Days ago, I asked on Math.StackExchange the question: Describe, as a direct sum of cyclic groups, the cokernel of the map: , where after some hints and understand modules over PID (see handouts), I finally got an answer. However, the answer suggests me to use Smith Normal Form (SNF), so I decided to understand: What does SNF truly represent?, so let’s start with a basic definition.
Definition: Let be a ring. Every left -module is a quotient of a free left -module such that is finitely generated if and only if can be chosen to be finitely generated.
We need to remember that the rank of any free -module is uniquely determined, later this will be important in our deductions, now let’s see what a group presentation is.
Definition: Let be a set and a subset of the free group such that is generated by . Indeed, a group presentation defines the quotient group of the free group by the normal subgroup generated by
Now, let’s suppose we would like to get the best presentation of a given module , this is possible thanks to the SNF, which gives the answer for a finitely generated -modules, to see this, let’s see how SNF builds this approach.
Let be a -map of commutative ring , where and are free -modules. Let be the basis of and , respectively. We build a matrix , called the presentation matrix, over such that
Lemma: Let an exact sequence of -modules
where is a -dimensional vector space over a field , then its matrix presentation is with respect to , where is the standard basis.
Proof: For we write in terms of , then
where is the Kronecker delta. Indeed, the presentation matrix . Now, let’s recall what SNF says us.
Smith Normal Form
Every nonzero matrix with entries in a Euclidean ring , which is equivalent to a matrix of the form
where and are nonzero, such matrix is called the SNF of .
Now that we have how the presentation matrix is involved with the bases, we can do the conntection between them.
Definition: Let be a euclidean ring and let be the presentation matrix associated to an -map , relative to some choice of bases, and let .
Theorem: If is -equivalent to a SNF , then those that are not units are the invariant factors of .
Proof: Suppose that is in fact the SNF of , then the bases of of and of with and for all , if any. We see that and if is a unit. If is the first that is not a unit, then
a direct sum of cyclic modules for which . Indeed are the invariant factors of .
The proof of the theorem clears why is necessary to get the SNF for answering to the question that I posted on Math.StackExchange
PS: Most of the ideas posted here were obtained of the book Advanced Modern Algebra Third Edition, Part 1 by Joseph J. Rotman, especially the proof of the theorem.