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.

Cochain shifts preserve coherent limits and colimits

PDF p. 4: Motivic dimension vanishing

Statement

Let $D (Z)$ be the unbounded derived infinity-category of cochain complexes of abelian groups, with homotopy category $D (Z)$ and fixed universe and mapping-space conventions. For every integer $k$ , the cochain shift $S_{k}$ given on complexes by $(S_{k} C)^{n} = C^{n + k}$ and $d_{S_{k} C}^{n} = (- 1)^{k} d_{C}^{n + k}$ , and on cochain maps by $(S_{k} f)^{n} = f^{n + k}$ , induces an equivalence of $D (Z)$ with inverse $S_{- k}$ . Naturally in $C$ , it satisfies $H^{n} (S_{k} C) = H^{n + k} (C)$ . It preserves every existing small coherent limit and colimit, including zero objects and derived fibers.

Facts & Assumptions

Given: An integer $k$ and the unbounded derived infinity-category in the Statement; smallness is relative to the fixed universes.

Goal: Justify preservation in the coherent target, rather than only in its homotopy category.

[F1]

External stable mapping and derived linear target interfaces supplies the enhanced localization of unbounded complexes at quasi-isomorphisms, its ordinary homotopy category, and the mapping-space universal properties of coherent limits and colimits. Cohomology descends to the derived category supplies cohomology on the homotopy category and the cochain-to-chain reindexing $C_{n}^{♯} = C^{- n}$ ; quasi-isomorphisms induce isomorphisms in every cohomology degree.

[F2]

Shift is an additive autoequivalence of the complex and homotopy categories gives mutually inverse shifts on complexes. Under the reindexing in [F1], its chain shift by $k$ is exactly the signed cochain shift in the Statement.

[F3]

The source record supplies the functor-category universal property of infinity-localization, including natural transformations.

External input

Import only [F3] with the hypotheses and reading limits in the source record. The shift construction and preservation argument below are local deductions from these explicit inputs.

Proof

1.1F1F2F3

By [F2], $S_{k}$ and $S_{- k}$ are inverse functors on complexes. The displayed differential changes only by a sign, so its kernels and images in degree $n$ give $H^{n} (S_{k} C) = H^{n + k} (C)$ , naturally on cochain maps. Consequently both functors preserve quasi-isomorphisms as defined in [F1]. Compose each with the localization functor and apply [F3] to obtain functors on $D (Z)$ . Full faithfulness in [F3] transfers their inverse identities on complexes to inverse natural equivalences on the localization, including their coherence. Cohomology factors through its homotopy category by [F1], so the same natural cohomology identity holds there.

2.1F1step 1.1∎

Write $S = S_{k}$ and $T = S_{- k}$ . For a small coherent diagram $(D_{i})$ with limit $L$ and any object $E$ , the inverse equivalences of step 1.1 and the universal property in [F1] give $Map (E, S L) ≃ Map (T E, L) ≃ \lim_{i} Map (T E, D_{i}) ≃ \lim_{i} Map (E, S D_{i})$ , naturally with the induced cone. Hence $S L$ is its limit. If $Q$ is a colimit, the dual calculation is $Map (S Q, E) ≃ Map (Q, T E) ≃ \lim_{i \in I^{o p}} Map (D_{i}, T E) ≃ \lim_{i \in I^{o p}} Map (S D_{i}, E)$ for a diagram indexed by $I$ . This proves colimit preservation. Empty diagrams give preservation of zero, and the pullback defining a fiber is therefore preserved as well.

Depends on

Used by · 1

Normalizing the comparison fiber for the dimension criterion

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 Algebra, September 18, 2017 · Definition 1.3.4.1, printed p.108, zero-based PDF index 107, including its ordinary-category convention.

Precomposition with an infinity-localization is fully faithful on functor infinity-categories, with essential image the functors carrying the inverted arrows to equivalences.

Hypotheses: An infinity-category and a specified collection of arrows; for an ordinary category use its nerve. All categories are taken in suitable fixed universes.

Reading limits: Read Definition 1.3.4.1 completely, including its ordinary-category convention, in the actual primary PDF text during this assignment. Only this universal property is imported. The existing enhanced-mapping supplier supplies the unbounded derived localization; its existence and model foundations are not reconstructed here.

Sources

Jacob Lurie, Higher Algebra, September 18, 2017 · Definition 1.3.4.1, printed p.108, zero-based PDF index 107, including its ordinary-category convention.