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Higher-cycle localization including nonseparated schemes

PDF p. 13: Supported Chow group; PDF p. 16: Curve-supported motivic groups

Statement

Let $k$ be a field, $U$ a smooth finite-type $k$ -scheme, $V \subseteq U$ open, and $r \geq 0$ . No separatedness of $U$ over $k$ is required. The cokernel complex of the unnormalized simplicial restriction $z^{r} (U, ∙) \to z^{r} (V, ∙)$ is acyclic.

Statement references

Bloch cycle complexes and higher Chow groups

Open restriction of admissible cycles

External input

Use Levine's raw simplicial localization theorem in the field case of §7 of the identified primary preprint. Its introductory Theorem 1.7 says dimension one, but §1.6 and the actual proof in §7 explicitly treat bases of dimension at most one. Section 2.2 does not impose separatedness, and Theorem 6.12's proof synchronizes the same moving map and homotopy over a finite affine cover. Deep moving and blow-up results remain external inputs. This is neither a cubical replacement nor a definition by sheaf hypercohomology.

Facts & Assumptions

Given: $k, U, V, r$ as in the statement.

Goal: Apply the imported theorem with the actual dimension index and cycle convention.

[F1]

Levine §7 proves acyclicity of $z_{q} (V, ∙) / im (z_{q} (U, ∙))$ for finite-type schemes over a semilocal regular base of dimension at most one. The proof uses Theorem 6.12, including its finite-affine-cover reduction. Lemma 7.1 and the finite-flat degree pull/push argument treat finite residue fields.

[F2]

The local unnormalized simplicial Bloch complex, its multiplicities and open restriction are those of Bloch cycle complexes and higher Chow groups and Open restriction of admissible cycles.

Proof

1.1F1F2given

On an integral clopen component $U_{a}$ of dimension $d_{a}$ , put $q = d_{a} - r$ . Levine's dimension- $(q + n)$ generators on $U_{a} \times Δ^{n}$ are exactly the codimension- $r$ generators of [F2]. Its face dimension inequalities mean proper intersection with every coordinate face; in the regular equidimensional ambient scheme a nonempty intersection has the expected codimension, since each successive face equation lowers dimension by at most one. The simplicial face multiplicities and alternating differential agree. A field is a zero-dimensional semilocal regular base, so [F1] makes the restriction cokernel acyclic on this component, for finite as well as infinite fields.

2.1F1F2step 1.1∎

A smooth finite-type scheme over a field is regular and Noetherian. Its finitely many irreducible components are disjoint: an intersection would give a regular local ring with two minimal primes, whereas a regular local ring is a domain. Thus the components are integral and clopen. Cycle complexes, restrictions and their cokernels split as finite direct sums over them. Apply step 1.1 with their respective $d_{a} - r$ indices; a finite direct sum of acyclic complexes is acyclic. The empty scheme and empty open give the same conclusion.

Source notes

The earlier MVW citation uses that book's separated-scheme convention and did not alone justify the stated scope. Levine's separate broader simplicial input supplies it. On preprint p. 1 the introductory codimension conversion prints $d - p$ in place of $d - q$ ; the correct conversion used above is $z^{r} (U_{a}, n) = z_{d_{a} - r} (U_{a}, n)$ . On p. 43 the final target of the finite-extension pull/push display incorrectly remains over the extension; pushforward returns to the original quotient over $U$ . These two display typos do not affect the dimension-indexed localization argument. No local reconstruction of its deep moving constructions is claimed.

Depends on

Used by · 2

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Marc Levine, Techniques of localization in the theory of algebraic cycles · Primary 53-page preprint: §1.6–1.8 pp. 2–4; §2.2 p. 6; Theorem 6.12 and finite affine-cover reduction pp. 37–38; §7 pp. 43–44

For a finite-type scheme X over a field k and open V, the raw simplicial restriction cokernel z_q(V,*)/im(z_q(X,*)) is acyclic. The general finite-type proof uses one simultaneous simplex-factor moving map/homotopy for a finite affine cover.

Hypotheses: Schemes essentially finite type in §2.2, without a separatedness convention. The actual application is finite type over a field. Section7 explicitly takes a regular noetherian base of dimension at most one; a field is its zero-dimensional semilocal PID case. Infinite residue fields are used for the first moving statement; Theorem6.12 plus Lemma7.1 and finite-flat pull/push of coprime degrees handle finite fields. Smoothness and pure dimension are needed here only to rewrite the homological index q componentwise as dim(X_component)-r.

Reading limits: This adjudication read the cached 53-page primary preprint, SHA256 0b4192a0fc037cdd5f8c93f74472ce0924db43fb65d86668239e7d414b2f0504: pp. 1–6, 37–44, including the scheme convention, Theorem 6.12, its finite-affine-cover reduction, and the complete localization argument in §7. Pages 1 and 43 were also visually inspected. Deep constructions in §§3–5 and the companion blow-up theorem were not reconstructed. The remote Citeseer URL returned 403 on this reading; the cached primary PDF was used. Journal bytes and their pagination were not read.

Sources

Marc Levine, Techniques of localization in the theory of algebraic cycles · Primary 53-page preprint: §1.6–1.8 pp. 2–4; §2.2 p. 6; Theorem 6.12 and finite affine-cover reduction pp. 37–38; §7 pp. 43–44