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External canonical field restriction identifies the BEM and EM cdh models

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Statement

For every integer j, let Dj=Z(j)cdh be BEM Definition 7.8: the cdh sheafification of its universal lisse motivic complex on all qcqs schemes. For a field B, Dj restricts canonically to the cdh-local left Kan extension of classical motivic cohomology from smooth B-schemes. In particular, for B=Q and j≥0 this is the very complex Cj,Q=Z(j)cdh in EM v1 §3.3, definition(3.5), used by the supplied EM comparison. This is a natural equivalence with the specified classical-input maps and their sheafification units, not just agreement of groups. Restriction further to qcqs schemes over a characteristic-zero field k preserves it.

External input

Import BEM Theorem7.5 and Remark7.9 (with Definition7.8). Theorem7.5 is a canonical multiplicative Nisnevich-local base-independence result; applying cdh localization gives the exact restriction stated in Remark7.9. The smooth-field input is classical motivic cohomology, as in Theorem1.1(8); EM definition(3.5) uses precisely its cdh-local left Kan extension on qcqs prime-field schemes. The completed supplier External prime-field cdh comparison and cdh dimension bound fixes this actual EM target. Only this model identification is imported; no bound or unrelated assertion from that supplier is needed.

Depends on

Used by · 4

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

Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · Theorem7.5 and its full proof (following Proposition7.4); Definition7.8; Remark7.9; Theorem1.1(8) classical smooth-field identification. Compare EMv1 §3.3 definition(3.5), pp14–15.

Restriction of BEM D_j to qcqs B-schemes is canonically L_cdh of its classical smooth-B motivic left Kan extension (Remark7.9, by cdh sheafifying Theorem7.5). For B=Q this is exactly EMv1 definition(3.5). The identification is one of functors with actual maps, not a group-only comparison.

Hypotheses: For restriction: B a field or mixed-characteristic Dedekind domain; every qcqs B-scheme. Here only B=Q is needed, followed by restriction to any characteristic-zero field k. BEM field restriction allows every integer j; the supplied EM target comparison uses j>=0.

Reading limits: Read full Theorem7.5 and its short proof, Proposition7.4 proof immediately preceding it, Definition7.8 and Remark7.9. Read Theorem1.1(8) statement; EMv1 §3.3 definition(3.5) and opening construction were read directly. Did not reconstruct Lemma2.26 or the deeper smooth-field motivic comparison theorem; these are imported within the precise canonical restriction result. The HTML served at the v1 URL has title-page date August 24, 2026; this records the actually served text, not verification of an immutable August 2025 v1 snapshot.

Elden Elmanto and Matthew Morrow, Motivic cohomology of equicharacteristic schemes · §3.3, definition(3.5), pp14–15

Z(j)_cdh is the cdh sheafification of classical smooth-prime-field motivic cohomology left Kan extended to qcqs schemes.

Hypotheses: Qcqs schemes over a prime field; weight j>=0.

Reading limits: Read definition(3.5) and its surrounding construction directly; no reconstruction of foundational classical motivic comparison.

Sources