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Exact primary unit, Picard and smooth-purity inputs for field compatibility

PDF p. 17: Corollary 6.9; integral Chern and support comparison

Statement

The following established inputs are used integrally, with the versions, hypotheses and reading limits in the source records.

For any qcqs scheme $X$ with structure map $g : X \to Spec Z$ , the unit induces an equivalence of motivic $E_{\infty}$ rings $s^{0} 1_{X} \to s^{0} K G L_{X}$ , compatible with the equivalences obtained by base change from $Z$ . Over $Z$ , the canonical framed vector-bundle map is an equivalence $Σ_{f r}^{\infty} V e c t ≃ f_{0} K G L$ . Its rank-induced Bott-cofiber map to $Σ_{f r}^{\infty} Z$ is an equivalence. The latter object is naturally Spitzweck's absolute integral $H Z$ as an $E_{\infty}$ ring, by the adjunction construction from constant integers; constant $1$ is the ring unit. The required condition $f_{1} H Z = 0$ holds over the Dedekind base $Z$ . No characteristic inversion or $1 / 2$ assumption is imposed on this KGL assertion.

For a prime field $F$ , EM supplies natural multiplicative augmented comparison maps from its motivic K-filtration to its cdh KH-filtration, hence natural graded coefficient maps $η : Z (⋆)_{m o t} \to Z (⋆)_{c d h}$ . Each $η_{j} (X)$ is an equivalence when $X / F$ is regular Noetherian. On smooth $F$ -schemes the composite of the classical-to-EM map with $η$ is the canonical classical left-Kan/cdh comparison. EM's actual first-Chern map is obtained from the classical weight-one map on smooth $F$ -algebras by left Kan extension and Nisnevich sheafification, as in Lemma 5.1.

BEM supplies natural multiplicative graded maps from cdh coefficients through its base-changed represented coefficients to $Z (⋆)^{A}$ represented by the KGL slices. Their composite is an equivalence on equal-characteristic schemes. Its cdh coefficients restrict along every field extension by the canonical base-independent left-Kan/cdh construction. The represented theory has zero negative weights, the projective bundle formula and the Picard first class induced by $1 - [L^{\lor}]$ . On regular Noetherian bases its spectral Picard map is the determinant/unstable-slice-edge map of Construction 4.35; the cdh-local left Kan extension in Theorem 4.39(4), followed by Remark 4.45, supplies the indicated map on all qcqs schemes. For an $A^{1}$ -invariant Nisnevich graded ring theory with zero negative weights in every degree and PBF after every smooth base change, the mapping-limit argument in the proof of BEM Lemma 3.27 detects the Picard-space maps in (3.28) by the reduced $P^{1}$ class: the projective-space tower has zero $\lim^{1}$ obstruction. This is a statement about these Picard-space maps, not arbitrary higher spectral maps.

Over a perfect field, Spitzweck's integral field spectrum identifies with ordinary Voevodsky motivic cohomology and the Bloch cycle model, compatibly with the product and actual $G_{m}$ bonding map. Its canonical orientation classifies its tautological Picard line. Integral positive-characteristic specialization and arbitrary field extension have the hypotheses of Theorem 9.16 and Lemma 9.17. The occurrence of the tautological subline $O (- 1)$ in its line-projectivization convention does not replace the class of a given line by the class of its dual.

For a closed immersion $Z ↪ V$ between smooth separated finite-type schemes over a perfect field, smooth homotopy purity gives the coefficient-independent pointed equivalence $V / (V ∖ Z) ≃ Th (N_{Z / V})$ by the canonical deformation maps. On these regular smooth pairs the refined motivic Gysin map is the Thom isomorphism followed by purity; its forget-support map is the quotient projection. Navarro's supported Chern maps arise from the actual universal pointed K-space map and commute with forgetting support. Only this smooth-pair Gysin construction is used in the field-comparison application.

External input

The stated zero-slice, regular comparison, base-independence and smooth homotopy-purity theorems are imported on their exact hypotheses. The five local application lemmas prove the local compatibility consequences. BEM(6.4) is deliberately not the multiplicative bridge. The coefficient functors in this proposal use the canonical structured maps described in EM Constructions4.34/4.36 and BEM(4.19)/(7.14), with the canonical field-identification counits. Equivalences are inverted in the functor category; no unrelated objectwise inverses are selected.

Integral unit-arrow supplement

The first zigzag arrow is proved with the canonical rank-induced equivalence over Spec Z from Bachmann Theorem1.1(1)/Lemma2.1, Hoyois Theorem21 (including its actual E∞ unit), and the base-change square of BEM Corollary9.7. No integral characteristic-p conclusion is inferred solely from the classical field coefficient model or from Spitzweck Remark11.2.

Depends on

Used by · 5

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Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text.

External dependencies

Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv1207.4078v3 · Lemma6.6 and Theorem6.7 pp55–56; Theorem9.16/Lemma9.17 p77; Proposition11.1/Remark11.2 p79

The specified unit/Gm cycle bonding maps agree (boundary t=0 has coefficient +1); the field cycle spectrum identifies with classical HZ with products. The canonical orientation classifies the tautological Picard line, and the orientation/unit maps factor through the zero slices as in Remark11.2.

Hypotheses: Integral spectra; perfect field for the classical product model, arbitrary field extensions; positive-characteristic specialization is integral E∞.

Reading limits: Read the supplied arXiv:1207.4078v3 primary PDF text: introduction pp.3–4, complete Lemma 6.6 proof and Theorem 6.7 pp.55–56, Theorem 9.16/Lemma 9.17 with short proofs p.77, Proposition 11.1/Remark 11.2 p.79. Opened the primary URL. The deeper specialization and cycle-product foundations are imported; no separate published edition was read. Current assigned-concern recheck: Independently read the cached arXiv:1207.4078v3 cover and PDF pp.3–4,55–56,77,79, including Lemma 6.6, Theorem 6.7, Theorem 9.16/Lemma 9.17 and Proposition 11.1/Remark 11.2 with their printed arguments. The cycle boundary is +1; the product comparison is used on the perfect-field scope; positive-characteristic specialization respects E∞ structures. The tautological subline convention is retained. Deeper cited results and the remaining paper were not reconstructed. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/tmp/third-point-spitzweck.pdf; SHA-256 7da11119650a216f3722152c6b2e6b6b9b9feb1b28bf1f1f335e7e4ab20c6e74. Accessible primary-page extract: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/Spitzweck-primary-extract.pdf; SHA-256 ba52d9e5826f96436392ea34821638ac2668f79a9f49447d27bbc4e33e0a1a12.

Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv2309.08463v1 · Theorem 3.5/Remark 3.6, printed pp.14–16; Constructions 4.34–4.36/Remark 4.37, pp.31–32; Lemma 5.1/Definition 5.2, pp.33–34; Lemmas 5.10/5.12 and Construction 5.11, p.36; Theorem 6.1(2), Remark 6.3/Corollary 6.4 and footnote 12, pp.44–45

The actual graded comparison is induced by the augmented multiplicative filtered K→KH map. On smooth prime-field schemes its classical composite is the canonical Kan-extension/cdh comparison. The actual EM Picard map is defined from (4.9) by the Nisnevich-local left-Kan construction. Lemma 5.12 identifies the actual graded image of 1-[O(-1)] with the first Chern class of O(1), under the classical-to-EM comparison.

Hypotheses: The functors and comparison are on all qcqs schemes over a prime field. The comparison to cdh is an equivalence on regular Noetherian schemes; this includes smooth finite-type schemes over any extension field.

Reading limits: Read primary arXiv:2309.08463v1 web PDF text: Theorem 3.5/Remark 3.6; Constructions 4.34–4.36/Remark 4.37; complete Lemma 5.1 proof/Definition 5.2; Construction 5.11 and complete Lemma 5.12 proof; Theorem 6.1(2), Remark 6.3/Corollary 6.4 with proof and footnote 12. The regular-comparison proof is imported. EM v1 attributes its cdh/slice foundations to then-in-preparation [7]; no reading of a proof of [7] in that version is claimed. Current assigned-concern recheck: Independently read the cached arXiv:2309.08463v1 cover and PDF pp.14–16,31–34,36,44–45: Theorem 3.5/Remark 3.6; Constructions 4.34–4.36/Remark 4.37; complete Lemma 5.1 proof/Definition 5.2; Lemmas 5.10/5.12 and Construction 5.11; Theorem 6.1(2), Remark 6.3, Corollary 6.4 with proof and footnote 12. Theorem 6.1 is imported as an established input; its later proof was not reconstructed. The then-in-preparation [7] qualification is retained. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/EM-v1.pdf; SHA-256 15de979b7f1a58cc49c49e6c2d25fff8627e67372d33031bd8bb37d9c5cd90ff. Accessible primary-page extract: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/EM-primary-extract.pdf; SHA-256 654964debe348c3e4a32a7c29e097162b6acfaa1c05aee27a8669325c3fb844e.

Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, arXiv2508.09915v1 · Proposition 3.15/Definition 3.16; Remarks 3.21/3.23; Lemma 3.27 proof (3.28); §4.1.2; Definitions 4.15/4.17, Remarks 4.18/4.20–4.23, Theorem 4.25/Remark 4.28; Lemma 4.33 and Construction 4.35/(4.36); Theorem 4.39/Remark 4.45; Corollaries 1.11/6.26; Theorem 6.3 coefficient-comparison paragraphs and (6.4) qualification; Proposition 7.4, Theorem 7.5/Remarks 7.6–7.10; Theorem 7.12/(7.14); Corollary 9.7/(9.8)

The unit induces canonical E∞ zero-slice equivalences. The structured graded coefficients retain products and the unit-adjoint P¹ class. The proof of Lemma3.27 identifies actual Picard-space maps from their normalized P¹ class, with lim¹ controlled. The canonical cdh→represented comparison and field restriction are multiplicative, and the determinant/edge square specifies the Picard map.

Hypotheses: Structured coefficient constructions on qcqs schemes; Lemma3.27 requires A¹-local Nisnevich graded ring theory, zero negative weights and PBF; Construction4.35 on regular Noetherian bases, extended by Theorem4.39(4). Field base-independence for every field.

Reading limits: Read the listed statements and constructions in the supplied primary HTML text, including Proposition 3.15, Lemma 3.27 mapping-limit proof, Lemma 4.33 through its determinant computation, Theorem 4.39, Proposition 7.4/Theorem 7.5, Theorem 7.12 and Corollaries 6.26/9.7 with their printed arguments. Read the coefficient-comparison paragraphs of Theorem 6.3 and opened the primary URL. Corollary 1.11 and deeper foundations are imported as stated, not reconstructed. The equivalence (6.4) expressly carries no multiplicative/bonding claim and is not the bridge here. The supplied arXiv:2508.09915v1 HTML displays August 24, 2026; its cached bytes have SHA-256 98d10fba0a574dc9d82876855a7dac72428186893218590b2056601faaee839d. No identity with an unread PDF edition is asserted. Current assigned-concern recheck: Independently read the cached arXiv:2508.09915v1 HTML passages listed in this reference: Proposition 3.15/Definition 3.16, Remarks 3.21/3.23, complete Lemma 3.27 mapping-limit proof (3.28); §4.1.2; Definitions 4.15/4.17, Remarks 4.18/4.20–4.23, Theorem 4.25/Remark 4.28; Lemma 4.33 and Construction 4.35/(4.36), Theorem 4.39/Remark 4.45; Corollary 1.11 statement, Theorem 6.3 coefficient-comparison paragraphs including the explicit limitation of (6.4), Corollary 6.26 with proof; Proposition 7.4/Theorem 7.5 and Remarks 7.6–7.10, Theorem 7.12 including (7.14) and its printed proof; Corollary 9.7/(9.8) with proof. Also opened the primary HTML URL. The cached HTML displays August 24, 2026; no identity with an unread PDF edition is asserted. Deep results invoked by these passages are imported, not reconstructed. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/integral-field-model/BEM.html; SHA-256 98d10fba0a574dc9d82876855a7dac72428186893218590b2056601faaee839d. Focused adjudication adjudicate-c74bc045892b: independently read Lemma 2.26, Definition 7.1, Lemma 7.3, Proposition 7.4 and Theorem 7.5 with their proofs, and Definition 7.8/Remarks 7.6–7.10 in this cached HTML; opened the primary HTML. Checked the canonical cdh identification and filtered use in the current EM Picard-square argument. No finiteness condition on the field extension is required. Deeper cited foundations are imported; earlier broader reading records are retained as history.

Navarro, On the higher Riemann–Roch without denominators · Navarro §§2.1–2.2 p4; §2.6 p6, final regular-pair paragraph; Theorem3.1 pp7–8; smooth homotopy purity as cited there

Supported Chern operations are induced by the actual universal map and commute with forgetting support. On regular smooth pairs deformation/purity identifies refined Gysin of one with the refined normal Thom class.

Hypotheses: Only regular immersions between smooth finite-type schemes over the fixed perfect field in the Gysin application. All closed supports allowed for pointed Chern fibers.

Reading limits: Read the supplied arXiv:1901.09337v1 primary PDF text pp.3–8: Example 1.1, §§2.1–2.2, Theorem 2.3/Definition 2.4, §2.6, Theorem 2.7 and Theorem 3.1 with its printed proof/Lemmas 3.2–3.3. Opened the primary URL. The RR theorem and cited universal computations are imported. The current supported-RR background-error qualifications are preserved; no use of the malformed Key formula, Thom expansion or unequal-free-rank assertion. Current assigned-concern recheck: Independently read the cached arXiv:1901.09337v1 cover and PDF pp.3–8: orientation and Example 1.1, §§2.1–2.2, Theorem 2.3/Definition 2.4, §2.6, Theorem 2.7 and Theorem 3.1 with its printed argument/Lemmas 3.2–3.3. Only the supported-map construction and the final regular-pair paragraph of §2.6 are inputs here. Preserve the existing source-error qualifications: no use of the malformed Key formula, Thom expansion or unequal-free-rank assertion. The RR theorem and its deep prerequisites are not reconstructed. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/tmp/third-point-navarro-2019.pdf; SHA-256 60c56daa6d43ef1d90a6bd589a67f8953b1275cb699eb504b516190f656a1b63. Accessible primary-page extract: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/Navarro-primary-extract.pdf; SHA-256 dbfe0e1294e1862a39426e37970cc49a9119e5e7b44a6aa2d6d8ea807aeb331b.

Morel–Voevodsky, A¹-homotopy theory of schemes, Publications Mathématiques IHES90(1999) · §3.2 Theorem2.23, printedp115/PDFp72; canonical blowup/purity maps in its proof; Proposition2.24

There is a canonical pointed motivic equivalence X/(X−Z)≃Th(N), given by the two specified deformation/blowup maps. These are maps before applying a coefficient theory.

Hypotheses: Apply the primary smooth homotopy-purity theorem only to closed immersions of smooth separated finite-type schemes over Spec k, for a perfect field k. This is a sufficient subscope of its stated smooth finite-type setting. Nonseparated smooth pairs in the application are handled on affine opens and then by descent.

Reading limits: Read the supplied Numdam primary PDF text: finite-type smooth-site convention, printed pp.94–95/PDF pp.51–52; complete Theorem 2.23 statement, its canonical blowup maps and Proposition 2.24 at printed p.115/PDF p.72. Remaining local-coordinate proof imported. No fresh live Numdam PDF reading is claimed. Current assigned-concern recheck: Independently read the cached Numdam journal PDF cover, printed pp.94–95 (PDF pp.51–52) and printed p.115 (PDF p.72). Read the complete Theorem 2.23 statement, its two canonical blowup/deformation maps and Proposition 2.24; visually inspected printed p.115 from the primary PDF as well as its extracted text. The remaining local-coordinate proof on following pages is imported as an established theorem, not reconstructed. The live web PDF still exceeded the browser size limit; shell DNS access also failed. Neither failure prevented reading the cached primary pages. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/integral-field-model/MV.pdf; SHA-256 170d4a97b4ac3d0b11319b7bdbc04060e6c6daf93fbd1662534d47ecd2d63216. Accessible primary-page extract: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/MV-primary-extract.pdf; SHA-256 147fe0887f18781d1d41f200cd988f0f17e011ee8442bd3e47347a352f0cf3fe.

Tom Bachmann, The very effective covers of KO and KGL over Dedekind schemes, EMS online-first PDF, DOI 10.4171/JEMS/1458 · Online-first PDF p.2: Theorem 1.1(1), Remark 1.2 and Example 1.3; p.3: Lemma 2.1 with proof and proof of Theorem 1.1(1)

The canonical framed vector-bundle map identifies its suspension with f0KGL under the hypothesis. The actual rank-induced Bott-cofiber map to framed integers, identified with Spitzweck HZ, is an equivalence. Bott periodicity identifies f0KGL/beta with s0KGL.

Hypotheses: Theorem1.1(1): f1(HZ)=0; explicitly satisfied for schemes essentially smooth over a Dedekind scheme, including Spec Z. No 1/2 or p-inversion in the KGL assertion. The 1/2 condition appears only in the separate KO assertion. Lemma2.1 is stated for schemes generally.

Reading limits: Read the supplied EMS online-first primary PDF, DOI 10.4171/JEMS/1458, PDF pp.2–3: Theorem 1.1(1), Remark 1.2, Example 1.3, Lemma 2.1 with proof, and proof of Theorem 1.1(1). Opened the primary URL. Its perfect-field reduction is an established external input; no separate final journal layout was read. The separate KO hypothesis involving 1/2 is not used. Current assigned-concern recheck: Independently read the cached EMS online-first primary cover (DOI 10.4171/JEMS/1458), PDF pp.2–3: Theorem 1.1(1), Remark 1.2, Example 1.3, Lemma 2.1 with proof and the proof of Theorem 1.1(1). No separate final journal layout or deep referenced perfect-field proof was read. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/integral-field-model/Bachmann-JEMS.pdf; SHA-256 b026a33d1b2926caf111c241a31e25a5ffbc0106f2a41927cc9ae43991ac7bad. Accessible primary-page extract: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/Bachmann-primary-extract.pdf; SHA-256 bfade953963aaffc77e5e2fc8f8dc1648eb4a4356f01f603131ae09f042ea27c.

Marc Hoyois, The localization theorem for framed motivic spaces, Compositio157(2021),1–11 · Theorem18 and §4 introduction PDFp7; Lemma20 and Theorem21 with full proof PDFpp8–9

There is a natural equivalence of motivic E∞ rings HZ_S≃gamma_*Sigma_infty_fr Z_S. It is constructed by adjunction from the ring isomorphism Omega_infty HZ≃Z, hence preserves the actual unit, and is functorial in S.

Hypotheses: Every base scheme S. HZ is explicitly Spitzweck’s absolute motivic spectrum. Integral coefficients throughout Theorem21; no characteristic inversion.

Reading limits: Read the supplied author primary PDF dated July 2, 2020, PDF pp.7–9: Theorem 18 with proof, §4 setup, Lemma 20, and Theorem 21 with complete proof and continuation. Opened the primary URL. Framed recognition and base-change foundations are imported, not reconstructed; no separate journal layout was read. Current assigned-concern recheck: Independently read the cached author primary PDF cover (dated July 2, 2020) and PDF pp.7–9: Theorem 18 with proof, §4 setup, Lemma 20 with proof and Theorem 21 with its complete proof/continuation. The framed recognition and base-change foundations are imported; no separate journal layout was read. Primary cache: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/integral-field-model/Hoyois-framedgluing.pdf; SHA-256 7cc948428a46879bdfdb7cb39f1fa3697d88a0a67d2fb1d6ab93d17579ee6641. Accessible primary-page extract: /home/lazyinspirit/Projects/workflow-engines/.research-workflow/vector-bundles-2610-22679-20261109/scratch/purity-source-resolution/Hoyois-primary-extract.pdf; SHA-256 54140cef1212536cf5471921548469245683aed1912c8b715e3f1a79c6b3366c.

Sources

Spitzweck, A commutative P¹-spectrum representing motivic cohomology over Dedekind domains, arXiv1207.4078v3 · Lemma6.6 and Theorem6.7 pp55–56; Theorem9.16/Lemma9.17 p77; Proposition11.1/Remark11.2 p79
Elmanto–Morrow, Motivic cohomology of equicharacteristic schemes, arXiv2309.08463v1 · Theorem 3.5/Remark 3.6, printed pp.14–16; Constructions 4.34–4.36/Remark 4.37, pp.31–32; Lemma 5.1/Definition 5.2, pp.33–34; Lemmas 5.10/5.12 and Construction 5.11, p.36; Theorem 6.1(2), Remark 6.3/Corollary 6.4 and footnote 12, pp.44–45
Bachmann–Elmanto–Morrow, A¹-invariant motivic cohomology of schemes, arXiv2508.09915v1 · Proposition 3.15/Definition 3.16; Remarks 3.21/3.23; Lemma 3.27 proof (3.28); §4.1.2; Definitions 4.15/4.17, Remarks 4.18/4.20–4.23, Theorem 4.25/Remark 4.28; Lemma 4.33 and Construction 4.35/(4.36); Theorem 4.39/Remark 4.45; Corollaries 1.11/6.26; Theorem 6.3 coefficient-comparison paragraphs and (6.4) qualification; Proposition 7.4, Theorem 7.5/Remarks 7.6–7.10; Theorem 7.12/(7.14); Corollary 9.7/(9.8)
Navarro, On the higher Riemann–Roch without denominators · Navarro §§2.1–2.2 p4; §2.6 p6, final regular-pair paragraph; Theorem3.1 pp7–8; smooth homotopy purity as cited there
Morel–Voevodsky, A¹-homotopy theory of schemes, Publications Mathématiques IHES90(1999) · §3.2 Theorem2.23, printedp115/PDFp72; canonical blowup/purity maps in its proof; Proposition2.24
Tom Bachmann, The very effective covers of KO and KGL over Dedekind schemes, EMS online-first PDF, DOI 10.4171/JEMS/1458 · Online-first PDF p.2: Theorem 1.1(1), Remark 1.2 and Example 1.3; p.3: Lemma 2.1 with proof and proof of Theorem 1.1(1)
Marc Hoyois, The localization theorem for framed motivic spaces, Compositio157(2021),1–11 · Theorem18 and §4 introduction PDFp7; Lemma20 and Theorem21 with full proof PDFpp8–9