LemmaStatement: AI-generatedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

Actual EM coefficient mapping and smooth-site comparison

PDF p. 17: Corollary 6.9; coefficient mapping and topology bridge for stable universal detection

Statement

Assume the Axiom of Choice and compatible nested Grothendieck universes containing $ω$ , with the site and coefficient size bounds of the supplied models. Let $k$ be a field and let $C$ be the smooth finite-type $k$ -site, allowing nonseparated schemes. Write $S_{τ} = {Shv}_{τ} (C, S p a c e s)$ for space-valued sheaves with covering-sieve descent, for $τ = Z a r$ or $N i s$ , and $a_{τ}$ for sheafification. Use the actual EM model $E_{k}$ and its coefficient spectra $A_{i} (U) = H (Z (i)_{m o t} (U) [2 i])$ for $i \geq 0$ . Here $H$ is Eilenberg–Mac Lane realization, with $π_{r} H C = H^{- r} (C)$ and $H^{a} (C [b]) = H^{a + b} (C)$ .

Let $D$ be the supplied native qcqs smooth derived $k$ -site. Discrete realization $j : C ≃ D$ identifies the Zariski sites and their coefficient, pointed-stack and motivic constructions by the supplied native-field-site comparison. Throughout, ${\tilde{E}}_{k}^{p, q} (X)$ for $X \in S_{Z a r, *}$ means native cohomology of the corresponding stack $(j^{*})^{- 1} X$ ; the coefficient levels on $C$ are the restrictions $j^{*} A_{i}$ . This specifies the site comparison before using the native suspension/evaluation adjunction.

For a pointed space-valued presheaf $P$ , define the ordinary mapping spectrum $T_{i} (P) = {map}_{Fun (C^{o p}, S p)} (Σ^{\infty} P, A_{i}),$ where $Σ^{\infty}$ is reduced circle suspension, applied objectwise. Then $A_{i}$ has Nisnevich descent, and there are natural identifications $Ω^{\infty} T_{i} (P) ≃ {Map}_{S_{τ, *}} (a_{τ} P, Ω^{\infty} A_{i}), π_{r} T_{i} (P) ≅ {\tilde{E}}_{k}^{2 i - r, i} (a_{Z a r} P) (r \geq 0) .$ Thus the same mapping spectrum computes the pointed target maps for both topologies, in particular their degrees zero and one. For an unpointed presheaf, use its disjointly pointed version in this formula. If it already has a chosen point, evaluation there splits off $A_{i} (k)$ from the unpointed mapping spectrum, leaving $T_{i} (P)$ as the reduced summand.

Let $P_{n} (U) = B (G L_{n} (Γ (U, O_{U})))$ , pointed at its sole object; $G L_{n}$ is the group of invertible $n$ -by- $n$ matrices, and $B$ is its nerve, whose degree- $q$ simplices are $q$ -tuples of matrices. Then $a_{Z a r} P_{n} = {V e c t}_{n, k}$ , the stack of rank- $n$ vector bundles and isomorphisms pointed by $O^{n}$ . Consequently the bigraded cohomology computation for this actual oriented PBF model computes these mapping groups in degrees zero and one, on both sites. All these identifications preserve source stabilization by addition of a trivial line and restriction arrows. No comparison theorem limited to separated schemes is used.

Statement references

An actual non-A¹ P¹-spectrum for the supplied EM complexes

Actual EM orientation and projective bundle maps on that model

External symmetric P¹ model, orientation criterion and fundamental stabilization

Elmanto–Morrow motivic complex input

Source-specific non-A¹ motivic spectrum framework and hypotheses

Nisnevich covers and distinguished squares

Stable spectra and exact triangles

The Axiom of Choice

Grothendieck universe closure convention

Classical realization of the native smooth field site

Facts & Assumptions

[A1]

Assume The Axiom of Choice and Grothendieck universe closure convention, including the separate existence and size bounds stated above. These are inherited by the coefficient, spectral and sheaf constructions; no choice-free version is asserted.

[F1]

The actual model has level $i$ equal to $Ω^{\infty} A_{i}$ , and the actual orientation satisfies PBF. These are the model and orientation, not an abstract cohomology theory. An actual non-A¹ P¹-spectrum for the supplied EM complexes Actual EM orientation and projective bundle maps on that model

[F2]

The motivic suspension/evaluation adjunction and the projective-line shift evaluate $Σ_{P^{1}}^{i} E_{k}$ at level zero as $Ω^{\infty} A_{i}$ . The framework also supplies circle stabilization and its adjunction, reduced cohomology, the smooth site and the vector-bundle stack; ordinary mapping spectra and their homotopy groups use the stable interface. External symmetric P¹ model, orientation criterion and fundamental stabilization Source-specific non-A¹ motivic spectrum framework and hypotheses Stable spectra and exact triangles

[F3]

The actual complexes are coherent derived Nisnevich sheaves on all size-bounded qcqs field-schemes, hence on this smooth site, including its nonseparated objects. The supplied geometric clause makes every smooth finite-type scheme over a field Noetherian and qcqs, without separatedness, so this restriction is within that scope. The Nisnevich covering convention includes Zariski covers. Elmanto–Morrow motivic complex input External actual augmented EM and smooth slice tower comparison Nisnevich covers and distinguished squares

[F4]

Reference 0 supplies sheafification as a reflective localization of space-valued presheaves for a small site, with sheaves a full subcategory; its sheafification is left exact. Apply this separately to the two topologies on the same bounded category.

[F5]

Reference 1 identifies the unbounded derived category of abelian groups with modules over $H Z$ , and says that module limits are created on underlying spectra. Use its canonical Eilenberg–Mac Lane realization with cochain grading; it preserves coherent limits and has $π_{r} H C = H^{- r} (C)$ .

[F6]

The comparison Classical realization of the native smooth field site gives the actual equivalence $j : C ≃ D$ over the discrete field, including nonseparated schemes. It transports arbitrary presheaves, pointed Zariski sheafification, coefficient levels, the usual pointed projective line and its spectrum suspension/evaluation and shift rules. It also identifies the native vector-bundle stack with the classical one, preserving its trivial point, addition of a trivial line and restriction maps. Its two geometric instances are separate; in spectral geometry it uses the supplied flat, fiber-smooth convention.

External input

The categorical premises in [F4] and [F5] are imported only at the locators and hypotheses in references 0 and 1. Their foundational proofs are not reconstructed. The EM model, orientation, PBF, coefficient descent and motivic evaluation are supplied by [F1]–[F3]. The native-to-classical comparison, including its precise geometric inputs, is supplied by [F6]. The mapping comparison below is a local composition of those inputs; the whole lemma is not an imported statement.

Proof

Given: The field, the two bounded smooth sites, the actual model, $i \geq 0$ , the presheaf $P$ , Choice and the stated universes.

Goal: Identify the coefficient mapping groups and the site change needed by the universal Chern application.

1.1A1F1F2F3F5F6

First use [F6] to transport the native actual model and its adjunctions along $j^{*}$ . This is an equivalence on all presheaves and pointed Zariski sheaves, so the native cohomology of $(j^{*})^{- 1} a_{Z a r} P$ has the meaning fixed in the Statement for every $P$ . By [F3] the complexes have Nisnevich descent; [F5] transports those coherent limits to spectra, and $Ω^{\infty}$ preserves them, so $A_{i}$ and $Ω^{\infty} A_{i}$ are Nisnevich sheaves and hence Zariski sheaves. For Zariski sheaves, the motivic suspension/evaluation adjunction and shift in [F2] identify maps into $Σ_{P^{1}}^{i} E_{k}$ with pointed maps into its actual zeroth level $Ω^{\infty} A_{i}$ . Circle suspension and [F1]–[F2] identify the latter mapping space with the zeroth space of the ordinary mapping spectrum into $A_{i}$ . Taking $π_{r}$ for $r \geq 0$ gives exactly ${\tilde{E}}_{k}^{2 i - r, i}$ by the framework grading, and [F5] gives the cochain sign convention stated above. The model's orientation and PBF are supplied by [F1], independently of this adjunction. The map $P_{n} \to {V e c t}_{n, k}$ sends the sole object to $O^{n}$ and its automorphisms to invertible matrices. It is fully faithful and Zariski locally essentially surjective, since vector bundles are locally free. Because [F6] identifies this target with the native vector-bundle stack of [F2], its descent glues precisely these local free objects and their transition isomorphisms, proving $a_{Z a r} P_{n} = {V e c t}_{n, k}$ .

2.1F2F3F4F5F6step 1.1∎

By [F4], mapping from $a_{τ} P$ into the $τ$ -sheaf $Ω^{\infty} A_{i}$ is mapping from $P$ in presheaves. The adjunction holds for pointed maps by taking the fiber over the specified basepoint. Circle suspension then identifies this with $Ω^{\infty} T_{i} (P)$ . The same reasoning applies to every shift $A_{i} [m]$ , since shifting a derived sheaf preserves its limit diagrams and [F5] preserves these limits. Thus it also identifies the corresponding mapping spectra for the two topologies, without replacing ordinary sheaf descent by hyperdescent. For a presheaf with a chosen point, evaluation at that point has the section given by constant maps pulled back from $Spec k$ . In the stable category this splits the unpointed mapping spectrum into $A_{i} (k)$ and its fiber, which by the pointed suspension adjunction is $T_{i} (P)$ . The site equivalence of [F6] preserves these constructions and their coefficient levels. All maps used are adjunction units, coefficient maps, evaluation, or the fixed trivial-bundle inclusion; hence they commute with source maps, addition of a trivial line and restriction diagrams. Together with step 1.1 this proves the asserted bridge on the full stated site.

Depends on

Used by · 3

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

Jacob Lurie, Higher Topos Theory (2009) · Definition 6.2.2.6 and Proposition 6.2.2.7, printed p. 579; proof of Proposition 6.2.2.7, pp. 582–583 (PDF pages 597, 600–601)

Space-valued sheaves on a small site form a full reflective localization of presheaves, with left exact sheafification.

Hypotheses: A small category with a Grothendieck topology; apply separately to Zariski and Nisnevich topologies on the fixed universe-bounded smooth category.

Reading limits: Freshly read Definition 6.2.2.6, Proposition 6.2.2.7 and its complete displayed proof in the primary PDF extraction for this repair. Import the localization theorem; its transfinite and higher-category foundations are not reconstructed. Only covering-sieve descent is used.

Jacob Lurie, Higher Algebra (18 September 2017) · Theorem 7.1.2.13 and proof, printed pp. 1212–1213; Corollary 4.2.3.3(1)–(2), p. 527

The derived infinity-category of modules over a commutative ring is canonically equivalent to module spectra over that discrete ring spectrum. Module limits are created in underlying spectra.

Hypotheses: Take the commutative ring to be Z and the ambient monoidal category to be spectra. All complexes are unbounded in the fixed universes; translate source homological grading to cochain grading.

Reading limits: Freshly read the full Theorem 7.1.2.13 statement and proof and the complete Corollary 4.2.3.3 statement in the primary PDF extraction for this repair. Import the equivalence and module-limit theorem; their model-category and operadic prerequisites are not reconstructed. Use the supplied cochain convention.

Sources

Jacob Lurie, Higher Topos Theory (2009) · Definition 6.2.2.6 and Proposition 6.2.2.7, printed p. 579; proof of Proposition 6.2.2.7, pp. 582–583 (PDF pages 597, 600–601)
Jacob Lurie, Higher Algebra (18 September 2017) · Theorem 7.1.2.13 and proof, printed pp. 1212–1213; Corollary 4.2.3.3(1)–(2), p. 527