nLab
diagram chasing

Context

Diagram chasing lemmas

Homological algebra

homological algebra

and

nonabelian homological algebra

Context

Basic definitions

Stable homotopy theory notions

Constructions

Lemmas

diagram chasing

Homology theories

Theorems

Contents

Idea

Diagram chasing is a common technique in homological algebra for proving properties of and constructing morphisms in abelian categories, where one traces elements in various ways around commutative diagrams.

Examples

Many basic lemmas in homological algebra, such as the five lemma, the 3x3 lemma and the snake lemma, are typically proven by diagram chases. See for instance the proof at five lemma or any book on homological algebra.

The salamander lemma can sometimes be used to give more conceptual proofs.

Diagram chases in general abelian categories

There are at least five approaches to performing diagram chases in general abelian categories (not assumed to be concrete like Ab or RRMod), roughly listed in increasing order of sophistication:

  1. Directly use the universal properties of the objects involved (kernels, cokernels etc.).
  2. Use equivalence classes of generalized elements to mimic naive (element-based) proofs, as detailed at element in an abelian category. These can only be used to show properties of already-given morphisms and not to construct morphisms (for example the connecting homomorphism of a short exact sequence) by giving their action on elements.
  3. Use genuine generalized elements, see element in an abelian category. With these, the naive way of constructing morphisms ABA \to B (first showing that for every aAa \in A there exists a certain bBb \in B, then showing that this is independent of any choices made) carries over to the general setting.
  4. Interpret constructive element-based proofs written in the fragment of regular logic using categorical semantics, see internal logic. The naive way of constructing morphisms works because the “axiom of unique choice” is valid in abelian categories (see Johnstone, Prop. D1.3.12).
  5. Use the Freyd-Mitchell embedding theorem.

References

Revised on September 18, 2012 23:41:52 by Urs Schreiber (82.169.65.155)