Five lemma

The five lemma is a vital tool in homological algebra. It is used to study how different mathematical structures connect. This lemma is most useful when working with commutative diagrams. A commutative diagram is a collection of objects and maps where following different paths leads to the same result.

The five lemma works in many settings. It is valid for all abelian categories. These include the category of abelian groups and the category of vector spaces. It also works in the category of groups. The lemma provides a way to prove that a specific map is an isomorphism. An isomorphism is a map that is both injective and surjective, creating a perfect one-to-one connection.

To understand the five lemma, one must first understand the four lemmas. The five lemma is actually a combination of these two smaller theorems. The four lemmas are dual to each other, meaning they are mathematical opposites.

One four lemma helps prove that a map is surjective, or an epimorphism. The other four lemma helps prove that a map is injective, or a monomorphism. An epimorphism is a map that covers every element in the target set. A monomorphism is a map where every element in the target has at most one source. By proving both four lemmas, mathematicians can prove the full five lemma.

Mathematicians prove these theorems using a technique called diagram chasing. This method involves tracing the path of individual elements through the diagram. To make this possible, we assume we are working in a category of modules over a ring. In this setting, we can treat maps as functions that act on specific elements. This allows us to use concepts like kernels and images. A kernel is the set of elements that a map sends to zero. An image is the set of all possible results from a map. Even if we are not in a module category, Mitchell's embedding theorem ensures the proof still works. This theorem states that any small abelian category can be represented as a category of modules.

The first part of the proof focuses on surjectivity. We assume that the maps labeled $m$ and $p$ are surjective. We also assume the map $q$ is injective. To show the middle map $n$ is surjective, we start with an element in the target group. We use the commutativity of the diagram to move through the objects. We use the fact that the rows are exact to find specific elements. Exactness means the image of one map is exactly the kernel of the next. By following these steps, we can eventually find a source element for any target element. This proves the map is an epimorphism.

The second part of the proof focuses on injectivity. Here, we assume that $m$ and $p$ are injective. We also assume that $l$ is surjective. We start by picking an element that the map $n$ sends to zero. We then trace this element back through the diagram using the properties of the other maps. Because $p$ is injective, we can conclude certain elements must be zero. Because the rows are exact, we can find elements in the previous groups that map to them. Eventually, we show that the original element must have been zero. This confirms the map is a monomorphism.

The five lemma has many important applications in higher mathematics. It is frequently used when working with long exact sequences. These sequences are often used to compute homology or cohomology. Homology is a way to study the shape and structure of mathematical objects. Often, a mathematician will study a simpler subobject first. They can find the homology of that simple object quite easily.

They can then use a morphism to create a long exact sequence. The five lemma helps them bridge the gap between the known and the unknown. It allows them to determine the unknown homology groups of the original, more complex object.

This tool is also essential for comparing different mathematical theories. For example, it can prove that two different ways of measuring a shape are actually the same. This includes comparing simplicial homology to singular homology. It can also be used to show that de Rham cohomology and singular cohomology coincide. By using the five lemma, mathematicians can ensure that different mathematical languages are describing the same underlying truths. It acts as a bridge between different ways of seeing the same mathematical world.