Alphabeta MathHow statement and proof provenance work
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A cartesian square induces an isomorphism on the kernels of its parallel legs
Statement
In a cartesian square in an abelian category
the induced morphism is an isomorphism.
Facts & Assumptions
Given: The displayed cartesian square.
The square is a pullback (A square is cartesian exactly when a short sequence is exact).
In a pullback square, the induced map on the kernels of the parallel legs is an isomorphism (In a pullback square, the induced map on the kernels of the two parallel arrows is an isomorphism).
Proof
Since the square is cartesian, [L1] identifies it as a pullback square.
Applying [L2] to that pullback gives the claimed isomorphism .
Depends on
Used by
Nothing in the library uses this result yet.
Dependency tree · two levels
11 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
- The Stacks Project, Section 12.5, Lemma 12.5.12 (standard reference, not scraped)