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Naturality of the homology connecting morphism
Statement
A morphism of short exact sequences of complexes induces a commutative square for every .
Facts & Assumptions
Given: A morphism of short exact sequences of complexes.
A morphism of short exact sequences of complexes is a commutative three-column ladder of chain maps (A morphism of short exact sequences of complexes).
The arrow category of an abelian category is abelian, with kernels, cokernels, and the relevant diagrams computed componentwise (The arrow category of an abelian category).
The homology connecting morphism is the connecting morphism attached to the quotient-kernel snake diagram of a short exact sequence of complexes (The connecting morphism in homology).
The weaker Stacks snake construction applies in every abelian category (Snake lemma under the weaker Stacks hypotheses).
Proof
By [L1], each degree of the given ladder is a morphism of short exact sequences. Passing to the quotient-kernel diagrams used on this page therefore gives a morphism between two weaker Stacks snake diagrams. Regard every comparison arrow as an object of the arrow category. By [L2], the resulting diagram is itself weaker Stacks snake data in that abelian category.
Apply [L4] in the arrow category to the diagram from step 1.1. Its connecting morphism is an arrow object whose two components are the connecting morphisms of the original weaker snake diagrams; being a morphism in the arrow category says exactly that the square between those components commutes. Under the identifications in [L3], this is the displayed homology connecting square.
Depends on
Used by
- The long exact homology sequence is natural Corollary
- Naturality of a connecting map under a map of coefficient sequences Example
- FALSE: naturality of the long exact sequence follows without checking the connecting square False statement
- The left derived connecting map is independent of the horseshoe resolution and lifts Lemma
- Natural transformations of base functors give morphisms of derived delta functors Proposition
- Short exact sequences of filtered complexes give compatible exact couples Proposition
- The homology effect of surgery away from the middle dimensions Proposition
- Functoriality and coefficient long exact sequences for Hochschild homology Theorem
- Naturality of de rham mayer vietoris for maps of covered manifolds Theorem
- Naturality of the cohomology connecting morphism Theorem
- The long exact Tor sequence in the left-module variable Theorem
- The long exact Tor sequence in the right-module variable Theorem
Dependency tree · two levels
22 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)