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Exact hypercomplete arc descent for finite constructible coefficients

PDF p. 6: Proposition 2.6 proof

Statement

For every finite commutative ring Λ, the functor X↦Dconsb(Xet,Λ) on all qcqs schemes, with derived pullback, is a hypercomplete arc sheaf. Here bounded constructible means a derived complex with bounded constructible cohomology sheaves. On the qcqs F-scheme cdh site, for a prime field F, an ordinary D(Z)-valued cdh sheaf is precisely a presheaf which takes the empty scheme to 0 and carries Nisnevich squares and the stated finitely presented abstract blowup squares to cartesian squares. Here ordinary means covering-family/Čech descent; this square criterion alone asserts no unrestricted hypercompletion comparison.

Source qualification

The empty-scheme condition corrects an omission in the literal source criterion. A constant presheaf with nonzero value has cartesian values on every square but fails the empty covering. The categorical arc theorem and its finite-ring scope are unaffected.

External input

Use the hypercomplete constructible-category theorem only for finite constant coefficients; it does not silently assert hypercomplete descent for arbitrary torsion coefficients merely from Theorem5.4. The latter is read only for context. No restriction to characteristic zero is needed for these finite-coefficient assertions.

Facts & Assumptions

Given: The hypotheses in the statement.

Goal: Establish the stated exact map or descent conclusion.

[F1]

The exact external source records.

Proof

1.1F1

Import BM Theorem5.13 with Notation5.8 and the full finite-ring hypotheses.

2.1F1step 1.1∎

Its assertion is hypercomplete, which is stronger than the Čech descent formulation of Theorem5.4. Use the EM §3.3 square criterion on the stated prime-field site with the empty-scheme condition: the empty covering forces value zero, while square descent alone omits that condition. These are established source inputs; their foundational descent proofs are not locally reconstructed.

Depends on

Used by · 2

Dependency tree · 1 level

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Bhargav Bhatt and Akhil Mathew, The arc-topology · Definition1.2/Remark1.4 p.3; Definition1.5 p.4; §5.1 pp.37–38; Notation5.8 p.41; Proposition5.11 pp.41–42; Theorem5.13 pp.42–43

For a finite ring Λ, X↦D^b_cons(X_et,Λ), and each bounded-amplitude subcategory, is a hypercomplete arc sheaf on all qcqs schemes. The coefficient complexes have their derived pullback functors.

Hypotheses: Λ finite; X any qcqs scheme, including nonnoetherian/singular/nonreduced; no invertibility or dimension hypotheses. Bounded constructible complexes are as in Notation5.8.

Reading limits: Read full listed definitions, Theorem5.13 and complete proof, and Proposition5.11 and proof. Read §5.1 through Theorem5.4 and proof for its torsion/variance context. The deeper categorical descent and proper-base-change results invoked there are imported, not reconstructed. The source expressly notes at p.41 that taking derived endomorphisms of the constant sheaf recovers étale cohomology.

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §3.3 p.14, paragraph defining qcqs cdh topology and following square criterion

On qcqs schemes the cdh topology is generated by Nisnevich covers and the specified finitely presented abstract blowup squares; derived cdh sheaves send these squares to cartesian squares.

Hypotheses: The exact qcqs site with finitely presented closed immersion and finitely presented proper map, not only ordinary blowups.

Reading limits: Read the entire paragraph and its square, including the stated nonnoetherian generalizations cited to [40,Prop2.1.5] and [9,App.A]. Those underlying proofs are not reconstructed; the square criterion is used with the necessary empty-scheme condition made explicit. Ordinary sheafification is distinguished from hypercompletion.

Sources