Snake lemma

The snake lemma is a fundamental tool in homological algebra. This branch of mathematics studies the relationships between different algebraic structures. Mathematicians use the lemma to construct long exact sequences. An exact sequence is a chain of objects and maps where the output of one map perfectly matches the input of the next. The lemma is valid in any abelian category. Examples of these categories include the category of abelian groups or the category of vector spaces over a field.

To understand the mechanism, consider a commutative diagram with two rows. Each row is an exact sequence, meaning the maps flow into each other without gaps or overlaps. The diagram also features three vertical maps, often labeled $a$, $b$, and $c$. These vertical maps connect the top row to the bottom row. The lemma focuses on the kernels and cokernels of these three maps. A kernel consists of all elements that a map sends to zero. A cokernel represents the remaining part of a structure after a map has been applied.

The lemma's most important result is the existence of a connecting homomorphism. This map, often called $d$, bridges the gap between the kernel of $c$ and the cokernel of $a$. This connection completes a long exact sequence. The process of finding this map is called a diagram chase. In the case of abelian groups, you pick an element in the kernel of $c$. You then move through the diagram using the horizontal maps. You must find an element that lands in the correct spot in the cokernel. This specific path creates the link between the two ends of the sequence.

There are specific properties regarding the types of maps involved. If the middle vertical map $f$ is a monomorphism, then the map from the kernel of $a$ is also a monomorphism. A monomorphism is a type of map that is injective, meaning it does not collapse different elements into the same one. Similarly, if the map $g$ is an epimorphism, then the map to the cokernel of $c$ is also an epimorphism. An epimorphism is a surjective map, meaning it covers the entire target set. These details ensure the resulting long sequence maintains its mathematical integrity.

Historically and theoretically, the lemma is known for its naturality. This means that the sequence produced by the lemma behaves predictably when you change the maps in a consistent way. If you have a diagram of two different commutative diagrams with exact rows, the snake lemma can be applied to both. This results in two long exact sequences that are related to each other. This relationship is captured in a larger commutative diagram. This property is vital for researchers working in complex fields like algebraic topology.

One notable application involves the tensor product. In the category of vector spaces, the tensor product is a way to combine spaces. When you apply a tensor product to a short exact sequence, the result might not be exact. This is known as the failure of the tensor product to be exact. The snake lemma provides a way to measure this failure. By applying the lemma, mathematicians can construct a sequence that accounts for the missing pieces. This helps explain exactly why the original sequence lost its exactness.

However, the lemma does not work in every mathematical setting. It can fail in the category of groups depending on how you define a cokernel. If you use the standard categorical definition of a cokernel for groups, the resulting sequence might only be a chain complex. A chain complex is a sequence where the compositions of maps are zero, but it is not necessarily exact. For example, using the alternating group $A_5$ can provide a counterexample. In such cases, the sequence fails to be exact because the middle column is not exact. To make it work in groups, one must use right cosets instead of cokernels. This turns the sequence into a sequence of pointed sets rather than groups.