Alphabeta MathHow 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.
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These labels describe origin, not correctness: citations and verification chips remain separate evidence.
Homology object of a chain complex
Definition
Let be a chain complex and let be the canonical morphism supplied by The boundary subobject factors through the cycle subobject.
The th homology object of is the cokernel
Equivalently, one writes with the understanding that this quotient notation abbreviates the cokernel of the canonical monomorphism from boundaries to cycles.
Depends on
Used by
- A two-term complex has kernel and cokernel homology Example
- A zero-differential complex has homology equal to each term Example
- A chain map induces a well-defined map on homology Theorem
- A complex is exact at n exactly when its nth homology is zero Theorem
- An exact functor commutes with homology Theorem
- Euler-Poincare formula for finite free complexes Theorem
Dependency tree · two levels
8 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)