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Finite-rank detection of stable pointed motivic classes

PDF p. 17: Corollary 6.9; stable universal Chern and support comparison prerequisite

Statement

Assume the Axiom of Choice and the fixed site and universe bounds. Let $k$ be a perfect field, let $B_{n} = B G L_{n, k}$ be pointed by the trivial bundle, and use the stabilization $E \mapsto E \oplus O$ . Let $B_{\infty} = {colim}_{n \geq 1} B_{n}$ in pointed local sheaves on the smooth finite-type $k$ -site. For $i \geq 1$ , let $Y_{i} = Ω^{\infty} (Z (i)_{c l} [2 i])$ be the derived target for classical integral motivic cohomology. Then restriction is injective: $[B_{\infty}, Y_{i}]_{*} ⟶ {\underset{\leftarrow}{l i m}}_{n} [B_{n}, Y_{i}]_{*} .$ Here brackets denote homotopy classes of maps in that sheaf category, not maps of groups of bundle classes. The assertion also holds for the actual EM target through the supplied natural smooth-field equivalence. Consequently, two existing stable pointed universal maps which agree at every finite rank admit a homotopy as maps of sheaves. Restricting this one homotopy to a smooth base and then taking derived sections gives homotopies compatible with its restriction arrows. Both Zariski and Nisnevich sites are permitted, with the same derived targets.

Statement references

A unit-induced integral field coefficient zigzag

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

External formal universal Chern maps and supported construction

Annala–Iwasa, April 14, 2025, Corollary 4.4.7

Stacks Project, Lemma 15.88.5

Facts & Assumptions

[A1]

Assume The Axiom of Choice and the supplied universes. Choice is inherited from the coefficient constructions and permits the recursive choices in step 3.1.

[F1]

The supplied integral field zigzag and actual Picard square identify the classical and actual EM coefficient targets as natural derived targets on the smooth site. They retain products, the unit and the actual orientation. The universal-Chern supplier supplies the existing stable pointed maps and their common source model; no uniqueness of their coherent stabilization is assumed here. A unit-induced integral field coefficient zigzag The actual EM comparison commutes with the Picard first-Chern map External formal universal Chern maps and supported construction

[F2]

Universal Chern cohomology and countable derived limits supplies the cohomology computation of Annala–Iwasa, April 14, 2025, Corollary 4.4.7, with its hypotheses and reading limits in the source record. Applied to classical integral $H Z$ , it gives $H^{*, *} (B_{n}) = H^{*, *} (k) [⁣ [c_{1}, \dots, c_{n}] ⁣]$ , with the actual universal coefficients. Its Lemma 4.4.3 supplies Whitney. The source's qcqs smooth convention includes nonseparated schemes. This is an established external computation, not a local reconstruction.

[F3]

The stable interface supplies derived mapping spectra, coherent limits and products, and homotopy exactness of a fiber sequence. For the countable tower of derived abelian mapping complexes used below, Universal Chern cohomology and countable derived limits supplies Stacks Project Definition 13.34.1 and Lemma 15.88.5 supply the product-difference triangle and its exact sequence; Lemma 15.88.1 describes its $R^{1} \underset{\leftarrow}{l i m}$ term. Stable spectra and exact triangles Homotopy exactness of a spectral fiber sequence

Provenance of application

The universal Chern computation and countable derived-limit results are imported through Universal Chern cohomology and countable derived limits. The detection statement and numbered application argument are locally authored (ai-generated); they do not reconstruct the imported proofs. The cited literature is not asserted to state the detection lemma.

Proof

Given: $k$ , $i \geq 1$ , the pointed universal stacks and the derived coefficient targets on the fixed smooth site.

Goal: Remove the possible inverse-limit obstruction to comparing two already constructed stable maps.

1.1F1F2

Zariski local sheafification of the simplicial classifying presheaf $B G L_{n}$ is ${V e c t}_{n}$ : a vector bundle is Zariski locally trivial and its automorphisms are the invertible matrices. Thus [F2] computes the mapping cohomology of our actual source. Under stabilization, Whitney gives $c_{j} \mapsto c_{j}$ for $j \leq n$ and $c_{n + 1} \mapsto 0$ . Consequently restriction on cohomology is evaluation at $c_{n + 1} = 0$ , which is surjective in every fixed bidegree: lift a series by the identical expression in the first $n$ variables. Evaluation at the trivial bundle is its constant coefficient. Removing that split constant summand shows that restriction on the reduced groups ${\tilde{H}}^{2 i - 1, i} (B_{n + 1}) \to {\tilde{H}}^{2 i - 1, i} (B_{n})$ is also surjective. The classical coefficient target is Nisnevich local, so adjunction for sheafification computes the same mapping groups on the Nisnevich site.

2.1F3step 1.1

Let $T_{n}$ be the derived spectrum of pointed maps from $B_{n}$ to the coefficient spectrum $H (Z (i)_{c l} [2 i])$ ; equivalently use reduced free abelian chains of $B_{n}$ and their derived Hom complex into $Z (i)_{c l} [2 i]$ . Then $Ω^{\infty} T_{n} = {Map}_{*} (B_{n}, Y_{i})$ and $π_{1} T_{n} = {\tilde{H}}^{2 i - 1, i} (B_{n})$ . The stabilized general-linear source of [F1] is this colimit: each simplex is a finite tuple of matrices and lies in some $G L_{n}$ , so filtered colimits commute with the finite products in the nerve, and local sheafification preserves this colimit. The actual coherent colimit defining $B_{\infty}$ then gives ${Map}_{*} (B_{\infty}, Y_{i}) ≃ Ω^{\infty} {\underset{\leftarrow}{l i m}}_{n} T_{n}$ . In this enhanced setting the product-difference triangle of [F3], or its derived-complex version, therefore gives the exact sequence $\prod_{n} π_{1} T_{n} \overset{1 - s}{\to} \prod_{n} π_{1} T_{n} \to π_{0} {\underset{\leftarrow}{l i m}}_{n} T_{n} \to \prod_{n} π_{0} T_{n}$ , where $s ((a_{n}))_{n} = f_{n} (a_{n + 1})$ for the restriction maps $f_{n}$ . This uses genuine coherent mapping objects before taking homotopy classes.

3.1A1F3step 1.1step 2.1

The first arrow in step 2.1 is surjective. Given $(b_{n})$ , set $a_{1} = 0$ and recursively choose $a_{n + 1}$ with $f_{n} (a_{n + 1}) = a_{n} - b_{n}$ , using the surjectivity in step 1.1 and [A1]. Then $(1 - s) (a_{n}) = (b_{n})$ . Exactness now makes $π_{0} {\underset{\leftarrow}{l i m}}_{n} T_{n} \to \prod_{n} π_{0} T_{n}$ injective; its image lies in the compatible inverse limit. This proves the claimed injectivity and kills precisely the potential $R^{1} \underset{\leftarrow}{l i m}$ ambiguity, rather than assuming compatible finite-rank homotopies.

4.1F1step 2.1step 3.1∎

Two existing stable maps with equal finite-rank restrictions consequently belong to the same component of ${Map}_{*} (B_{\infty}, Y_{i})$ , so there is a pointed homotopy in the sheaf category itself. The splitting into the basepoint value and the reduced mapping object also shows that finite-rank equality of their unpointed coefficient classes suffices when both maps are pointed. Transport by the equivalence in [F1] proves the assertion for the EM target. A homotopy of sheaf maps restricts as a single natural transformation to the site over any smooth base, hence gives a homotopy of its entire restriction diagram after derived sections. This claims existence, not a contractible space of choices, and uses no affine- $K_{0}$ detection.

Depends on

Used by · 1

Projective relations transport the actual universal Chern operations

Dependency tree · 3 levels

Arrows run from prerequisites to the items that use them. External source nodes link directly to the cited text. The view shows up to three dependency steps; open a prerequisite to continue.

External dependencies

Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · §2.1.1–2.1.4 p. 14; Lemma 3.2.3 p. 23; Definition 4.4.1 and Lemma 4.4.3 pp. 30–31; Corollaries 4.4.5 and 4.4.7 and Theorem 4.4.6 pp. 31–32

Corollary 4.4.7 identifies the bigraded cohomology of Vect_n with formal series over base cohomology in its actual universal Chern classes. Lemma 4.4.3 identifies their pullbacks under addition of a trivial line.

Hypotheses: A homotopy-commutative oriented motivic E1-ring spectrum satisfying the projective bundle formula, in the multiplicative category of stacks over the qcqs base Spec(k). Apply to classical integral HZ; its orientation and PBF are those in the supplied field comparison. Smooth means qcqs smooth and does not require separatedness.

Reading limits: Read the cached primary PDF at these passages, including the proof of Theorem 4.4.6 and the adjacent corollaries; opened the live primary URL to check the version. The deeper lifting and categorical results used in that proof are accepted as established inputs, not reconstructed.

Stacks Project, derived limits and the Milnor exact sequence · Lemma 15.88.5 (tag 0919), statement and proof; Definition 13.34.1 (tag 08TB); Lemma 15.88.1(2)–(3) (tag 07KW)

The kernel of Hom(L,Rlim T_n) to lim Hom(L,T_n) is R^1 lim Hom(L,T_n[-1]). For a surjective countable system this derived-limit obstruction vanishes.

Hypotheses: A countable inverse system with its derived limit in a triangulated category with the required products; here the derived mapping complexes are in D(Ab), with a genuine enhanced limit.

Reading limits: Read the statements and proofs of Lemmas 15.88.1 and 15.88.5 and the product-difference definition in 13.34.1 on the primary live site. Only these results are imported; no assertion that an arbitrary diagram in a homotopy category determines canonical coherent data is made.

Stacks Project, derived limits: product-difference triangle · Definition 13.34.1

A derived limit fits into the distinguished triangle from the limit to the countable product, then by one minus the transition shift to that product, then to the shifted limit. Such limits exist when the product exists, including in D(Ab); uniqueness in the triangulated category is only up to a nonunique isomorphism.

Hypotheses: A countable inverse system in a triangulated category with the required product. Here the system comes from actual coherent derived mapping complexes in D(Ab), before passage to the homotopy category.

Reading limits: Read Definition 13.34.1 and its following explanation on the primary live site in this adjudication. Only that definition and explanation are imported; no canonical coherence is inferred from an arbitrary diagram in a triangulated category.

Stacks Project, Rlim of abelian groups · Lemma 15.88.1(2)–(3), statement and proof

For a countable inverse system of abelian groups, the derived limit is represented by the product-difference complex in degrees zero and one. Its first derived-limit group vanishes for a Mittag–Leffler system, in particular for a system with surjective transition maps.

Hypotheses: A countable inverse system of abelian groups; the vanishing assertion assumes the Mittag–Leffler condition. In this application the groups are the first homotopy groups of the derived mapping tower, and step 1.1 supplies surjective transition maps.

Reading limits: Read the statement and proof on the primary live site in this adjudication. Only parts (2)–(3), the product-difference description and Mittag–Leffler vanishing, are imported; the deeper derived-functor prerequisites cited there are not reconstructed.

Sources

Annala–Iwasa, Motivic spectra and universality of K-theory, author PDF dated April 14, 2025 · §2.1.1–2.1.4 p. 14; Lemma 3.2.3 p. 23; Definition 4.4.1 and Lemma 4.4.3 pp. 30–31; Corollaries 4.4.5 and 4.4.7 and Theorem 4.4.6 pp. 31–32
Stacks Project, derived limits and the Milnor exact sequence · Lemma 15.88.5 (tag 0919), statement and proof; Definition 13.34.1 (tag 08TB); Lemma 15.88.1(2)–(3) (tag 07KW)
Stacks Project, derived limits: product-difference triangle · Definition 13.34.1
Stacks Project, Rlim of abelian groups · Lemma 15.88.1(2)–(3), statement and proof