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Divisibility of curve-supported top cohomology

PDF p. 13: Supported Chow group and normalization

Statement

Assume the Axiom of Choice and the completed size conventions. Let X be an affine finite-type threefold over an arbitrary algebraically closed field k, U=Xsmooth, and T⊂X a closed integral curve contained in U. Then the actual support fiber has Hmot6,3(X/(X∖T),Z)≅CHT3(U), and this group and its image in Hmot6,3(X,Z) are divisible. On the three-dimensional smooth stratum the right side is the length-order divisor quotient of k(T) on closed points of T and receives the actual surjection from Pic⁡(T~), with residue degrees one. No smoothness or normality of T, characteristic restriction on divisibility, or injectivity of support-to-global is assumed.

Facts & Assumptions

Given: The hypotheses in the statement, with the Axiom of Choice and fixed universe bounds.

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

[A1]

Assume The Axiom of Choice and the fixed compatible site and universe bounds of the supplied constructions. These are used when applying the actual geometry, comparison and support inputs in step 1.1 and the normalization and divisibility inputs in step 2.1.

[F1]

The actual smooth open pair and its choice scope supplies the qcqs two-open geometry and the smooth separated finite-type pairs. A coherent comparison of the particular open restriction supplies the actual weight-three motivic-to-cycle restriction diagram over the stated arbitrary algebraically closed field.

[F2]

Excision and the exact fixed-support cycle fiber identifies the unshifted cohomological support fiber with zT3(U,∙)[−6], after reindexing homological degree n as cochain degree −n. In particular its degree-six cohomology is H0zT3(U,∙)=CHT3(U), with the actual support maps.

[F3]

The divisor presentation and actual Picard surjection gives the surjection from the divisible Picard group of the normalization in the dimension-three stratum, and zero in a lower-dimensional smooth stratum. Divisibility survives the quotient and the support image proves divisibility of the support group and its actual support-to-global image.

Proof

1.1A1F1F2given

Under [A1], apply [F1] to the stated X,U,T. All schemes in its two-open diagram are qcqs and the smooth pair is separated finite type over the perfect field k. The compatible comparison and excision in [F2] identify the actual unshifted support fiber with zT3(U,∙)[−6]. Taking degree-six cohomology gives H0zT3(U,∙), proving the displayed supported Chow isomorphism with the declared cohomological indexing.

2.1A1F3step 1.1∎

Apply [F3] to this same curve and field. On the three-dimensional smooth stratum its actual normalization map sends each point to its image with residue degree one and surjects from a divisible Picard group to the divisor quotient. In a lower-dimensional stratum the support group is zero. In either case it is divisible; [F3] also gives divisibility of its image under the actual support-to-global homomorphism. This proves every stated assertion without assuming that homomorphism injective.

Depends on

Used by · 1

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