DefinitionDefinition: Literature-sourcedProof: Not applicablejudge pass (gpt-5.6-terra)audited 2026-08-30

How statement and proof provenance work

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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.

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Zero complex and stalk complex

Definition

Fix a zero object $0$ of an abelian category (Initial object, terminal object, and zero object).

The zero complex is the chain complex with $C_{n} = 0$ for every $n$ and every differential equal to the zero endomorphism of $0$ .

For an object $A$ and an integer $n$ , the stalk complex $S^{n} (A)$ is the chain complex with $S^{n} (A)_{k} = {\begin{cases} A, & k = n, \\ 0, & k \neq n, \end{cases}$ and every differential zero. It is concentrated in degree $n$ .

Depends on

Used by

A three-term zero-composite diagram that is not distinguished Counterexample
A zero-differential complex has homology equal to each term Example
An octahedron for two composable maps of stalk complexes Example
Euler class of a two-term mapping cone Example
The cone triangle of multiplication by m Example
Collapse solves all extension problems False statement
Every triangulated subcategory is thick False statement
The third map in a morphism of triangles is unique False statement
Associator, symmetry and unitors of the abelian sheaf tensor product Lemma
Koszul coherence of derived sheaf tensor Lemma
Morphisms from the constant sheaf are global sections Lemma
Sheaf cohomology classes as derived morphisms Lemma
Collapse does not in general split the abutment Proposition
G0 of an abelian category equals triangle K0 of its bounded derived category Theorem
Triangle K0 of perfect complexes equals split K0 of finite projectives Theorem

Dependency tree · two levels

5 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

Charles A. Weibel, Chapter 1 of An Introduction to Homological Algebra (standard reference, not scraped)
The Stacks Project, Section 12.13: Complexes (standard reference, not scraped)