Diagram chasing is a method of mathematical proof used especially in homological algebra. Homological algebra studies, in particular, the homology of chain complexes in abelian categories – therefore the name. From a modern perspective, homological algebra is the study of algebraic objects, (such as groups, rings or Lie algebras, or sheaves of such objects), by ‘resolving them’, replacing them by more stable objects whose homotopy category is the derived category of an abelian category. Given a commutative diagram, a proof by diagram chasing involves the formal use of the properties of the diagram, such as injective or surjective maps, or exact sequences. A syllogism is constructed, for which the graphical display of the diagram is just a visual aid. It follows that one ends up "chasing" elements around the diagram, until the desired element or result is constructed or verified. Examples of proofs by diagram chasing include those typically given for the Snake Lemma, Four Lemma, Five Lemma, Nine Lemma, and Zig-Zag Lemma.
Formally, an exact sequence is a sequence of maps
between a sequence of spaces which satisfies
where denotes the image and the group kernel. That is, iff for some . It follows that The notion of exact sequence makes sense when the spaces are groups, modules, chain complexes, or sheaves. The notation for the maps may be suppressed and the sequence written on a single line as
- A short exact sequence:
beginning and ending with zero, meaning the zero module .
- A long exact sequence:
Special information is conveyed when one of the spaces is the zero module. For instance, the sequence
is exact iff the map is injective. Similarly,
is exact iff the map is surjective.
In homological algebra, given a short exact sequence of -modules, there is a canonical long exact sequence
where the are certain "connecting homomorphisms" (or "snake maps"). This can be deduced from the Snake Lemma. For the proof the latter, one can engage in "diagram chasing". One can see, in the movie It’s My Turn (1980) a proof given for the Snake Lemma using diagram chasing. To define : given , lift " to , push it into by , then check that the image has a preimage in . Then verify that the result is well-defined, et cetera.
with exact rows. We wish to prove that the sequence
First we claim that if any square
is commutative, then there are well-defined morphisms and . For example, if , then the square
must commute, and so the image of in the top row must be in . The proof of the claim for cokernels is similar. Thus we have two sequences,
each of which inherits being a complex from the original diagram.
Suppose is sent to . By exactness, has a preimage . Because the diagram
is commutative and the bottom morphism is injective, and so . So the sequence
is exact. The proof of the claim for the cokernel sequence is similar.
So now all we need to do is find a connecting morphism such that the resulting sequence is exact at both of those points.
Suppose . Then has at least one preimage in . So let and be preimages of . Thus and so by exactness has a preimage . By commutativity of the diagram, has a preimage , which is unique by injectivity of the morphism . But we know that the square
is commutative. We wish to define by . Observe that
and so the choice of preimage of does not affect which cokernel element we ultimately select. So now we have our connecting morphism. By applying this definition we see that
is a complex.
Suppose is sent to by the connecting morphism. Thus we have a diagram
which is commutative. Let be the image of under the morphism . Exactness of the diagram implies that is a preimage of . But . So the kernel-cokernel sequence is exact at . The proof that it is exact at is similar.