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External BEM big-site finite Beilinson–Lichtenbaum comparison

PDF p. 6: Singular Beilinson–Lichtenbaum comparison

Statement

Fix a prime $p$ and $S = Spec Z [1 / p]$ . On $C = {S c h}_{S}^{q c q s}$ , with the big cdh topology generated by finite Nisnevich covers and finitely presented abstract blowup covers, BEM Corollary7.20 gives compatible natural equivalences $D_{j} / p^{r} ≃ L_{C} τ^{\leq j} R Γ_{e t} (-, μ_{p^{r}}^{\otimes j}) (j \geq 0),$ where $D_{j} = Z (j)^{c d h}$ is BEM’s cdh-motivic complex, and quotient is the objectwise derived quotient. BEM Corollary2.9 states that stable-valued cdh sheaves are closed under presheaf colimits, so the finite quotient is indeed computed objectwise. These are components of the stated equivalence of E∞-algebras in graded presheaves; this draft uses only the component $j = 3, r = 1$ and its actual comparison map. $L_{C}$ denotes BEM’s complex-valued sheafification, with derived Čech descent for the covers just specified. This is the big qcqs site, not the older separated finite-type site fixed in def-cdh-derived-localization.

External input

Import precisely Corollary7.20 and its ambient-site conventions. Its proof cdh-sheafifies the actual map of Theorem7.16 and the syntomic-to-étale identification of Example5.45(1), so no map is chosen from group agreement. The deep motivic/syntomic comparison is an established external input, not reconstructed here. This item does not assert a primitive root exists on all ambient schemes.

Depends on

Used by · 2

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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 · §2.2, Definition2.5, abstract blowup square(2.6), Remark2.7 and Proposition2.8; Corollary7.20 and its full proof, with Theorem7.16 and Example5.45(1)

There are compatible equivalences of E∞-algebras in graded presheaves D(⋆)/p^r ≃ L_cdh τ^{<=⋆} RΓ_et(−,µ_(p^r)^{⊗⋆}). At r=1,j=3 this is the exact twisted comparison used below. Cor7.20 is obtained by cdh sheafifying Theorem7.16 and Example5.45(1).

Hypotheses: All qcqs Z[1/p]-schemes; p a prime, exponent r>=0; weight j>=0. Big cdh topology generated by finite Nisnevich covers and finitely presented abstract blowup covers. No field, reducedness, smoothness, finite-type or dimension hypothesis.

Reading limits: Directly read all of §2.2 through Corollary2.9, full Corollary7.20 and its short proof, Theorem7.16 statement and comparison-map construction at the start of its proof, and full Example5.45(1). Imported the substantial comparison; did not reconstruct the preceding syntomic/BL construction or Theorem6.21. Reading refers to the served HTML bytes, hashed below, whose displayed title page is dated24August2026 despite the v1 URL.

Sources

Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes · §2.2, Definition2.5, abstract blowup square(2.6), Remark2.7 and Proposition2.8; Corollary7.20 and its full proof, with Theorem7.16 and Example5.45(1)