Exact Sequences

In Homology and Cohomology we introduced the idea of chain complexes to help us obtain information about topological spaces. We recall that a chain complex is made up of abelian groups of spaces C_{n} and boundary homomorphisms \partial_{n}: C_{n}\rightarrow C_{n-1} such that for all n the composition of successive boundary homomorphisms \partial_{n-1}\circ \partial_{n}: C_{n}\rightarrow C_{n-2} sends every element in C_{n} to the zero element in C_{n-2}.

Chain complexes can be expressed using the following diagram:

...\xrightarrow{\partial_{n+3}}C_{n+2}\xrightarrow{\partial_{n+2}}C_{n+1}\xrightarrow{\partial_{n+1}}C_{n}\xrightarrow{\partial_{n}}C_{n-1}\xrightarrow{\partial_{n-1}}C_{n-2}\xrightarrow{\partial_{n-2}}...

We now abstract this idea, generalizing it so that the groups C_{n} do not necessarily have to be topological spaces, and show an example of a chain complex that is ubiquitous in mathematics.

First we recall some ideas from Homology and Cohomology. Our “important principle” was summarized in the following statement:

All boundaries are cycles.

Boundaries in C_{n} are elements of the image of the boundary homomorphism \partial_{n+1}. Cycles in C_{n} are elements of the kernel of the boundary homomorphism \partial_{n}. Therefore, we can also state our “important principle” as follows:

\text{Im }\partial_{n+1}\subseteq \text{Ker }\partial_{n} for all n

This is of course just another restatement of the defining property of all chain complexes that two successive boundary functions when composed send every element of its domain to the zero element of its range.

There is an important kind of chain complex with the following property:

\text{Im }\partial_{n+1}=\text{Ker }\partial_{n} for all n

Such a chain complex is called an exact sequence. Sometimes we just say that the chain complex is exact. We will show some simple examples of exact sequences, but for these examples we will drop the notation of the boundary homomorphism \partial_{n} to show that many properties of ordinary functions can be expressed in terms of exact sequences.

Consider, for example, abelian groups A, B, and C. The identity elements of A, B, and C will be denoted by 0, writing 0\in A, 0\in B, and 0\in C if necessary. We will also write 0 to denote the trivial abelian group consisting only of the single element 0. Let us now look at the exact sequence

0\rightarrow A\xrightarrow{f} B

where 0\rightarrow A is the inclusion function sending 0\in 0 to 0\in A. The image of this inclusion function is therefore 0\in A. By the defining property of exact sequences, this is also the kernel of the function f:A\rightarrow B. In other words, f sends 0\in A to 0\in B. It is a property of group homomorphisms that whenever the kernel consists of only one element, the homomorphism is an injective, or one-to-one, function. This means that no more than one element of the domain gets sent to the same element in the range. Since this function is also a homomorphism, it is also called a monomorphism.

Meanwhile, let us also consider the exact sequence

B\xrightarrow{g} C\rightarrow 0

where C\rightarrow 0 is the “constant” function that sends any element in C to 0. The kernel of this constant function is therefore the entirety of C. By the defining property of exact sequences, this is also the image of the function B\rightarrow C. In other words, the image of the function g is the entirety of C, or we can also say that every element of C is assigned by g to some element of B. Such a function is called surjective, or onto. Since this function is also a homomorphism, it is also called an epimorphism.

The exact sequence

0\rightarrow A\xrightarrow{f} B\xrightarrow{g} C\rightarrow 0

is important in many branches of mathematics, and is called a short exact sequence. This means that f is a monomorphism, g is an epimorphism, and that \text{im }f=\text{ker g} in B. As an example of a short exact sequence of abelian groups, we have

0\rightarrow 2\mathbb{Z}\xrightarrow{f} \mathbb{Z}\xrightarrow{g} \mathbb{Z}/2\mathbb{Z}\rightarrow 0

(see also Modular Arithmetic and Quotient Sets). The monomorphism f takes the abelian group of even integers 2\mathbb{Z} and “embeds” them into the abelian group of the integers \mathbb{Z}. The epimorphism g then sends the integers in \mathbb{Z} to the element 0 in \mathbb{Z}/2\mathbb{Z} if they are even, and to the element 1 in  \mathbb{Z}/2\mathbb{Z} if they are odd. We see that every element in \mathbb{Z} that comes from 2\mathbb{Z}, i.e. the even integers, gets sent to the identity element or zero element 0 of the abelian group \mathbb{Z}/2\mathbb{Z}.

In the exact sequence

0\rightarrow A\xrightarrow{f} B\xrightarrow{g} C\rightarrow 0

The abelian group B is sometimes referred to as the extension of the abelian group C by the abelian group A.

We recall the definition of the homology groups H_{n}:

H_{n}=\text{Ker }\partial_{n}/\text{Im }\partial_{n+1}.

We can see from this definition that a chain complex is an exact sequence (we can also say that the chain complex is acyclic) if all of its homology groups are zero. So in a way, the homology groups “measure” how much a chain complex “deviates” from being an exact sequence.

We also have the idea of a long exact complex, which usually comes from the homology groups of chain complexes which themselves form a short exact sequence. In order to discuss this we first need the notion of a chain map between chain complexes. If we have a chain complex

...\xrightarrow{\partial_{A, n+3}}A_{n+2}\xrightarrow{\partial_{A, n+2}}A_{n+1}\xrightarrow{\partial_{A, n+1}}A_{n}\xrightarrow{\partial_{A, n}}A_{n-1}\xrightarrow{\partial_{A, n-1}}A_{n-2}\xrightarrow{\partial_{A, n-2}}...

and another chain complex

...\xrightarrow{\partial_{B, n+3}}B_{n+2}\xrightarrow{\partial_{B, n+2}}B_{n+1}\xrightarrow{\partial_{B, n+1}}B_{n}\xrightarrow{\partial_{B, n}}B_{n-1}\xrightarrow{\partial_{B, n-1}}B_{n-2}\xrightarrow{\partial_{B, n-2}}...

a chain map is given by homomorphisms

f_{n}: A_{n}\rightarrow B_{n} for all n

such that the homomorphisms f_{n} commute with the boundary homomorphisms \partial_{A, n} and \partial_{B, n}, i.e.

\partial_{B, n}\circ f_{n}=f_{n-1}\circ \partial_{A, n} for all n.

A short exact sequence of chain complexes is then a short exact sequence

0\rightarrow A_{n}\xrightarrow{f_{n}} B_{n}\xrightarrow{g_{n}} C_{n}\rightarrow 0 for all n

where the homomorphisms f_{n} and g_{n} satisfy the conditions for them to form a chain map, i.e. they commute with the boundary homomorphisms in the sense shown above.

In the case that we have a short exact sequence of chain complexes, their homology groups will then form a long exact sequence:

...\rightarrow H_{n}(A)\xrightarrow{f_{*}}H_{n}(B)\xrightarrow{g_{*}}H_{n}(C)\xrightarrow{\partial}H_{n-1}(A)\xrightarrow{f_{*}}...

Long exact sequences are often used for calculating the homology groups of complicated topological spaces related in some way to simpler topological spaces whose homology groups are already known.

References:

Chain Complex on Wikipedia

Exact Sequence on Wikipedia

Algebraic Topology by Allen Hatcher

A Concise Course in Algebraic Topology by J. P. May

Abstract Algebra by David S. Dummit and Richard M. Foote

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3 thoughts on “Exact Sequences

  1. Pingback: More on Chain Complexes | Theories and Theorems

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