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Universal Chern cohomology and countable derived limits

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

Statement

For a homotopy-commutative oriented motivic E1-ring spectrum satisfying the projective bundle formula over a qcqs base, Annala–Iwasa Corollary 4.4.7 computes the bigraded cohomology of Vectn as formal series over base cohomology in the universal Chern classes; Lemma 4.4.3 supplies their Whitney behavior, including pullback under addition of a trivial line.

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. For a countable inverse system in a triangulated category with the required products, the derived limit fits into the product-difference triangle, and the associated Milnor exact sequence holds for derived Hom as in Stacks Project Lemma 15.88.5. These statements have the precise hypotheses in the source records.

External input

The Annala–Iwasa computation and Whitney behavior, and the Stacks Project derived-limit statements, are imported from the cited sources with the stated version and reading limits. Their proofs are literature-derived; no local reconstruction is claimed.

Depends on

Used by · 1

Dependency tree · 1 level

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

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.

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