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Independent Navarro and EM Chern operations on the smooth separated site

Statement

Assume the Axiom of Choice and compatible site and universe bounds, with $ω$ in the scheme-data universe. Let $k$ be a perfect field and $i \geq 1$ . On smooth separated finite-type $k$ -schemes put $P (T) = Ω^{\infty} K_{P e r f} (T)$ and let $e_{i} : Y_{i}^{c l} ≃ Y_{i}^{E M}$ be the specified integral coefficient equivalence. Here $Y_{i}^{a} (T) = Ω^{\infty} H (D_{i}^{a} (T) [2 i])$ for the classical or EM weight- $i$ complex. Let $C_{i}^{E M} : P \to Y_{i}^{E M}$ be the supplied actual operation and $N_{i} : P \to Y_{i}^{c l}$ Navarro's independently specified represented Chern operation, precomposed with the canonical ordinary-K-to-KH source map. There is a pointed natural homotopy $e_{i} N_{i} ≃ C_{i}^{E M} .$ For every closed subset $Z \subseteq V$ in this site, the induced operations on the restriction fibers for $(V, V ∖ Z)$ agree under $e_{i}$ , compatibly with forgetting support. This includes the supported third Chern operation on the positive residue-sheaf class in the smooth affine threefold application. The comparison is integral in every characteristic. It asserts no extension of Navarro's independent endpoint to nonseparated or singular schemes.

Definition

Use the size-bounded category ${S m}_{k}^{s e p}$ of smooth separated finite-type schemes over $k$ , with all $k$ -morphisms and Nisnevich covers. Smoothness and finite type concern the structure morphism to $Spec k$ ; separated means that its diagonal is a closed immersion. Perfect means that every irreducible nonconstant polynomial over $k$ is separable. Fix nested universes containing the schemes, sites, complexes and diagrams; their existence is separate from Choice.

The targets and $e_{i}$ are the actual coefficient functors of the Chern-transport and field-zigzag suppliers. Write $H (k)$ for Riou's $A^{1}$ -localized Nisnevich homotopy category and $K_{i}^{R} = K (Z (i), 2 i)$ for his represented space. Use derived Nisnevich- and $A^{1}$ -local representatives; their full infinity-category is denoted $H (k)$ , whose homotopy category is $H (k)$ . Put $R_{i} (T) = {Map}_{H (k)_{*}} (T_{+}, K_{i}^{R})$ , where $T_{+}$ is $T$ with a disjoint basepoint. The bridge $b_{i} : Y_{i}^{c l} ≃ R_{i}$ is constructed in Proof 1.1 from the actual shifted spectrum, the field spectrum comparison and Riou's projective-line loop equivalences in [F8]. It is an equivalence of ordinary Nisnevich sheaves, before any optional further localization of the actual coefficients. Use $b_{i}$ to express both existing Chern maps in Riou's target; it does not define Navarro's map by transport from EM. A natural homotopy means a path in the derived mapping space of ordinary sheaves. The derived motivic local objects form a full subcategory there, so equality in $H (k)$ between their maps is equality of components of this mapping space.

Write $P^{s h}$ for ordinary Nisnevich sheafification of $P$ , with its canonical source unit. The affine Q/perfect comparison identifies $P^{s h}$ with the ordinary sheafified Quillen model, retaining the actual bundle classes and pointing. Riou's genuine regular-site model identifies that source with $Z \times G r$ . The actual EM map, its transported partner $C_{i}^{t r} = e_{i}^{- 1} C_{i}^{E M}$ , and $N_{i}$ are evaluated using these source units. No equality of global perfect classes with differences of global bundles is required. Source record 1 defines $N_{i}$ through the independently specified represented map and the canonical ordinary-K-to-KH comparison.

For $d, r \geq 0$ , ${G r}_{d, r}$ is Riou's finite Grassmannian of rank- $d$ locally direct-summand subbundles of $O^{d + r}$ , with universal bundle $U_{d, r}$ . Its component $n \in Z$ in $Z \times G r$ represents $u_{d, r, n} = [U_{d, r}] - d [O] + n [O]$ . These are classes in the supplied perfect $K_{0} = π_{0} K_{P e r f}$ under the canonical model comparison, including every negative integer $n$ .

For $U = V ∖ Z$ put $K (V / U) = fib (K (V) \to K (U))$ and $M_{i}^{a} (V / U) = fib (H D_{i}^{a} (V) \to H D_{i}^{a} (U))$ . Fibers are coherent pullbacks over zero, with their projection maps. The supported operation has target $Ω^{\infty} (M_{i}^{a} (V / U) [2 i])$ . At a smooth closed point $x$ , the positive residue class is the class of $i_{*} κ (x) [0]$ , where $κ (x) = O_{V, x} / m_{x}$ ; its perfectness and supported class are supplied separately by the point-class lemma.

Facts & Assumptions

Given: The field, positive index, actual operations, coefficient comparison and size assumptions in Statement.

Goal: Identify the independent represented Navarro map before taking support fibers.

[A1]

Assume The Axiom of Choice and the independent compatible universe bounds, with closure convention Grothendieck universe closure convention. These are inherited by the supplied actual operations, sheaf and spectral constructions, and source imports; they supply no independent acceptance.

[F1]

Projective relations transport the actual universal Chern operations supplies $e_{i}$ , the classical transported map $C_{i}^{t r} = e_{i}^{- 1} C_{i}^{E M}$ , their bundle-coefficient comparison and whole restriction diagrams. External formal universal Chern maps and supported construction supplies the actual EM operation, its ordinary sheafified-source map, source unit and vanishing on trivial integer ranks. A unit-induced integral field coefficient zigzag specifies the classical shifted mapping-spectrum coefficient functor and its actual maps; the additional comparison with Riou's space is supplied in [F8]. Smooth motivic comparison identifies its degree- $2 i$ , weight- $i$ values with $C H^{i}$ .

[F2]

Coefficientwise perfect Cartan on every qcqs field-scheme gives actual coefficientwise Cartan and a multiplicative formal Chern series on perfect $K_{0}$ . Its global values are the original coefficient values before hypercompletion. Full affine K-space comparison and its sheafification supplies the canonical full affine Q/perfect comparison, including actual object maps and ordinary sheafification. Restricted to the present site, affine open covers give the same sheaf comparison; their intersections are objects of this site.

[F3]

Source record 0 imports Riou's canonical genuine-model comparison and Theorem 6.2.1.2 for $K (Z (i), 2 i)$ . Its proof verifies condition (K): the inverse-limit phantom obstruction for finite Grassmannians vanishes. Lemma 1.2.6 and Remark 1.2.7 give detection by all $u_{d, r, n}$ , including affine vector-bundle torsors and finite disjoint unions. The general Theorem 1.1.6 is not applied without its target hypothesis. The source also gives Nisnevich descent and $A^{1}$ locality of the genuine K-model, with the canonical bundle-class comparison. The target bridge required before applying its uniqueness theorem is [F8], not the degree-zero Chow identification alone.

[F4]

Source record 1 specifies Navarro's represented positive Chern map from the ordinary Chern transformation, the actual-bundle Whitney and trivial-bundle normalization, the ordinary-K-to-KH comparison and the supported postcomposition convention. The ordinary virtual transformation here is the multiplicative total-series extension of those bundle classes: Extending a Whitney class to K zero justifies this standard extension and its inverse-series formula on differences. Navarro's oriented motivic coefficient theory has the required projective bundle and exact-sequence Whitney properties recorded in source record 1. This step is only applied to the explicit virtual bundle classes on the finite Grassmannians. The genuine-model source comparison of [F3], together with [F2], fixes the same bundle and integer-rank classes. No RR identity or Gysin theorem is used to prove the present comparison.

[F5]

Stable spectra and exact triangles supplies coherent mapping spaces, homotopy fibers and their functorial projections. The classical coefficient target is an infinite-loop space, so its mapping spaces admit addition and inverses. Scheme sites and sheafification fixes the site, sieve and sheafification-unit terminology; the space-valued local model used here is the separately imported one in [F3], not a consequence of set-valued sheafification alone.

[F6]

Perfect fields: every irreducible polynomial is separable, Smooth morphism of schemes, Locally finite type and finite type morphisms and Separated morphism of schemes fix the geometric hypotheses. fields as regular local rings makes the field a regular noetherian local ring of dimension zero, so $Spec k$ is Riou's regular noetherian separated base. Finite-type field schemes lie in the qcqs source domain gives the qcqs domain. The support-domain check in [F1]'s transport supplier gives smooth finite-type qcqs open complements. Such a complement remains separated: its open immersion is separated by Open and closed immersions are separated, and compose with $V \to Spec k$ using Separated morphisms compose.

[F7]

Perfect residue classes at smooth closed points supplies the parameter-independent perfect residue-sheaf class at a smooth closed point over a perfect field. Its residue field is that of The residue field at a point of an affine scheme. Every algebraic extension of a perfect field is separable passes from the irreducible-polynomial perfectness convention to the all-algebraic-extensions convention of that supplier. For the qcqs scheme and quasi-compact complement checked in [F6], Perfect K theory localization and open excision identifies category-defined supported perfect K-theory with the restriction fiber and preserves its map to global K-theory; thus it matches the residue-sheaf class with the corresponding restriction-fiber class; the equality of operations below holds on all such fiber classes.

[F8]

A unit-induced integral field coefficient zigzag specifies the shifted classical coefficient spectrum as $A_{i}^{c l} (T) = {map}_{S H (k)} (M_{k} (T), H_{S p i} \otimes T_{k}^{\otimes i})$ , where $M_{k} (T) = Σ_{P^{1}}^{\infty} T_{+}$ and $T_{k} = Σ_{P^{1}}^{\infty} (P^{1}, \infty)$ ; hence $T_{k}^{\otimes i}$ is the combined twist and shift $(i) [2 i]$ , not the twist alone. Exact primary unit, Picard and smooth-purity inputs for field compatibility supplies the integral field comparison $H_{S p i} ≃ H Z$ with the classical product and bonding model. Source record 2 checks its cycle/product and bonding identifications and the separate generic-point, positive-characteristic and field-extension hypotheses. Source record 0, Definition 6.2.1.1 and the opening of section 6.2.3, identifies Riou's $K_{i}^{R}$ as the levels of this classical $H Z$ , with equivalences $K_{i}^{R} ≃ R {Hom}_{*} ((P^{1}, \infty), K_{i + 1}^{R})$ . These are the projective-line Omega-spectrum identifications used by the suspension/evaluation adjunction, not a conclusion from equality of cohomology groups. Actual EM coefficient mapping and smooth-site comparison supplies ordinary Nisnevich descent for the actual $Y_{i}^{E M}$ , its cochain-shift convention $π_{r} H (C) = H^{- r} (C)$ , and the actual coefficient mapping-space interpretation. The equivalence $e_{i}$ in [F1] gives the same ordinary descent for $Y_{i}^{c l}$ . Derived Nisnevich-local representatives of Riou's levels are also ordinary sheaves; this does not identify all ordinary sheaves with hypercomplete ones. The spectrum comparison and the adjunction preserve coefficient evaluation and restriction arrows. Under the classical cycle/product identification and [F1]'s projective-relation normalization, bundle classes have the same classical Chow coordinates; no line is replaced by its dual. Navarro's target in source record 1 is the zeroth motivic space of this same shifted Spitzweck spectrum, so postcomposition by $b_{i}$ merely expresses his independently specified map in Riou's model.

Proof

1.1A1F1F2F3F4F6F8given

Work under [A1] on the site and genuine source model of [F3] and [F6]. First construct the bridge in [F8]. Postcompose its actual classical mapping spectrum with the specified field equivalence $H_{S p i} ≃ H Z$ , tensoring with the same $T_{k}^{\otimes i}$ . Apply $Ω^{\infty}$ and the motivic suspension/evaluation adjunction. Riou's Omega-spectrum description then gives $Y_{i}^{c l} (T) ≃ {Map}_{H (k)_{*}} (T_{+}, Ω_{P^{1}}^{\infty} (H Z \otimes T_{k}^{\otimes i})) ≃ R_{i} (T)$ . This is the natural equivalence $b_{i}$ : postcomposition and the adjunction are functorial on the entire scheme category and its coherent diagrams, so this compares full spaces and all restriction maps. Derived evaluation of a local representative computes the displayed right-hand mapping space. By [F8], both sides are ordinary Nisnevich sheaves, making $b_{i}$ an equivalence in that full sheaf category; its right-hand side is also $A^{1}$ -local, hence so is its left-hand side. The combined $(i) [2 i]$ shift is fixed by $T_{k}^{\otimes i}$ , and the common cycle/product normalization identifies the bundle coordinates. Navarro's independent map is the source-record-1 represented map expressed through this coefficient adjunction, so $b_{i} N_{i}$ is that same map in Riou's model. No comparison of the two Chern operations has been assumed in constructing $b_{i}$ . On ${G r}_{d, r}$ , the actual Chern series of $u_{d, r, n}$ is the series of $U_{d, r}$ times those of the trivial integer classes by [F2]. Every trivial nonnegative rank has series $1$ by [F1]; its additive negative also has series $1$ by the multiplicative inverse in [F2]. Thus the same calculation holds for every $n \in Z$ , including $d = 0$ , and the result is precisely the series of $U_{d, r}$ . Apply the product-preserving comparison in [F1] to identify it with the classical bundle series. Navarro's ordinary virtual transformation in [F4] has this same value by Whitney and trivial-bundle normalization. Hence the maps $b_{i} C_{i}^{t r}$ and $b_{i} N_{i}$ , on the common genuine K-source fixed by [F2]–[F4], induce equal degree- $i$ transformations on every $u_{d, r, n}$ . Riou's universal-class detector in [F3] identifies their transformations on all the $K_{0}$ presheaf values. This argument uses the canonical source comparisons and actual Cartan; it makes no assumption that global perfect classes are differences of global bundles.

2.1F1F2F3F4F5F6F7F8choosestep 1.1∎

Apply Riou's bijection of [F3] to the maps $b_{i} C_{i}^{t r}$ and $b_{i} N_{i}$ : step 1.1 gives equality of their unpointed classes with target $K_{i}^{R}$ on the genuine K-model. By [F2]–[F3], the ordinary sheafified source $P^{s h}$ is equivalent to the derived local genuine K-model. The target $R_{i}$ is a derived local representative by [F8] and step 1.1. Full inclusion of these local objects in ordinary sheaves identifies the relevant mapping-space components with Riou's morphism sets. Postcomposition with the ordinary sheaf equivalence $b_{i}$ is an equivalence of mapping spaces, so this equality pulls back to equality of components for the original maps $N_{i}$ and $C_{i}^{t r}$ into the actual $Y_{i}^{c l}$ . Choose a path between these original maps. Both positive-index endpoints are pointed. Its restriction to the zero section is a loop at zero in the additive target; subtract the constant-in-source extension of this loop from the chosen path. Addition and inverse in [F5] leave the endpoints unchanged and make that restriction constant. Equivalently, evaluation at zero splits the mapping space into its pointed fiber and the constant-map factor. We obtain a pointed natural homotopy $N_{i} ≃ C_{i}^{t r}$ . Precompose with the common ordinary source unit and postcompose with $e_{i}$ , using its inverse homotopy, to obtain $e_{i} N_{i} ≃ C_{i}^{E M}$ on actual perfect K-values. For $U = V ∖ Z$ , [F6] puts both $V$ and $U$ in this same separated smooth site, including the empty complement. Evaluate this one pointed homotopy on the entire restriction arrow and take homotopy fibers. Functoriality in [F5] gives the asserted supported comparison and forget-support square. In the represented local models the fiber is the mapping object from the cofiber $V / U$ , so this is exactly Navarro's postcomposition convention in [F4], with the same ordinary-K-to-KH source comparison. For the positive residue-sheaf class, the localization equivalence and geometric checks in [F6]–[F7] identify its category-defined support class with this restriction-fiber class. In particular the comparison applies to that same positive class, and hence to $i = 3$ in the smooth affine threefold application. No source class is changed by the coefficient comparison, and no assertion is extended to nonseparated or singular schemes.

External input

Riou's genuine K-model, universal-class detection and motivic-target uniqueness are imported at source record 0. Its section 6.2.3 supplies the projective-line Omega-spectrum levels needed for the target bridge. Navarro's independent Chern construction and supported postcomposition are imported at source record 1. The precise Spitzweck field, product and bonding comparisons are recorded at source record 2 and supplied by the declared primary-input item. Their hypotheses and reading limits are explicit. Proof 1.1 locally composes those spectrum and adjunction inputs into the ordinary sheaf equivalence $b_{i}$ , then compares universal virtual classes; Proof 2.1 pulls back Riou's equality through that equivalence and takes support fibers. Neither the imported foundations nor the RR theorem is reconstructed. This locally proved composition is not marked as an external-input item.

Depends on

Used by · 1

Integral oriented field coefficient comparison with actual Picard and support maps

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

Joël Riou, Algebraic K-theory, A1-homotopy and Riemann–Roch theorems, arXiv:0907.2710v2 · Conventions §1.1, p.3; Theorem 1.1.6 and Definition 1.2.2, p.5; Lemma 1.2.6 and Remark 1.2.7, pp.6–7; Propositions 3.1.2, 3.1.3, 3.1.5 and Corollary 3.1.4 with their proofs, pp.11–13; Definition 6.2.1.1 and Theorem 6.2.1.2 with proof, p.31. Opening of §6.2.3, p.33 (HZ levels and projective-line loop equivalences).

Over a regular noetherian separated base the genuine K-models agree with Z times Gr, retaining bundle classes and Nisnevich descent and A1 locality. For a perfect field and nonnegative weight, Theorem 6.2.1.2 detects maps to K(Z(i),2i) by K0-to-Chow transformations; its proof checks condition (K). Section 6.2.3 identifies these spaces as the projective-line Omega-spectrum levels of classical HZ.

Hypotheses: The base is regular, noetherian and separated for the genuine-model results; Riou defines Sm/S to consist of smooth separated finite-type schemes. The motivic-target theorem requires a perfect field and nonnegative i. Apply only for i at least one. Match the actual perfect source by its ordinary affine Q/perfect sheaf comparison; no global vector-bundle presentation on arbitrary schemes is assumed.

Reading limits: This repair opened the live author-hosted v2 PDF and read the cached primary PDF text at pp.3,5–7,11–13,31,33, including Theorem 6.2.1.2 and the opening Omega-spectrum description of §6.2.3. No new image inspection, reconstruction of cited motivic foundations, whole-paper review or acceptance is claimed. The ordinary-sheaf target bridge in Proof 1.1 is a local composition, not an asserted verbatim theorem of Riou.

Alberto Navarro, On the higher Riemann–Roch without denominators, arXiv:1901.09337v1 · §1 conventions, p.2; oriented Chern properties and Example 1.1, p.3; §§2.1–2.2, p.4; §3.5, p.9.

Navarro specifies the Chern map from the ordinary virtual Chern transformation via the represented Z times Gr KGL model and the ordinary-K-to-KH comparison. On regular schemes this is the ordinary K-model. Positive supported operations are postcomposition on the pointed quotient, compatible with forgetting support.

Hypotheses: The paper uses finite-dimensional noetherian schemes. Here use only the smooth separated finite-type site over a perfect field and its closed-support pairs. Riou Theorem 6.2.1.2 supplies the motivic target condition (K); no unrestricted version of Navarro’s quoted general comparison or of his RR theorem is imported.

Reading limits: This repair opened the live arXiv v1 PDF and read cached primary PDF pp.2–4,9: site conventions, the suspension/evaluation adjunction, oriented bundle classes, the represented Chern construction, supported postcomposition and relative fiber formulas. No new image inspection or reconstruction of the cited foundations or RR proof is claimed. Apply Riou uniqueness only on its verified motivic target scope.

Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · §6, equation (18) and Lemmas 6.5–6.6, pp.54–56; Theorem 6.7, p.56; Theorem 7.6, p.58; Theorem 9.16 and Lemma 9.17, p.77.

The field cycle model agrees with classical motivic HZ, preserving products and the Gm bonding class. The absolute spectrum restricts to classical HZ at the generic field and at positive-characteristic residue fields; the latter comparison respects E-infinity structures. The natural field-extension comparison is an equivalence.

Hypotheses: Theorem 6.7 uses its field model; apply over the fixed perfect field. Theorem 7.6 uses the fraction field of a Dedekind domain. Theorem 9.16 uses a positive-characteristic closed point of a mixed-characteristic Dedekind base; take Spec Z and then extend from its prime field by Lemma 9.17. Use the same integral product and bonding coordinates as the existing primary-input and field-zigzag suppliers; no prime is inverted.

Reading limits: Opened the live arXiv v3 PDF extraction at equation (18), Lemmas 6.5–6.6, Theorems 6.7,7.6,9.16 and Lemma 9.17, including their displayed proofs. Also read the supplied cached primary extract at pp.55–56,77. The cited cycle-product, base-change and cancellation foundations remain imported. No full-paper reading, new visual inspection or independent acceptance is claimed.

Sources

Joël Riou, Algebraic K-theory, A1-homotopy and Riemann–Roch theorems, arXiv:0907.2710v2 · Conventions §1.1, p.3; Theorem 1.1.6 and Definition 1.2.2, p.5; Lemma 1.2.6 and Remark 1.2.7, pp.6–7; Propositions 3.1.2, 3.1.3, 3.1.5 and Corollary 3.1.4 with their proofs, pp.11–13; Definition 6.2.1.1 and Theorem 6.2.1.2 with proof, p.31. Opening of §6.2.3, p.33 (HZ levels and projective-line loop equivalences).
Alberto Navarro, On the higher Riemann–Roch without denominators, arXiv:1901.09337v1 · §1 conventions, p.2; oriented Chern properties and Example 1.1, p.3; §§2.1–2.2, p.4; §3.5, p.9.
Markus Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv:1207.4078v3 · §6, equation (18) and Lemmas 6.5–6.6, pp.54–56; Theorem 6.7, p.56; Theorem 7.6, p.58; Theorem 9.16 and Lemma 9.17, p.77.