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A specified étale-natural comparison with supported cycles

PDF p. 17: Corollary 6.9; specified étale transport of the positive supported point class

Statement

Assume the Axiom of Choice and the compatible nested Grothendieck universes and site bounds of the supplied coefficient constructions, with $ω$ in the smaller universe. Fix a perfect field $k$ and an integer $j \geq 0$ . On smooth separated finite-type $k$ -schemes there is a specified equivalence of unshifted coefficient spectra with the realized Bloch complexes, natural for étale maps and open immersions. Its construction is the inverse of the specified classical-to-EM field comparison followed by Spitzweck's flat-natural cycle comparison. For a closed subset $T \subset V$ , it induces $Ψ_{V, T} : M_{j, T}^{E M} (V) ≃ H (z_{T}^{j} (V, ∙) [- 2 j]),$ compatibly with support-to-global maps and pullback of pairs by étale maps and open immersions. In particular, if $q : W \to V$ is étale and the scheme-theoretic inverse image of a reduced rational closed point $y$ is exactly a reduced rational closed point $x$ , and both point immersions have codimension $j$ , then the classes defined by $Ψ_{V, {y}}^{- 1} [y]$ and $Ψ_{W, {x}}^{- 1} [x]$ satisfy $q^{*} Ψ_{V, {y}}^{- 1} [y] = Ψ_{W, {x}}^{- 1} [x]$ . This fixes a comparison used to define positive classes; it makes no assertion that arbitrary unnormalized fixed-ambient cycle equivalences give that same class.

Statement references

A unit-induced integral field coefficient zigzag

The actual EM comparison commutes with the Picard first-Chern map

Cycle complexes with fixed support

Definition

Use $k$ -schemes and $k$ -morphisms with the supplied structure-morphism meaning. Smoothness, finite type, separatedness, étaleness and open immersion have the declared scheme-theoretic meanings. A rational closed point $y$ has closed singleton and residue field $κ (y) = O_{V, y} / m_{y} = k$ . A reduced point is the subscheme $Spec k$ with this closed immersion. Codimension has the local-ring convention of the Bloch-complex definition; the point-class assertion assumes codimension $j$ so that the literal point is an admissible degree-zero cycle of weight $j$ . For a pair map put $T^{'} = q^{- 1} (T)$ as a closed subset; the open complement pulls back to $W ∖ T^{'}$ . All products and inverse images are scheme fibre products over the fixed field or the indicated scheme, with the supplied universal property and existence theorem. Closed immersions have the declared closed-subspace and structure-sheaf meaning. A reduced singleton inverse image in the Statement is an assumption on its scheme structure, not a consequence of rationality alone.

Write $S_{j}$ for the unshifted Spitzweck/classical coefficient spectrum and $A_{j}$ for the unshifted represented KGL-slice coefficient spectrum. Let $ρ_{j} : S_{j} ≃ A_{j}$ be the coefficient equivalence induced by the unit zigzag, and let $θ_{j} : M_{j}^{E M} ≃ A_{j}$ be the actual structured EM comparison $γ η$ . Inverses are taken in the functor category, on the same smooth site and with the same Tate coordinate. If $λ_{j} : S_{j} ≃ H (z^{j} (-, ∙) [- 2 j])$ is the fixed flat-natural comparison of [F2], define $Ψ_{j} = λ_{j} ρ_{j}^{- 1} θ_{j}$ on the subcategory of étale and open maps. The realization $H$ is the coherent limit-preserving functor of the declared realization supplier. All assertions here are about spectra and their cohomology; no lift of an arbitrary map of spectra to a map of derived abelian complexes is required. Put $M_{j, T}^{E M} (V) = fib (M_{j}^{E M} (V) \to M_{j}^{E M} (V ∖ T))$ . Use the unnormalized supported cycle subcomplex and the cochain convention that homological degree $n$ is cochain degree $- n$ , and $C [m]^{a} = C^{a + m}$ with differential $(- 1)^{m} d_{C}$ .

Facts & Assumptions

[A1]

Choice and universe existence are assumed separately. They discharge the inherited coefficient, realization and cycle-construction hypotheses; no independent inverse is chosen at each ambient. The Axiom of Choice Grothendieck universe closure convention

[F1]

The unit zigzag supplies $ρ_{j}$ as an equivalence of coefficient functors. The actual EM Picard-square supplier supplies the natural equivalence $θ_{j} = γ η$ on smooth finite-type schemes; its coefficient equivalence and common first-Chern normalization are used here. Their underlying coefficient spectra, common Tate coordinate and actual maps have the meanings of the primary-input supplier. A unit-induced integral field coefficient zigzag The actual EM comparison commutes with the Picard first-Chern map Exact primary unit, Picard and smooth-purity inputs for field compatibility

[F2]

Reference 0 supplies two distinct comparisons. Corollary 5.19 compares Zariski-localized complex diagrams on the flat subcategory through Lemmas 5.17–5.18; it does not alone compare represented coefficient spectra. Section 5.3 gives the linear $G$ -spectrum with level $M (r)_{k} [r]$ , where $G = Z [G_{m}, {1}]$ , and Proposition 5.29 gives its adjoint structure equivalences. Proposition 6.1 identifies $M (r)_{k}$ with ${\tilde{z}}^{r} [- 2 r]$ . The unital pairings (15), the isomorphisms (18), Lemmas 6.5–6.6 and the proof of Theorem 6.7 identify the ordinary field Eilenberg–Mac Lane multiplication with these pairings and fix the bonding coordinate. Write $B = R (E)$ for the realization of this field model on the Nisnevich site of [F7]. Give $B$ the unit $η_{B}$ and binary multiplication $μ_{B}$ transported from the ordinary field spectrum through the Theorem 6.7 comparison; its global cohomology products are (15). In particular $M (0)_{k} = Z$ . Theorem 7.18, Lemma 7.5/Theorem 7.6, Corollary 9.13/Theorem 9.16 and Lemma 9.17 give a spectrum equivalence $f_{k} : H_{S p i} ≃ B$ by the indicated absolute, generic or closed-fiber, and field-extension comparisons. They are used here for this underlying equivalence, without attributing multiplicativity of the entire composite to Theorem 7.18. Fix $e_{k}$ to be the unique unit-preserving homotopy class of maps $H_{S p i} \to B$ , represented by the sign-normalized $f_{k}$ ; step 1.1 proves its existence, equivalence, uniqueness and binary-product compatibility using [F10] and A unit-induced integral field coefficient zigzag. Choose a representative with its unit homotopy in the supplied enhancement. Use the same $T = (P^{1}, \infty) ≃ (G_{m}, 1) \land S^{1}$ throughout. Proposition 11.1 fixes the line convention. The coefficient evaluation and flat-cycle composition below use this one normalized map in every weight. Only its unit and global cohomology products are asserted; no coherent algebra structure on the raw cycle complexes is prescribed. In the field construction at the beginning of §6 the entire §5 construction is retained, with its global-sections replacement functor omitted. In particular, §5.2.1 constructs an interval diagram ${\tilde{γ}}_{j}^{'}$ in the homotopy category of complexes indexed by ${\hat{S}}_{f l} \times [1]$ , not just an arrow between individual derived-category values. Here $S$ is the category of smooth affine coordinate algebras with finite generating lists, flat arrows form $S_{f l}$ , and the hatted category is the auxiliary resolution of §5.1. Its projection induces the localization equivalence of Proposition 5.13. Lemmas 5.17–5.18 identify the two endpoint diagrams with the restricted strictification $γ_{j}^{'}$ and the actual normalized flat-cycle diagram. The interval arrow is an objectwise quasi-isomorphism on these affine diagrams: at an auxiliary object $t$ , the pullback $N z^{j} (A_{i}) \to N z^{j} (C_{t})$ along the polynomial extension is a quasi-isomorphism by homotopy invariance, and it factors through the good-position subcomplex, whose inclusion is a quasi-isomorphism by Theorem 5.8; two-out-of-three applies. The field construction omits the global-sections replacement, so this assertion is before Zariski localization. Section 6 constructs a second interval diagram $γ_{e, j}^{'}$ , whose source is the equidimensional cycle complex and whose target is the affine-space version of the same strictification; its Zariski localization is Proposition 6.1. These actual interval diagrams, with the natural transformations of §5.2.2 for the affine-space projections, are the specified comparison data used below.

[F3]

Bloch's unnormalized complex supplies cycle generators and face differentials. The supported-cycle definition supplies the natural inclusion of its kernel complex into the restriction fiber, with cochain form $a \mapsto (a, 0)$ . The support/indexing definitions give the actual motivic fiber and its source projection. The coherent realization preserves these fibers and shifts and has $π_{i} H (C) = H^{- i} (C)$ . Bloch cycle complexes and higher Chow groups Cycle complexes with fixed support Support fibers with explicit cohomological indexing Motivic and filtered K theory with support A coherent spectral realization of derived abelian complexes Stable spectra and exact triangles

[F4]

The formulation uses the declared perfect-field, base-scheme, smoothness, finite-type, separated, étale, open-immersion, closed-point and residue-field definitions. Étale maps are flat by their definition. Under [A1], the finite-type field lemma makes every open complement quasi-compact and qcqs; [F5] checks finite type and separatedness, so all vertices are in the domains of [F1]–[F3]. Perfect fields: every irreducible polynomial is separable Schemes and morphisms over a base Smooth morphism of schemes Locally finite type and finite type morphisms Separated morphism of schemes Étale morphism of schemes Open immersions of schemes Closed points of an affine scheme The residue field at a point of an affine scheme Finite-type field schemes lie in the qcqs source domain Fibre product of schemes Existence of all scheme fibre products Closed immersions of schemes

[F5]

Local finite type survives open source restriction by affine locality. For an open $U$ in a finite-type $k$ -scheme, [F4] makes $U$ quasi-compact. The only opens of $Spec k$ are the empty and whole subsets, so its structure morphism is quasi-compact and hence finite type. If the ambient is separated over $k$ , its open subscheme is separated: the open immersion is separated and separated morphisms compose. Finite type is affine-local on source and target Quasi-compact and quasi-separated morphisms Open and closed immersions are separated Separated morphisms compose

[F6]

Spitzweck uses normalized simplicial cycle chains (reference 0, p.37). For any simplicial abelian group, the natural inclusion of normalized chains into alternating-face chains is a chain homotopy equivalence; it is the identity in degree zero. Apply this to the simplicial cycle groups of reference 0, and to their flat pullback diagrams. Thus normalization can be removed naturally without changing a literal point generator. Normalized simplicial chains, prism homotopies and the abelian trivial-fibration criterion

[F7]

Reference 3, §5.2.25 and (5.3.35.1), supplies the derived integral adjunction $L : S H (k) ⇄ D_{A^{1}} (k) : R$ on the smooth Nisnevich site. Its left functor is reduced free abelian sheafification followed by normalized chains, motivic localization and stabilization; its right functor is the corresponding Eilenberg–Mac Lane realization. The target is the stabilization of the $A^{1}$ -localized derived category of ordinary abelian Nisnevich sheaves, without transfers. Reference 4, §3.2.3, states this same adjunction for the derived infinity-categories. We use these supplied model enhancements, with Choice, the fixed universe bounds and cofibrant suspension coordinates, rather than deriving motivic localization from objectwise Dold–Kan. The adjunction is monoidal on the left in the premotivic sense of reference 3, Definition 1.4.6. For a smooth $V$ , $L Σ_{T}^{\infty} V_{+} = Σ_{l i n}^{\infty} Z_{N i s} [V]$ , where $Z_{N i s} [V]$ is the sheafification of the free abelian presheaf on $U \mapsto {Hom}_{k} (U, V)$ . Put $G = Z_{N i s} [G_{m}] / Z_{N i s} [{1}]$ ; then $L (T) = G [1]$ . The linear $G$ -spectrum of [F2] is read in this Nisnevich model using [F8], and its realization is the field spectrum used there. Reference 3, §§5.3.20–5.3.24 and Remark 5.3.27, supplies derived suspension/evaluation and the equivalent choices of linear suspension coordinate. Reference 2, Theorem 7.10, supplies the derived shift: for a local fibrant $G$ -Omega spectrum $E$ , derived level zero of $E \otimes G^{j}$ is $E_{j}$ , $j \geq 0$ . A local sheaf-complex level evaluates by derived sections. Model and Quillen terminology, the ordinary normalization convention, and derived sections have the following declared meanings. Model categories and Quillen adjunctions Dold-Kan equivalence for simplicial modules with explicit inverse Derived global sections on a scheme site

[F8]

Reference 1, Proposition 13.10 and Example 13.11, supplies the natural change-of-topology quasi-isomorphism $R Γ_{Z a r} (V, C) \to R Γ_{N i s} (V, C)$ for a bounded-above complex of Nisnevich sheaves with transfers with homotopy-invariant cohomology over a perfect field. Apply it to the classical motivic-complex representative of $M (j)_{k}$ identified in [F2]; Example 13.11 verifies precisely these hypotheses. Proposition 14.8 and Corollary 14.9 show that this bounded classical representative has homotopy-invariant derived evaluations; hence Nisnevich/A1-local replacement retains its derived levels. Proposition 19.12 gives the natural descent map from cycle-complex evaluation to its Zariski derived sections as a quasi-isomorphism. Example 17.12 and Example 19.14 identify flat pullback with the cycle of the scheme-theoretic inverse image. Only étale and open maps between the stated smooth separated finite-type objects are used here. For the explicit comparison put $C_{j} = {\tilde{z}}^{j} [- 2 j]$ , $Z_{j} = N z^{j} (-, ∙) [- 2 j]$ and $Q_{j} = N z^{j} (- \times A^{j}, ∙) [- 2 j]$ . Here $N$ is normalized chains as in [F6], and ${\tilde{z}}^{j} = C_{*} z_{e q u i} (A^{j}, 0)$ is the normalized Suslin complex of equidimensional relative zero-cycles, as in reference 1, §16 and Lemma 19.4. Lemma 19.4 supplies the actual inclusion $i_{j} : C_{j} \to Q_{j}$ of simplicial cycle complexes. Projection supplies the actual flat pullback $p_{j} : Z_{j} \to Q_{j}$ . Both are natural for flat maps. Theorem 19.8 makes $i_{j}$ a Zariski-local quasi-isomorphism on smooth schemes over the perfect field. Homotopy invariance, Properties 17.4(1), makes $p_{j}$ an objectwise quasi-isomorphism. Proposition 19.12 applies to both $Z_{j}$ and $Q_{j}$ . Consequently the standard global comparison is induced by $(R Γ_{Z a r} p_{j})^{- 1} R Γ_{Z a r} i_{j}$ and the inverse descent map for $Z_{j}$ . Corollary 19.9 and the proof of Theorem 19.1 identify precisely this composite, with the stated shifts. These are statements about the actual inclusion, projection and descent arrows, not an arbitrary isomorphism of their cohomology groups.

[F9]

The enhanced mapping supplier gives the essentially unique finite-limit-preserving spectral lift of a finite-limit-preserving space-valued functor on a stable infinity-category. It also supplies the canonical mapping-space and derived-linear-target interfaces. Thus an adjunction between the supplied stable enhancements can be used on mapping spectra: its natural space-level adjunction, applied to all shifts, has a unique compatible spectral lift. The realization of the resulting derived abelian mapping complex is the fixed $H$ of [F3]. This uses an actual enhanced adjunction from [F7], not an enhancement inferred from an adjunction of homotopy categories. External stable mapping and derived linear target interfaces A coherent spectral realization of derived abelian complexes

[F10]

Reference 5, §3.3, supplies the following exact slice interfaces on $S H (k)$ . Put $C_{0} = S H (k)^{e f f}$ , the stable colimit-closed subcategory generated by motives of smooth $k$ -schemes, and $C_{1} = T \otimes C_{0}$ . The tensor unit $1$ is effective, $C_{0}$ is tensor-closed, and tensoring is separately exact and preserves colimits. Thus $C_{1} \otimes C_{0} \subseteq C_{1}$ . The effective covers are right adjoints to the inclusions, and $s^{0} X = cofib (f_{1} X \to f_{0} X)$ is right orthogonal to $C_{1}$ : $map (D, s^{0} X) = 0$ for every $D \in C_{1}$ . Here $map$ is the mapping spectrum in the supplied stable motivic enhancement, so this is vanishing in all suspension degrees. As $f_{0} 1 ≃ 1$ , the defining triangle is $f_{1} 1 \to 1 \overset{u}{\to} s^{0} 1$ . Remark 3.39 supplies the slice multiplication and its unit $u$ . The existing unit zigzag identifies $s^{0} 1$ with $H_{S p i}$ as a unital motivic ring, integrally over the perfect field. Exact primary unit, Picard and smooth-purity inputs for field compatibility A unit-induced integral field coefficient zigzag

[F11]

Reference 6, Theorem 3.1 and §3.1, imports product compatibility for the canonical integral comparison from ordinary motivic cohomology to higher Chow groups over a perfect field. Its formula (3.3) factors through equidimensional cycles, and the second isomorphism there preserves the equidimensional cycle pairing and Bloch's product. Apply it only to smooth separated finite-type $k$ -schemes, weights $r, s \geq 0$ and all cohomological degrees. Reference 7, Proposition 12.1 and its proof (printed pp.829–830), identifies the second comparison as the actual roof $C_{j} \overset{i_{j}}{\to} Q_{j} \overset{p_{j}}{\leftarrow} Z_{j}$ after Zariski sheafification. Thus the formula of [F8] is this second comparison: inclusion into cycles on the affine-space product, followed by inverse affine-space pullback and descent. The first factor has the products identified by Spitzweck (18) and Theorem 6.7. Thus the imported assertion concerns existing global cohomology products, without prescribing an algebra structure on raw cycle chains. The unit is the weight-zero whole-scheme cycle; here $i_{0}$ and $p_{0}$ are identities.

External input

References 0–2 supply the spectrum-model, actual interval diagrams, topology-change, equidimensional inclusion, affine-space homotopy invariance, descent and spectrum-evaluation inputs in [F2] and [F6]–[F8]. References 3–4 supply the integral motivic linear adjunction after Nisnevich and A1 localization and stabilization, including its actual enhanced models, as specified in [F7]. Apply all imports under their source-record hypotheses and reading limits. The actual coefficient maps come from [F1], support localization and coherent realization from [F3], and the spectral-lift interface from [F9]. Reference 5 supplies the effective tensor closure, slice triangle and orthogonality in [F10]. Reference 6 supplies the precise global product-comparison theorem in [F11], and reference 7 supplies its specific inclusion/projection roof; it is not used to produce a coherent spectral comparison. The interval-diagram application and identification with its classical global comparison, the normalization and binary-multiplication argument, the Tate-compatibility argument, coefficient evaluation, shift calculation, composition, support-fiber naturality and reduced-point application below are local arguments. This item is not imported as a whole.

Proof

Given: The field, nonnegative weight, size bounds, schemes, closed pairs and actual coefficient maps above.

Goal: Construct the stated natural coefficient and support equivalences and verify the stated pullback identity for point classes.

1.1A1F1F2F3F4F5F6F7F8F9F10F11given

By [F4]–[F5], all objects and open complements are in the supplied domains; smoothness survives open source restriction by its germwise definition. Use the integral enhanced adjunction $L ⊣ R$ of [F7], and the normalized spectrum equivalence $e_{k} : H_{S p i} ≃ R (E)$ of [F2], whose unit and binary-product compatibility are justified below; here $E$ is a local fibrant linear $G$ -Omega model of the field cycle spectrum, on the common Nisnevich site. To justify this normalization put $H = H_{S p i}$ and $B = R (E)$ . The local Omega model, the free-spectrum/evaluation adjunction of [F7], and $M (0)_{k} = Z$ in [F2] give $[1, B] = H_{N i s}^{0} (Spec k, Z) = Z$ , where brackets denote morphisms in $h S H (k)$ . The underlying equivalence $f_{k}$ gives $[1, H] ≅ Z$ too. In a unital ring with additive group $Z$ , the unit is a generator: if $g$ is an additive generator, $1 = n g$ and $g^{2} = m g$ , bilinearity and $1 g = g$ imply $n m = 1$ . Thus $f_{k} η_{H} = ϵ η_{B}$ with $ϵ \in {1, - 1}$ , and $e_{k} = ϵ f_{k}$ is a unit-preserving spectrum equivalence. This is one sign for the spectrum map, not choices for individual schemes or weights. By [F10] and the unital identification $s^{0} 1 ≃ H$ , the cofiber $C = cofib (η_{H} : 1 \to H)$ belongs to $C_{1}$ , while $H$ is effective and $map (D, H) = 0$ for $D \in C_{1}$ . The equivalence $f_{k}$ gives the same orthogonality for $B$ . Mapping the cofiber triangle into $B$ therefore gives an equivalence $map (H, B) ≃ map (1, B)$ . Consequently the unit condition determines $e_{k}$ uniquely up to homotopy, with contractible fiber over the unit in the mapping-space restriction. Tensor the triangle with $H$ . Since $C \otimes H \in C_{1}$ , restriction along $η_{H} \otimes {i d}_{H}$ is an equivalence $map (H \otimes H, B) ≃ map (H, B)$ . The two maps $e_{k} μ_{H}$ and $μ_{B} (e_{k} \otimes e_{k})$ restrict to $e_{k}$ , by the two unit laws and $e_{k} η_{H} = η_{B}$ ; hence they are homotopic. Tensoring this homotopy with Tate powers and evaluating on the diagonal of any smooth scheme proves compatibility with all the global cohomology products in nonnegative weights. The unit-adjoint reduced Tate class is preserved as well. This argument uses the existing multiplication on $B$ from Theorem 6.7 and proves compatibility of the entire normalized composite; it does not infer that compatibility from a spectrum isomorphism alone. This passage uses [F8] on the classical levels; their derived evaluations are preserved, and $E_{r}$ is represented by $M (r)_{k} [r]$ . For a complex level $C$ , derived internal Hom out of $G$ evaluates at $V$ as the fiber of $R Γ (V \times G_{m}, C) \to R Γ (V, C)$ induced by the unit section. The natural topology comparison in [F8] applies to both vertices and hence to this fiber. It therefore transports the adjoint structure equivalences of [F2] to the Nisnevich model, justifying the stated Omega model with those derived levels. Here is the required Tate compatibility without assuming that an arbitrary right adjoint preserves tensor products. For every motivic spectrum $X$ , invertibility of $T$ , monoidality of $L$ and the adjunction give the natural mapping-space equivalences $Map (X, R (E) \otimes T^{j}) ≃ Map (X \otimes T^{- j}, R (E)) ≃ Map (L (X) \otimes L (T)^{- j}, E) ≃ Map (L (X), E \otimes L (T)^{j})$ . These equivalences take place in the supplied enhancements. Both represented functors on $S H (k)^{o p}$ preserve finite limits: $L$ preserves finite colimits, and mapping into a fixed object takes these to limits. By [F9] their space-valued equivalence therefore lifts naturally to mapping spectra, compatibly with all suspensions; it is not only an isomorphism of cohomology groups. Set $X = Σ_{T}^{\infty} V_{+}$ . By [F7], $L (X) = Σ_{l i n}^{\infty} Z_{N i s} [V]$ and $L (T) = G [1]$ . Derived level zero of $E \otimes G^{j}$ is $M (j)_{k} [j]$ ; the further shift $[j]$ from $(G [1])^{j}$ makes it $M (j)_{k} [2 j]$ . Free-spectrum/evaluation adjunction then reduces the linear mapping spectrum to derived free-sheaf evaluation at $V$ . Concretely, on a K-injective model $I$ of that level, the free-sheaf adjunction identifies the entire mapping complex ${Hom}^{∙} (Z_{N i s} [V], I)$ with $I (V)$ , whose derived value is $R Γ_{N i s} (V, M (j)_{k} [2 j])$ ; normalized chains and [F9] identify its underlying mapping spectrum with the fixed realization [F3]. Thus the natural equivalence is ${map}_{S H (k)} (Σ_{T}^{\infty} V_{+}, H_{S p i} \otimes T^{j}) ≃ H (R Γ_{N i s} (V, M (j)_{k} [2 j]))$ . Remove $[2 j]$ on both sides, using [F3], to obtain $S_{j} (V) ≃ H (R Γ_{N i s} (V, M (j)_{k}))$ . All these operations are derived functors on fixed models; evaluation of this same $e_{k}$ , tensored with each Tate power, gives a single natural transformation on the smooth site. The motivic localized adjunction is the substantial imported premise in [F7]; objectwise free abelian groups and Dold–Kan alone are not asserted to supply it. Use the actual field interval diagram in [F2] to finish the construction coherently. Choose a representative of ${\tilde{γ}}_{j}^{'}$ in the category of complexes of affine flat diagrams; an isomorphism of either endpoint with its stated diagram may be represented by a roof of maps of entire diagrams. Localize these complexes objectwise. A strict interval diagram then gives a natural transformation into $D (Z)$ , and its objectwise quasi-isomorphisms become equivalences there. The supplied projection from the hatted index is an equivalence after localization, so descent along it transports this entire natural transformation to the affine flat category. Apply Zariski localization and derived evaluation to the entire interval diagram, using the fixed derived-section models. The endpoint defining $M (j)_{k}$ is the Zariski localization of $γ_{j}^{'}$ , by the definition on p.41 and its field version on p.51; the other endpoint is the actual cycle diagram. Cycle descent in [F8] identifies its derived evaluations with $Z_{j}$ . On nonaffine schemes use these same Zariski-localized objects and derived evaluation; Corollary 5.19 is exactly their comparison on the flat site. Thus this process gives a coherent equivalence $R Γ_{Z a r} (-, M (j)_{k}) ≃ Z_{j}$ , including every restriction arrow, rather than lifting an arbitrarily specified ordinary derived-category diagram. The actual topology-change map in [F8] gives a natural equivalence $R Γ_{Z a r} (-, M (j)_{k}) \to R Γ_{N i s} (-, M (j)_{k})$ through its classical representative; invert it in the functor category. Compose this inverse with the preceding equivalence, [F6]'s natural normalization inclusion and the already constructed coefficient evaluation. This defines one $λ_{j} : S_{j} ≃ H (z^{j} (-, ∙) [- 2 j])$ on the étale and open category, before taking supports. Fix the preceding coherent choices and identify the global map by one common diagram, rather than by independent rectifications of its arrows. Work first over the affine flat category and suppress the common shift $[- 2 j]$ . Put $P_{j} = k [t_{1}, \dots, t_{j}]$ . At an auxiliary index $t = (A, B, i)$ of a flat algebra chain $K$ , use the five complexes $Z_{K} (t) = N z^{j} (A_{i})$ , $Q_{K} (t) = N z^{j} (P_{j} \otimes_{k} A_{i})$ , $D_{K} (t) = N z_{F_{t}}^{j} (C_{t})$ , $D_{K}^{P} (t) = N z_{F_{t}^{P_{j}}}^{j} (P_{j} \otimes_{k} C_{t})$ and $C_{K} (t) = {\tilde{z}}^{j} (A_{i})$ . There are maps $a_{K} : Z_{K} \to D_{K}$ and $a_{K}^{P} : Q_{K} \to D_{K}^{P}$ given by the flat comparison of §5.2.1, projection pullbacks $p_{K} : Z_{K} \to Q_{K}$ and $d_{K} : D_{K} \to D_{K}^{P}$ , and the equidimensional inclusion $i_{K} : C_{K} \to Q_{K}$ . Both $d_{K} a_{K}$ and $a_{K}^{P} p_{K}$ pull a cycle along $A_{i} \to P_{j} \otimes_{k} C_{t}$ , so $d_{K} a_{K} = a_{K}^{P} p_{K}$ on generators. Also $c_{K} : = a_{K}^{P} i_{K} : C_{K} \to D_{K}^{P}$ is precisely the arrow $α_{K}^{e}$ of §6: after the flat polynomial pullback, the relative zero-cycle is quasi-finite over the auxiliary base and is in good position for each auxiliary closed test scheme, as in [F8], Example 19.5. These maps commute with cycle faces and the auxiliary transition maps by the actual transformations of §§5.2.1–5.2.2 and §6. Let $J$ be the finite category consisting of this commuting square together with the triangle $C_{K} \to Q_{K} \to D_{K}^{P}$ and its composite $c_{K}$ . We have thus constructed functors $E_{n} \times J \to C p x (Z)$ compatible with simplex faces. Apply the $T^{p}$ construction at the beginning of §5.2, the same cosimplicial resolution and Lemma 5.16 to this whole $J$ -diagram, on ${\hat{S}}_{f l} \times J$ . Choose its representative and endpoint roofs once, and restrict that choice to the constituent subdiagrams. The restrictions recover the flat interval diagrams and the restriction of $γ_{e, j}^{'}$ ; the projection diagram is the one of §5.2.2. This supplies compatible representatives, not an enhancement inferred from equality in an ordinary derived category. The projection of the hatted category is a localization equivalence by Proposition 5.13, so it transports the entire $J$ -diagram and its commuting relations together. Zariski localization and derived evaluation are then applied to this one diagram. Restore $[- 2 j]$ and let $D_{j}, D_{j}^{P}$ be its two strictification endpoints after sheafification; use $C_{j}, Z_{j}, Q_{j}$ for the sheafifications of [F8]'s complexes in the following sheaf-level calculation. The maps are $a_{j} : Z_{j} ≃ D_{j}$ , $a_{j}^{P} : Q_{j} ≃ D_{j}^{P}$ , $d_{j} : D_{j} ≃ D_{j}^{P}$ and $c_{j} : C_{j} ≃ D_{j}^{P}$ . The first two equivalences are the flat comparisons. The relation $d_{j} a_{j} = a_{j}^{P} p_{j}$ and homotopy invariance of $p_{j}$ prove that $d_{j}$ is an equivalence. The other relation is $c_{j} = a_{j}^{P} i_{j}$ , so [F8]'s local equivalence of $i_{j}$ makes $c_{j}$ an equivalence too. Under $D_{j} = M (j)_{k}$ and Corollary 5.22's identification of $D_{j}^{P}$ with $R \underset{}{Hom} (Z [A^{j}]_{Z a r}, M (j)_{k})$ , Corollary 5.24, applied to $k \to P_{j}$ , identifies $d_{j}$ with the actual projection pullback. Thus the canonical target identification used in Proposition 6.1 is $d_{j}^{- 1}$ , and its equidimensional comparison is $v_{j} = d_{j}^{- 1} c_{j} : C_{j} ≃ M (j)_{k}$ . The selected flat comparison is $a_{j}^{- 1}$ . Cancellation in this same localized diagram gives $a_{j}^{- 1} v_{j} = a_{j}^{- 1} d_{j}^{- 1} a_{j}^{P} i_{j} = p_{j}^{- 1} i_{j}$ . This is an equality of the selected derived-sheaf maps with their specified endpoint identifications. In particular it persists under derived evaluation on nonaffine schemes; equality of local cohomology classes is not being used to glue an unspecified global map. To display the descent endpoint as well, let ${des}_{Z, j} : Z_{j} (-) \to R Γ_{Z a r} (-, Z_{j})$ be [F8]'s actual cycle descent equivalence. Then the selected $b_{j} : R Γ_{Z a r} (-, M (j)_{k}) ≃ Z_{j} (-)$ satisfies $b_{j} R Γ_{Z a r} (v_{j}) = {des}_{Z, j}^{- 1} (R Γ_{Z a r} p_{j})^{- 1} R Γ_{Z a r} i_{j}$ . Reference 7, Proposition 12.1 and its proof, identifies exactly this inclusion/projection roof as the Friedlander–Suslin comparison named in reference 6, §3.1, not merely an isomorphism with the same source and target. The nonaffine descent here is supplied by [F8]; reference 7's Corollary 12.2 is only stated for quasiprojective schemes and is not used to enlarge that scope. Finally the topology-change transformation is natural in the very same $v_{j}$ . Transport it through the coherent equivalence $v_{j}$ to the bounded classical Nisnevich representative of [F8] and invert that natural transformation in the functor category. Thus the topology and descent maps in $λ_{j}$ are the ones in the just-identified comparison, with no independent choices at their endpoints. The coefficient evaluation uses the single normalized $e_{k}$ in every weight; the earlier argument proves its unit and binary-product compatibility. Spitzweck Theorem 6.7 identifies its target products with (15). The square just established identifies the remaining global comparison with the one in [F11], so that imported theorem proves compatibility with the existing Bloch products in every degree and nonnegative weight. It includes the canonical topology and descent identifications through [F8]; they are not separately asserted multiplicative simply because their models have the same name. The natural normalized-to-unnormalized comparison induces the canonical identification of the same higher Chow groups; apply the product theorem to these groups, without asserting a chain-level multiplicative structure on normalization. The weight-zero maps in [F11] preserve $1$ , so units are preserved as well. No multiplicative raw-cycle enhancement is needed. Finally compose with the unchanged actual coefficient maps to obtain $Ψ_{j} = λ_{j} ρ_{j}^{- 1} θ_{j}$ , with inverse $θ_{j}^{- 1} ρ_{j} λ_{j}^{- 1}$ . This constructs the specified map before taking supports.

2.1F3F6F8step 1.1∎

Adjoin zero to the restriction diagrams of step 1.1 and take their limits in spectra. Fibers are pullbacks along zero, so their equivalence commutes with projection to the source and with pullback of pairs. By [F3], localization gives the natural quasi-isomorphism from the supported cycle kernel to the cycle restriction fiber, including its source projection. Its naturality for these flat maps is explicit: restriction commutes with pullback by [F8], and the fiber map $(a, b) \mapsto (q^{*} a, q^{*} b)$ commutes with $a \mapsto (a, 0)$ . Invert that natural kernel-to-fiber equivalence and use the fiber-preserving realization of [F3]. This gives $Ψ_{V, T}$ and the stated projections. The codimension assumptions make both reduced point cycles admissible in weight $j$ . By [F8], flat pullback of $[y]$ is the cycle of its scheme-theoretic fiber. That fiber is the single reduced rational point $x$ ; its coefficient is the length of its field local ring, namely one. Thus $q^{*} [y] = [x]$ . By [F6], normalization is the identity on these degree-zero cycles. The commuting support square therefore gives $q^{*} Ψ_{V, {y}}^{- 1} [y] = Ψ_{W, {x}}^{- 1} [x]$ . By [F3], a degree-zero point cycle in $z_{T}^{j} [- 2 j]$ lies in cochain degree $2 j$ , hence in $π_{- 2 j}$ and $H^{2 j, j}$ ; this is $H^{6, 3}$ in codimension three.

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Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · §5.1 Theorem 5.8, Lemmas 5.11–5.16, pp.37–41; §5.2.1 Lemmas 5.17–5.18/Corollaries 5.19–5.20, pp.42–43; §5.2.2 Lemma 5.21/Corollary 5.22 and Lemma 5.23/Corollary 5.24, pp.43–45; §5.3 Proposition 5.29 and §6 Proposition 6.1, (15), (18), Lemmas 6.2–6.6/Theorem 6.7, pp.50–56; retained absolute-field inputs: Lemma 7.5/Theorem 7.6 p.58, §7.5 Theorem 7.18 pp.60–64, Definition 8.1/Theorem 8.2 pp.64–65, Corollary 9.13 p.76, Theorem 9.16/Lemma 9.17 p.77, Proposition 11.1/Remark 11.2 p.79.

The strictified cycle diagrams give the flat-map comparison of Corollary 5.19. The field linear G_m-spectrum has levels M(r)_k[r] with the adjoint structure equivalences of Proposition 5.29. Proposition 6.1 identifies its levels with the classical complexes; (15), (18), Lemmas 6.5–6.6 and Theorem 6.7 identify products and the standard bonding coordinate, whose localization boundary is +1. Theorem 7.18 identifies the constructed absolute spectrum; generic restriction, closed-fiber specialization and field extension give its field model. Section 8 supplies strong periodization and Proposition 11.1 the projective-line coordinate. The coefficient-evaluation composition is proved locally below, rather than attributed to Corollary 5.19 alone. Theorem 6.7 identifies the ordinary field multiplication with (15). The absolute-to-field passages are used for the underlying spectrum equivalence. Unit normalization and compatibility of the full composite with binary multiplication are proved locally using the separate slice input in reference 5, rather than attributed to Theorem 7.18. The comparison in §5.2.1 is constructed as a homotopy-category object of entire complex diagrams indexed by the auxiliary flat category times an interval. Its endpoint identifications and projection are Lemmas 5.17–5.18 and Proposition 5.13. Section 6 explicitly carries this construction to fields and constructs the equidimensional comparison by another interval diagram. These constructions, rather than only the concluding isomorphism of Corollary 5.19, supply the coherent input.

Hypotheses: Integral coefficients; smooth separated finite-type schemes over the fixed perfect field k; j>=0. Use the linear sheaf-spectrum model of §§5.3–6, its local fibrant replacements, and the same Tate coordinate (P1,infinity)=(G_m,1) smash S1. Generic restriction is used in characteristic zero, closed-fiber specialization in positive characteristic, followed by arbitrary field extension. Raw-cycle functoriality is restricted to flat maps, here étale maps and open immersions. Products are the supplied global cohomology pairings; no equivalence of independently prescribed E-infinity raw-cycle algebras is imported.

Reading limits: Historical incoming reading record (not fresh reading or acceptance by this repair): For adjudicate-ddfee9315338, freshly read text extracted with PyMuPDF from the cached primary v3 PDF tmp/third-point-spitzweck.pdf in the frozen article run: printed pp.50–56,58,60–65,75–77,79. This includes Proposition 6.1, the unital pairings (15), (18), Lemmas 6.5–6.6 and Theorem 6.7 with its proof, the canonical comparison construction in section 7.5, strong periodization, and field base change and orientation statements. Opened the live versioned arXiv PDF; no cached/live byte identity, visual inspection, whole-paper reading, reconstruction of cited foundations or independent acceptance is claimed. The other locator ranges and unchanged imports are retained from the incoming source record, not claimed as freshly checked in this repair. The repair uses the existing primary-input supplier for the absolute field comparison and sharpens which supplied comparison is evaluated; it does not construct a new multiplicative raw-cycle enhancement. For adjudicate-57a130121935, read the cached primary v3 PDF with PyMuPDF at printed pp.50–52,54–56,58,60–64,76–77,79, and opened the live versioned PDF text at Theorems 6.7 and 7.18 and their surrounding constructions. The earlier reading record is historical. Fresh reading verifies the field pairings, bonding comparison, underlying absolute/field identifications and the distinction between these assertions. No fresh verification of the other locator ranges, cached/live byte identity, foundational reconstruction, whole-paper review or independent acceptance is claimed. Fresh adjudicate-1c2cee9f9ebe reading: PyMuPDF text of cached tmp/third-point-spitzweck.pdf, printed pp.37–45 and 50–56, including the actual interval diagrams, hatted-index localization, affine-space natural transformations, field comparison, products and bonding calculation; opened the live versioned arXiv PDF. The other retained locator ranges are inherited inputs, not freshly reread here. No cached/live byte identity, visual inspection, whole-paper reading, reconstruction of cited foundations or independent acceptance is claimed. The identification of the selected flat comparison with the canonical global comparison is the local commuting-square argument in step 1.1. Fresh adjudicate-ed2ffd26ea2b reading: read PyMuPDF text of the cached primary v3 PDF tmp/third-point-spitzweck.pdf at printed pp.37–45 and 50–56, including the good-position complexes, hatted localization, strictification, finite-index T construction, both flat endpoints, affine-space projection diagrams, field interval comparison and product/bonding calculation. Opened the live versioned arXiv PDF. The local application below explicitly rectifies the common diagram and cancels its canonical projection endpoint. Other source ranges and prior reading descriptions are retained historical inputs; no cached/live byte identity, visual inspection, whole-paper reading, foundational reconstruction or independent acceptance is claimed.

Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology (2006) · Proposition 13.10/Example 13.11 pp.101–102; Proposition 14.8/Corollary 14.9 p.112; §16 pp.125–126, Theorem 16.7 pp.126–127; Properties 17.4 p.136, Example 17.12 pp.138–139; Lemma 19.4 p.159, Example 19.5/Proposition 19.6 p.160, Theorem 19.8/Corollary 19.9 p.161, Proposition 19.12 p.162, proof of Theorem 19.1 and Examples 19.13–19.14 p.163 (cached PDF page = printed page +12).

The natural change-of-topology map identifies Zariski and Nisnevich hypercohomology for bounded-above complexes of Nisnevich sheaves with transfers with homotopy-invariant cohomology over a perfect field. Example 13.11 applies this to classical motivic complexes; Proposition 14.8/Corollary 14.9 give A1-locality of the bounded homotopy-invariant transfer representative. Cycle complexes are actual flat-pullback diagrams and satisfy Zariski descent. For a flat map the pullback of a cycle is the cycle associated to its scheme-theoretic inverse image. The specific coefficient-spectrum evaluation is a local adjunction application, not a claim of Proposition 13.10. Lemma 19.4 defines the inclusion of the Suslin equidimensional cycle complex into cycles on the product with affine space. Theorem 19.8 makes it a Zariski-local quasi-isomorphism. Properties 17.4(1) makes projection pullback an objectwise quasi-isomorphism. Theorem 19.1 is constructed by these particular arrows and the descent maps, as specified in Corollary 19.9 and its proof.

Hypotheses: Perfect field and smooth separated finite-type schemes. Apply Proposition 13.10 only to the bounded-above classical motivic-complex model, with homotopy-invariant transfer cohomology as in Example 13.11. Apply Proposition 19.12 to the cycle complexes on finite-type field schemes. Only étale/open flat pullbacks are needed; no arbitrary nonflat raw-cycle pullback, resolution of singularities or Borel–Moore formula is used.

Reading limits: Historical incoming reading record (not fresh reading or acceptance by this repair): Fresh text-extraction reading for adjudicate-e1f21b2c8e18 of the cached author-hosted book PDF scratch/open-diagram-resolution/mvw.pdf in the frozen run, PDF112–114,121–124,148,150–151,172–176 (one-based). Read Proposition 13.10 with its displayed proof, Example 13.11 and Proposition 14.8/Corollary 14.9 with their proofs, the flat convention, small-site and transfer comparison, descent statement/proof and topology-change application. The author-hosted live URL timed out. No visual inspection, remote-byte identity, whole-book reading or reconstruction of Theorem 13.8 and deeper localization/transfer foundations is claimed. Fresh adjudicate-1c2cee9f9ebe reading: PyMuPDF text of the cached author-hosted scratch/open-diagram-resolution/mvw.pdf at printed pp.100–102,110–112,126–129,136,138–139,159–165. Read the topology comparison, A1-locality statements, equidimensional model, actual cycle inclusion, homotopy invariance, descent and the comparison formula. The application uses only the listed comparison/descent inputs, not the resolution-dependent Borel–Moore statements also present on these pages. No live-byte verification, visual inspection, whole-book reading or reconstruction of the imported localization and transfer foundations is claimed. Fresh adjudicate-ed2ffd26ea2b reading: read cached author-hosted PDF text at Proposition 13.10/Example 13.11, Proposition 14.8/Corollary 14.9, Theorem 16.7, Properties 17.4, Example 17.12 and printed pp.159–163 (Lemma 19.4 through the proof of Theorem 19.1 and Example 19.14). Checked the actual inclusion, projection, descent and topology-change application. The live author-hosted URL timed out. Other incoming reading descriptions are historical; no live-byte identity, visual inspection, whole-book reading, reconstruction of imported transfer/localization foundations or independent acceptance is claimed.

Mark Hovey, Spectra and symmetric spectra in general model categories, arXiv:math/0004051v3 · Definition 1.2 p.5; Theorems 3.4/3.8 pp.13–15; Definition 6.3 p.30; §7 setup p.30, Theorems 7.8/7.10 pp.32–33

Evaluation at level n has a free-spectrum left adjoint. Stably fibrant spectra are local Omega spectra. Derived tensor with the suspension coordinate agrees naturally with right shift, whose level n is the former level n+1. These adjunctions apply to derived mapping objects and preserve the natural maps of spectra.

Hypotheses: For sequential spectra: a left proper cellular model category and left Quillen suspension endofunctor. For symmetric spectra: a left proper cellular monoidal model category, a compatible left proper cellular module model category and a cofibrant suspension coordinate. Here use only the already supplied local linear sheaf-spectrum construction in Spitzweck §§5.3–6, with replacements and fixed universes; no new model-existence theorem for arbitrary sheaf categories is inferred from the abstract result.

Reading limits: Historical incoming reading record (not fresh acceptance by this repair): Freshly opened and read the versioned primary arXiv PDF through browser text: Definition 1.2 and surrounding definitions; Theorems 3.4 and 3.8 with proofs and Definition 3.7; Definition 6.3; §7 hypotheses, Definitions 7.6/7.9 and Theorems 7.8/7.10 with their stated proof reductions. Abstract model-existence and localization foundations are imported. The free-linear adjunction and application to the particular spectrum model and 2j shift are local arguments. No claim to whole-paper or source-foundation review. For the present adjudicate-d9e14f7afeb7 repair, freshly read the versioned primary PDF text at the symmetric-spectrum setup, Theorem 7.10 and Theorems 8.3–8.4. The earlier reading description is historical; the motivic localized adjunction is now imported explicitly from references 3–4, not claimed as an objectwise local derivation. No new source-foundation proof or independent acceptance is claimed. Fresh adjudicate-1c2cee9f9ebe browser-PDF-text reading: Definitions 7.6/7.9, Theorems 7.8/7.10 and their surrounding stable-model/shift construction on printed pp.32–33. Other ranges are retained inputs. No whole-paper or foundational reconstruction is claimed. Fresh adjudicate-ed2ffd26ea2b browser reading of the primary versioned PDF: Definition 6.3 and the §7 model hypotheses at p.30, Definitions 7.6/7.9 and Theorems 7.8/7.10 at pp.32–33, including the right-shift formula. Other ranges and incoming reading descriptions are historical inputs. No whole-paper reading, model-existence reconstruction or independent acceptance is claimed.

Denis-Charles Cisinski and Frédéric Déglise, Triangulated categories of mixed motives, arXiv:0912.2110v4 (2019) · Definition 1.4.6 p.29; Example 5.1.10 pp.170–171; §5.2.25 p.194; §§5.3.20–5.3.24 pp.207–209; Remark 5.3.27 p.210; §5.3.35 and (5.3.35.1) p.213 (PDF indices: printed page +39).

Reduced free abelian sheaves and normalized chains give a Quillen adjunction on pointed Nisnevich A1-local models, inducing the integral premotivic adjunction N:SH(S) ⇄ DA1(S):K after stabilization. The target uses ordinary abelian sheaves without transfers. Stable fibrant linear spectra have local A1-invariant levels and adjoint structure equivalences; suspension/evaluation derives. Changing from the reduced Gm coordinate to its suspension gives an equivalent linear stabilization.

Hypotheses: Use S=Spec(k), a perfect field, the smooth Nisnevich site, coefficients Z, fixed universes and Choice. Use the source model constructions and cofibrant coordinates. The adjunction (5.3.35.1) is integral; the rational equivalence (5.3.35.2) is not used.

Reading limits: Historical incoming reading record (not fresh acceptance by this repair): Fresh primary PDF-text reading of the listed passages and surrounding constructions; §5.3.35 explicitly invokes Hovey for stabilization. Model existence, descent and localization foundations are imported. No whole-book reading, visual inspection or independent acceptance; no claim that Hovey alone constructs the motivic adjunction. Fresh adjudicate-1c2cee9f9ebe browser-PDF-text reading: §§5.2.25 and 5.3.35, including the pointed model adjunction and integral stabilization (5.3.35.1); also §§5.3.14–5.3.19 around the free-spectrum/evaluation and Tate-coordinate models. Other locator ranges are retained inputs. No whole-book reading or independent acceptance is claimed. Fresh adjudicate-ed2ffd26ea2b browser reading of the primary versioned PDF: §§5.2.25 and 5.3.35 with the pointed integral model adjunction and stabilization, Definition 5.3.24 and Remark 5.3.27 with the linear Omega and coordinate conventions. Other ranges and incoming reading descriptions are historical. The established model and localization foundations remain imported; no whole-book reading or independent acceptance is claimed.

Frédéric Déglise, Motivic homotopy theory and stable homotopy groups, Séminaire Bourbaki no.1241 (June 2025) · §3.1 p.34; Definition 3.1 p.36; §3.2.3 p.39, including footnote 52 (PDF indices 33–35 and 38).

The derived infinity-category of ordinary abelian Nisnevich sheaves on smooth k-schemes, after A1-localization and P1-stabilization, is DA1(k). Dold–Kan extends to an adjunction Z_A1:SH(k) ⇄ DA1(k):H_A1. Smooth representables are linearized to free abelian sheaves. This is the infinity-category version of the linear adjunction, before adjoining transfers.

Hypotheses: A field k, here perfect; the smooth Nisnevich site and integral abelian sheaves in fixed universes. Use reduced P1 pointed at infinity, equivalently its Gm smash S1 coordinate. No equivalence with a category of modules over motivic HZ, no transfer equivalence, and no characteristic inversion is used.

Reading limits: Historical incoming reading record (not fresh acceptance by this repair): Fresh text reading of the classical stable Dold–Kan diagram, stabilization definition, and §3.2.3 with its footnote. This exposition states the enhanced adjunction; its model and higher-categorical foundations remain imported, complemented by reference 3. No whole-exposé reading or independent acceptance. Fresh adjudicate-1c2cee9f9ebe browser-PDF-text reading: §3.2.3, its derived-infinity-category construction and footnote 52 at printed p.39. The preceding full stabilization discussion is retained input, not freshly reread. No whole-exposé reading or independent acceptance is claimed. Fresh adjudicate-ed2ffd26ea2b browser reading of primary PDF §3.2.3 at printed p.39: the derived infinity-category construction and the enhanced linear adjunction. The preceding discussion and incoming reading descriptions are retained historical inputs. No whole-exposé or foundational reconstruction and no independent acceptance is claimed.

Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes, primary arXiv HTML at 2508.09915v1 · §3.1 symmetric monoidal stable setup; §3.3 opening effective-subcategory definitions, (3.36), properties (1)–(4), and Remark 3.39 (3.40)

Effective motivic spectra form a stable colimit-closed tensor subcategory; its Tate translate defines the j-effective objects. The effective-cover right adjoints give the slice cofiber triangle. The jth slice is right orthogonal, at the level of mapping spectra, to all (j+1)-effective objects. The slice filtration is lax symmetric monoidal, with the induced multiplication and unit on its graded pieces.

Hypotheses: A qcqs base scheme; applied only to Spec k for the fixed perfect field, in the existing universe-bounded stable symmetric monoidal motivic category. Use integral coefficients and the same invertible Tate coordinate. The identification of H_Spi with s^0 1 comes from the existing declared unit-zigzag supplier, not this general slice statement. No convergence, characteristic inversion or arbitrary-base comparison is inferred.

Reading limits: Historical incoming reading record (not fresh acceptance by this repair): For adjudicate-57a130121935, opened the primary HTML and read §3.3 from the effective-subcategory definition through properties (1)–(4) and Remark 3.39, including the displayed adjunction and cofiber formula. The stable monoidal setting is also the current declared primary-input interface. The HTML is the actual version served at this URL; no unread PDF identity or entire-paper reading is claimed. Slice existence and its monoidal refinement are imported; unit normalization, uniqueness and the binary-multiplication application are local arguments. No foundation reconstruction or independent acceptance is claimed. Fresh adjudicate-1c2cee9f9ebe primary HTML reading: §3.3 from the effective-subcategory definition through properties (1)–(4), Remark 3.39 and Remark 3.41. Verified the tensor closure, slice triangle, mapping-spectrum orthogonality and multiplicativity used locally. No reconstruction of the cited foundations or independent acceptance is claimed. Fresh adjudicate-ed2ffd26ea2b primary HTML reading of §3.3: effective tensor closure, effective covers, property (4) mapping-spectrum orthogonality and Remark 3.39 multiplicativity. The application is restricted to the already supplied perfect-field setting. Earlier descriptions are historical. No whole-paper or foundational reconstruction and no independent acceptance is claimed.

Satoshi Kondo and Seidai Yasuda, Product structures in motivic cohomology and higher Chow groups (author manuscript; journal version JPAA 215 (2011), 511–522) · Manuscript p.1 perfect-field convention; §3, (3.1), Theorem 3.1 p.8; §3.1, (3.3)/(3.5) p.9 and Proposition 3.3 p.10.

For a perfect field k and smooth X, the canonical integral comparison H^a_M(X,Z(r)) ≅ CH^r(X,2r−a) preserves products. Formula (3.3) factors through equidimensional cycles; §3.1 identifies its second factor as the product-preserving Friedlander–Suslin comparison. Proposition 3.3 treats its first factor.

Hypotheses: Apply only over the fixed perfect field to smooth separated finite-type schemes, integral coefficients, r,s≥0 and all cohomological degrees. No resolution, infinite-field or characteristic-inversion assumption is used. No coherent raw-cycle algebra or supported product comparison is imported.

Reading limits: Fresh adjudicate-1c2cee9f9ebe reading of browser-indexed primary-manuscript text at p.1 (perfect-field convention), p.8 ((3.1), Theorem 3.1), p.9 (§3.1, (3.3)/(3.5), the second comparison and its product assertion), and p.10 (Proposition 3.3). Direct opening of the mirror PDF returned 403. These actual indexed manuscript passages, not another proposal’s reading claim, are the evidence used. No complete PDF retrieval, journal-version identity, full-proof or cited-foundation reconstruction, or independent acceptance is claimed. The theorem is imported only for global cohomology products; the selected-map identification is local. Fresh adjudicate-ed2ffd26ea2b reading of primary-manuscript text obtained by opening the indexed CiteSeer PDF result: p.1 perfect-field convention and pp.8–10, including (3.1), Theorem 3.1, (3.3)/(3.5), the identification of its second factor with Friedlander–Suslin Proposition 12.1 and the product assertion for that factor. Initial direct opening returned 403; the indexed document subsequently opened as readable PDF text. Independently read the published Friedlander–Suslin Proposition 12.1 to check the named roof. No full-proof reconstruction, complete PDF download, journal/manuscript byte identity, whole-paper reading or independent acceptance is claimed.

Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 773–875 · §12, definition of the Suslin cycle complex and Proposition 12.1 with its proof, printed pp.829–830 (PDF pages 57–58); Corollary 12.2 p.830 for its stated quasiprojective scope.

For q≥0, on the small Zariski site of a smooth field scheme X, the Suslin complex C_*(Z_{Q,A^q}) and Bloch complex z^q are naturally isomorphic in the derived category by the roof C_*(Z_{Q,A^q}) → z^q(-×A^q,*) ← z^q(-,*). The first arrow is cycle inclusion and the second is projection pullback; both are quasi-isomorphisms of Zariski sheaf complexes. This is the comparison cited as the second factor of Kondo–Yasuda (3.3).

Hypotheses: Smooth schemes over a field F, applied only to smooth separated finite-type schemes over the fixed perfect field k, with q≥0 and integral coefficients. The proposition concerns complexes on the small Zariski site. Its Corollary 12.2 is restricted to smooth quasiprojective schemes; the present descent extension to all stated smooth separated finite-type schemes uses the independently cited MVW Proposition 19.12. No spectral enhancement, supported product, infinite-field or resolution-dependent K-theory theorem is imported.

Reading limits: Fresh adjudicate-ed2ffd26ea2b reading: opened the published primary Numdam PDF and read §12 at printed pp.829–830, including the definition, Proposition 12.1 and its proof, and Corollary 12.2 with its explicit quasiprojective scope. Checked the inclusion and projection arrows against Kondo–Yasuda §3.1 and MVW Lecture 19. The source foundations (homotopy invariance, field comparison and transfers) remain imported, not reconstructed. No whole-paper reading, proof of the K-theory spectral sequence, journal/manuscript identity for Kondo–Yasuda, or independent acceptance is claimed.

Sources

Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · §5.1 Theorem 5.8, Lemmas 5.11–5.16, pp.37–41; §5.2.1 Lemmas 5.17–5.18/Corollaries 5.19–5.20, pp.42–43; §5.2.2 Lemma 5.21/Corollary 5.22 and Lemma 5.23/Corollary 5.24, pp.43–45; §5.3 Proposition 5.29 and §6 Proposition 6.1, (15), (18), Lemmas 6.2–6.6/Theorem 6.7, pp.50–56; retained absolute-field inputs: Lemma 7.5/Theorem 7.6 p.58, §7.5 Theorem 7.18 pp.60–64, Definition 8.1/Theorem 8.2 pp.64–65, Corollary 9.13 p.76, Theorem 9.16/Lemma 9.17 p.77, Proposition 11.1/Remark 11.2 p.79.
Carlo Mazza, Vladimir Voevodsky and Charles Weibel, Lecture Notes on Motivic Cohomology (2006) · Proposition 13.10/Example 13.11 pp.101–102; Proposition 14.8/Corollary 14.9 p.112; §16 pp.125–126, Theorem 16.7 pp.126–127; Properties 17.4 p.136, Example 17.12 pp.138–139; Lemma 19.4 p.159, Example 19.5/Proposition 19.6 p.160, Theorem 19.8/Corollary 19.9 p.161, Proposition 19.12 p.162, proof of Theorem 19.1 and Examples 19.13–19.14 p.163 (cached PDF page = printed page +12).
Mark Hovey, Spectra and symmetric spectra in general model categories, arXiv:math/0004051v3 · Definition 1.2 p.5; Theorems 3.4/3.8 pp.13–15; Definition 6.3 p.30; §7 setup p.30, Theorems 7.8/7.10 pp.32–33
Denis-Charles Cisinski and Frédéric Déglise, Triangulated categories of mixed motives, arXiv:0912.2110v4 (2019) · Definition 1.4.6 p.29; Example 5.1.10 pp.170–171; §5.2.25 p.194; §§5.3.20–5.3.24 pp.207–209; Remark 5.3.27 p.210; §5.3.35 and (5.3.35.1) p.213 (PDF indices: printed page +39).
Frédéric Déglise, Motivic homotopy theory and stable homotopy groups, Séminaire Bourbaki no.1241 (June 2025) · §3.1 p.34; Definition 3.1 p.36; §3.2.3 p.39, including footnote 52 (PDF indices 33–35 and 38).
Tom Bachmann, Elden Elmanto and Matthew Morrow, A¹-invariant motivic cohomology of schemes, primary arXiv HTML at 2508.09915v1 · §3.1 symmetric monoidal stable setup; §3.3 opening effective-subcategory definitions, (3.36), properties (1)–(4), and Remark 3.39 (3.40)
Satoshi Kondo and Seidai Yasuda, Product structures in motivic cohomology and higher Chow groups (author manuscript; journal version JPAA 215 (2011), 511–522) · Manuscript p.1 perfect-field convention; §3, (3.1), Theorem 3.1 p.8; §3.1, (3.3)/(3.5) p.9 and Proposition 3.3 p.10.
Eric M. Friedlander and Andrei Suslin, The spectral sequence relating algebraic K-theory to motivic cohomology, Ann. Sci. École Norm. Sup. (4) 35 (2002), 773–875 · §12, definition of the Suslin cycle complex and Proposition 12.1 with its proof, printed pp.829–830 (PDF pages 57–58); Corollary 12.2 p.830 for its stated quasiprojective scope.