamental point cycle. Explicitly extend the nonnegative weight system by $C_{j} = 0$ and $H^{a, j} = 0$ for $j < 0$ , and likewise for its support fibers. With this convention, $H_{x}^{2 j, j} (X) = 0$ for every integer $j < 3$ . These are supported generators, even when their images in global groups vanish.

Facts & Assumptions

[F1]

Excision and K theory purity at a smooth closed point and Perfect residue classes at smooth closed points give the actual K-support fiber and its dévissage coordinate, normalized by the residue sheaf.

[F2]

A smooth point has the exact finite-type excision pair supplies a smooth affine pure-three-dimensional finite-type neighbourhood and actual motivic support excision. The actual smooth restriction arrow gives the point-supported cycle fiber compares the actual smooth restriction arrow and its fiber with the supported cycle complex. Point-supported cycles give all-weight chain purity and low-weight zero identifies that complex with $C_{0} (k (x)) [- 6]$ in weight three, carrying the literal reduced point cycle $[x]$ to $1$ , and proves zero for $0 \leq j < 3$ . Motivic purity and lower-weight vanishing at a smooth closed point gives the same lower-weight vanishing and permits the explicit negative-weight zero extension. Smooth motivic comparison supplies the smooth coefficient convention.

[F3]

Open excision for a descent theory preserves support-to-global arrows under restriction when its descent hypothesis holds. The actual K and motivic excision data here are supplied in [F1] and [F2]. AC, The Axiom of Choice, is used through those geometric and comparison suppliers.

Given: $X, k, x$ as stated and the completed support conventions.

Goal: Fix the local pair and its actual positive coordinates.

Proof

1.1F1F2

Choose the smooth affine pure-three-dimensional finite-type neighbourhood supplied by [F2], then apply [F1] on it. The local ring at $x$ is regular of dimension $3$ ; shrink so that its point ideal has a regular-sequence Koszul resolution. The residue sheaf is the globally defined perfect object, independently of the resolution, and its K-support coordinate is $+ 1$ .

2.1F2step 1.1

Apply the actual arrow/fiber comparison and chain isomorphism of [F2] with $d = j = 3$ . They identify $H_{x}^{6, 3} (V)$ with $H^{0} (C_{0} (k (x)))$ . In degrees zero and one, $z^{0} (k (x), ∙)$ is freely generated by the integral simplex; the two face maps in degree one both have coefficient $1$ , so their alternating difference is zero. Thus its degree-zero homology is $Z$ , positively generated by the reduced point. Define $δ_{x}$ on $V$ through this actual supported-cycle comparison. Its cycle coordinate is $+ 1$ ; this construction does not identify an unspecified abstract purity isomorphism with Gysin pushforward.

3.1F1F2F3step 1.1step 2.1∎

For $0 \leq j < 3$ , [F2] gives zero supported cycle complexes in all degrees. For $j < 0$ , the declared zero extension gives zero support fibers and hence zero groups. The actual K and motivic excision equivalences of [F1] and [F2] transport both positive generators from $V$ to $X$ , and preserve their support-forgetting arrows by [F3]. This proves the stated integer range and the coordinate assertions even when a global image vanishes.

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