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FALSE: the snake lemma is just a pair of short exact kernel and cokernel rows
Statement
The snake lemma says nothing beyond the separate kernel-row and cokernel-row statements for a morphism of short exact sequences.
Facts & Assumptions
Given: A morphism of short exact sequences.
The kernel and cokernel rows are only partially exact on their own (The kernel row and cokernel row of a morphism of short exact sequences are exact at two nodes each).
The kernel row need not be short exact (The kernel row of a morphism of short exact sequences need not be short exact).
The snake lemma adds the connecting morphism and the missing middle exactness (Snake lemma in an abelian category).
Refutation
The theorem [L1] only gives two-node exactness for the kernel row and two-node exactness for the cokernel row, and [L2] shows that nothing stronger is automatic.
By contrast, [L3] produces the connecting morphism and the exactness through it. So the snake lemma contains strictly more information than the separate kernel-row and cokernel-row statements.
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
14 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
- Saunders Mac Lane, Categories for the Working Mathematician, Lemma VIII.4.5 (standard reference, not scraped)