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Smooth adjunction identifies the absolute and relative augmented slice towers

Statement

Assume AC and fixed compatible nested Grothendieck universes containing $ω$ , the fields, schemes and diagrams below, with the site-size bounds of the supplied EM and perfect- $K$ constructions. Universe existence is a separate hypothesis. Let $k$ be a perfect field, let $W ↪ Y$ be an open immersion, and suppose that $Y$ and $W$ are smooth finite-type $k$ -schemes. Put $T = ∣ Y ∣ ∖ ∣ W ∣$ . For any spectrum-valued presheaf $A$ , write $A_{T} (Y) = fib (A (Y) \to A (W))$ .

Use the imported stable infinity-category $S H (k)$ of motivic spectra, its unit $1_{k}$ , and the suspension motive $M_{k} (Y)$ of $Y$ . The functor $f_{q}$ is the right-adjoint $q$ -effective cover, including its counit to the original spectrum; $s_{q} = cofib (f_{q + 1} \to f_{q})$ . Define the relative tower, with its actual augmentation, by $R_{k}^{q} (Y) = {map}_{S H (k)} (M_{k} (Y), f_{q} {K G L}_{k}) ⟶ {map}_{S H (k)} (M_{k} (Y), {K G L}_{k}) = K H (Y), q \geq 0.$ Here $map$ denotes the mapping spectrum and $K H$ is homotopy invariant $K$ -theory. The relative tower means this fixed-base construction, whereas the absolute tower evaluates the slice filtration of ${K G L}_{Y}$ in $S H (Y)$ .

The actual EM-to-cdh augmented comparison and the canonical smooth-pullback comparison give an open-natural augmented filtered equivalence $F_{m o t}^{∙} K (Y) ≃ R_{k}^{∙} (Y)$ over the canonical $K (Y) \to K H (Y)$ . Consequently the supported towers on $(Y, W)$ are equivalent, commuting with their augmentations and all stage-to-layer maps. On graded pieces use the fixed ordinary cycle comparison of the supplied open-natural layer interface. The equivalence just displayed induces its represented relative-slice leg, so the supported graded comparison is that same comparison, with its original positive cycle convention. In particular, whenever the supported stage- $q$ augmentations are equivalences, their inverse-augmentation leading maps commute under this comparison. No smoothness or finite-type hypothesis on $k$ over its prime subfield is imposed.

Facts & Assumptions

Given: AC, the stated universe and site bounds, the perfect field $k$ , and the smooth finite-type open pair $(Y, W)$ .

Goal: Supply the augmented comparison needed before a classical relative-slice projection can normalize the actual EM supported projection.

[A1]

The Axiom of Choice and Grothendieck universe closure convention specify the inherited foundational premises. All functors, their mapping spectra and the restriction diagram use compatible universe enlargements; no pointwise choice of inverses is made.

[F1]

External actual augmented EM and smooth slice tower comparison supplies the actual augmented EM-to-cdh equivalence on these regular Noetherian schemes and identifies the latter with the absolute evaluated KGL slice tower. Its geometric argument checks the qcqs domain for smooth finite-type schemes over any field. Its EM v1 Remark 3.6 source-version qualification remains binding. In particular, its prime-field classical-to-EM assertion alone is not used as a relative- $k$ comparison.

[F2]

Reference 0 imports $S H$ , effective covers, their canonical filtered base-change map, smooth adjunction and the absolute KGL spectrum. For $g : Y \to Spec k$ , write $g_{♯} ⊣ g^{*}$ with $g_{♯} 1_{Y} = M_{k} (Y)$ . These are coherent adjunctions; mapping spectra and counits are used before taking homotopy groups. The pullback comparison $g^{*} {K G L}_{k} ≃ {K G L}_{Y}$ is the canonical one representing $K H$ .

[F3]

Open-natural smooth cycle layers by separated-open descent fixes the standard cycle comparison on the whole open-restriction diagram. In its separated comparison, the represented-spectrum identification is BEM Remark 4.21, which explicitly uses Remark 3.41's smooth base-change maps. We use this very identification on the graded cofibers below and the same ordinary cycle leg; neither an independent cycle equivalence nor a sign automorphism is substituted. The nonseparated extension agrees with that comparison on separated opens.

[F4]

Augmented tower comparisons preserve supported graded projections transports natural augmented filtered equivalences to support fibers and their graded projections, and to inverse-augmentation maps whenever the corresponding augmentation is invertible. Decreasing spectral filtrations and associated graded and Stable spectra and exact triangles specify the coherent tower, cofiber, fiber and homotopy-group conventions. Motivic filtration and image groups fixes the actual EM tower; Nonconnective K theory of perfect complexes fixes the current perfect- $K$ model in [F1].

[F5]

Field, Perfect fields: every irreducible polynomial is separable and Schemes and morphisms over a base specify the field and structure morphisms. Smoothness and finite type refer to those morphisms as in Smooth morphism of schemes and Locally finite type and finite type morphisms. The arrow $W ↪ Y$ is the open-subscheme inclusion of Open immersions of schemes. The closed support here is the set-theoretic complement of this specified open; no additional scheme structure is used.

External input

Reference 0 supplies the motivic categories, smooth adjunction, effective covers and canonical smooth base-change equivalence used in [F2]. These substantial constructions are imported. The current EM-to-absolute-tower comparison is used through [F1], with its existing source qualification. The fixed classical cycle realization and its normalization are used through [F3]. The following proof composes these specified maps and checks their application to the open pair; the whole result is not marked external_input.

Proof

1.1A1F2F4F5givenconstruct

For $g : Y \to Spec k$ , [F2] gives the canonical equivalence $a_{g}^{q} : g^{*} f_{q} {K G L}_{k} \to f_{q} {K G L}_{Y}$ . It is the map obtained by factoring the pulled-back counit through the $q$ -effective cover, as in (3.42); it is an equivalence because $g$ is smooth. Apply smooth adjunction and then this map to form $θ_{Y}^{q} : R_{k}^{q} (Y) ≃ {map}_{S H (Y)} (1_{Y}, g^{*} f_{q} {K G L}_{k}) \overset{map (1_{Y}, a_{g}^{q})}{\to} {map}_{S H (Y)} (1_{Y}, f_{q} {K G L}_{Y})$ . This is an equivalence of augmented towers, not only of layers: $a_{g}^{q}$ was obtained over $g^{*} {K G L}_{k} ≃ {K G L}_{Y}$ , and the compatible right adjoints give all transitions. For an open immersion $h : W ↪ Y$ , the composites $h^{*} a_{g}^{q}$ followed by $a_{h}^{q}$ and $a_{g h}^{q}$ are the same canonical comparison: both factor the same pulled-back counit through its effective cover. Its universal mapping-space property supplies their compatibility and higher coherences. Adjunction respects this composition, so $θ$ is natural for the actual restriction arrow. The same argument applies to $W$ , which is in the stipulated domain. Its cofiber transformation is precisely (3.43), since cofibers are functorial in this coherent filtered map.

2.1A1F1F3F4F5step 1.1∎

Apply [F1] simultaneously to $Y$ and $W$ , whose regular Noetherian qcqs hypotheses it verifies from [F5]. Compose its EM-to-absolute-tower equivalence with the inverse of $θ$ from step 1.1. These are equivalences of entire coherent diagrams, so this gives the asserted augmented relative- $k$ comparison over the original $K \to K H$ map. Its graded relative-slice identification is the smooth-adjunction and (3.43) identification just constructed, namely Remark 4.21's identification used in [F3]. Composing with [F3]'s already fixed ordinary cycle leg therefore retains that very layer comparison and its positive convention on separated opens. Taking the restriction fiber retains it on the pair. Now apply [F4] to obtain the supported stage, augmentation and projection squares. If the augmentations are equivalences, invert their induced isomorphisms in those same squares to obtain the inverse-augmentation identity. This proves the comparison needed for the relative classical projection without applying smooth base change to $Spec k \to Spec F$ , where $F$ is the prime subfield; the only base-change maps used are the stipulated smooth maps over $k$ and their open restrictions.

Reading limits

The cited BEM passages were read in the primary served HTML. Its canonical smooth-base-change and adjunction interfaces, rather than a reconstruction of motivic homotopy theory, are the imported premises. The EM tower, open-natural cycle-layer and support-transport suppliers were inspected in their current mathematical bodies. EM v1's then-in-preparation qualification and the served BEM version/date discrepancy remain explicit. This local composition claims no independent review, publication or library audit.

Depends on

Used by · 1

Smooth rank, first Chern, determinant and supported positive leading cycle

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · §3.2.1 (3.29) and its following smooth-adjunction paragraph; §3.3 (3.36)–(3.43), especially Remark 3.41; Definition 4.2, Remark 4.3 and Remark 4.21.

Smooth pullback commutes with effective covers via the canonical augmented filtered map (3.42), and its cofiber map is (3.43). KGL is absolute. Smooth adjunction identifies relative evaluation with absolute evaluation; Remark 4.21 uses these same slice maps.

Hypotheses: Qcqs schemes and smooth structure maps; arbitrary field base. Apply only to smooth finite-type schemes over the fixed perfect field.

Reading limits: Read these served HTML passages, including the smooth-adjunction argument. The v1 URL labels August 2025 while its title dates August 2026; no immutable 2025 snapshot is claimed. Motivic foundations and KGL representability remain imported.

Sources

Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, served v1 HTML · §3.2.1 (3.29) and its following smooth-adjunction paragraph; §3.3 (3.36)–(3.43), especially Remark 3.41; Definition 4.2, Remark 4.3 and Remark 4.21.