Commutative Algebra/Diagram chasing within Abelian categories

Exact sequences of Abelian groups edit

Definition 4.1 (sequence):

Given   Abelian groups   and   morphisms (that is, since we are in the category of Abelian groups, group homomorphisms)

 ,

we may define the whole of those to be a sequence of Abelian groups, and denote it by

 .

Note that if one of the objects is the trivial group, we denote it by   and simply leave out the caption of the arrows going to it and emitting from it, since the trivial group is the zero object in the category of Abelian groups.

There are also infinite exact sequences, indicated by a notation of the form

 ;

it just goes on and on and on. The exact sequence to be infinite means, that we have a sequence (in the classical sense) of objects and another classical sequence of morphisms between these objects (here, the two have same cardinality: Countably infinite).

Definition 4.2 (exact sequence):

A given sequence

 

is called exact iff for all  ,

 .

There is a fundamental example to this notion.

Example 4.3 (short exact sequence):

A short exact sequence is simply an exact sequence of the form

 

for suitable Abelian groups   and group homomorphisms  .

The exactness of this sequence means, considering the form of the image and kernel of the zero morphism:

  1.   injective
  2.  
  3.   surjective.

Example 4.4:

Set  ,  ,  , where we only consider the additive group structure, and define the group homomorphisms

  and  .

This gives a short exact sequence

 ,

as can be easily checked.

A similar construction can be done for any factorisation of natural numbers   (in our example,  ,  ,  ).

Diagram chase: The short five lemma edit

We now should like to briefly exemplify a supremely important method of proof called diagram chase in the case of Abelian groups. We shall later like to generalize this method, and we will see that the classical diagram lemmas hold in huge generality (that includes our example below), namely in the generality of Abelian categories (to be introduced below).

Theorem 4.5 (the short five lemma):

Assume we have a commutative diagram

 ,

where the two rows are exact. If   and   are isomorphisms, then so must be  .

Proof:

We first prove that   is injective. Let   for a  . Since the given diagram is commutative, we have   and since   is an isomorphism,  . Since the top row is exact, it follows that  , that is,   for a suitable  . Hence, the commutativity of the given diagram implies  , and hence   since   is injective as the composition of two injective maps. Therefore,  .

Next, we prove that   is surjective. Let thus   be given. Set  . Since   is surjective as the composition of two surjective maps, there exists   such that  . The commutativity of the given diagram yields  . Thus,   by linearity, whence  , and since   is an isomorphism, we find   such that  . The commutativity of the diagram yields  , and hence  . 

Additive categories edit

Definition 4.6:

An additive category is a category   such that the following holds:

  1.   is an Abelian group for all objects   of  .
  2. The composition of arrows
 
is bilinear; that is, for   and  , we have
 
(note that, since no scalar multiplication is involved, this definition of bilinearity is less rich than bilinearity in vector spaces).
  1.   has a zero object.
  2. Each pair of objects   of   has a biproduct  .

Although additive categories are important in their own right, we shall only treat them as in-between step to the definition of Abelian categories.

Abelian categories edit

Definition 4.7:

An Abelian category is an additive category   such that furthermore:

  1. Every arrow of   has a kernel and a cokernel, and
  2. every monic arrow of   is the kernel of some arrow, and every epic arrow of   is the cokernel of some arrow.

We now embark to obtain a canonical factorisation of arrows within Abelian categories.

Lemma 4.8:

Let   be a category with a zero object and kernels and cokernels for all arrows. Then every arrow   of   admits a factorisation

 ,

where  .

Proof:

The factorisation comes from the following commutative diagram, where we call   and  :

 

Indeed, by the property of   as a kernel and since  ,   factors uniquely through  . 

In Abelian categories,   is even a monomorphism:

Lemma 4.9:

Let   be an Abelian category. If   and we have any factorisation  , then   is an epimorphism.

Proof:

Theorem 4.10:

Let   be an Abelian category. Then every arrow   of   has a factorisation

 ,

where   and  .

Exact sequences in Abelian categories edit

We begin by defining the image of a morphism in a general context.

Definition 4.12:

Let   be a morphism of a (this time arbitrary) category  . If it exists, a kernel of a cokernel of   is called image of  .

Construction 4.13:

We shall now construct an equivalence relation on the set   of all morphisms whose codomain is a certain  , where   is a category. We set

  for a suitable   (that is,   factors through  ).

This relation is transitive and reflexive. Hence, if we define

 ,

we have an equivalence relation (in fact, in this way we can always construct an equivalence relation from a transitive and reflexive binary relation, that is, a preorder).

With the image at hand, we may proceed to the definition of sequences, exact sequences and short exact sequences in a general context.

Definition 4.14:

Let   be an Abelian category.

Definition 4.15:

Let   be an Abelian category.

Definition 4.16:

Let   be an Abelian category.

Diagram chase within Abelian categories edit

Now comes the clincher we have been working towards. In the ordinary diagram chase, we used elements of sets. We will now replace those elements by arrows in a simple way: Instead of looking at "elements" " " of some object   of an abelian category  , we look at arrows towards that element; that is, arrows   for arbitrary objects   of  . For "the codomain of an arrow   is  ", we write

 ,

where the subscript   stands for "member".

We have now replaced the notion of elements of a set by the notion of members in category theory. We also need to replace the notion of equality of two elements. We don't want equality of two arrows, since then we would not obtain the usual rules for chasing diagrams. Instead, we define yet another equivalence relation on arrows with codomain   (that is, on members of  ). The following lemma will help to that end.

Lemma 4.18 (square completion):

Construction 4.19 (second equivalence relation):

Now we are finally able to prove the proposition that will enable us doing diagram chases using the techniques we apply also to diagram chases for Abelian groups (or modules, or any other Abelian category).

Theorem 4.20 (diagram chase enabling theorem):

Let   be an Abelian category and   an object of  . We have the following rules concerning properties of a morphism:

  1.   is monic iff  .
  2.   is monic iff  .
  3.   is epic iff  .
  4.   is the zero arrow iff  .
  5. A sequence   is exact iff
    1.   and
    2. for each   with  , there exists   such that  .
  6. If   is a morphism such that  , there exists a member of  , which we shall call   (the brackets indicate that this is one morphism), such that:
    1.  
    2.  
    3.  

We have thus constructed a relatively elaborate machinery in order to elevate our proof technique of diagram chase (which is quite abundant) to the very abstract level of Abelian categories.

Examples of diagram lemmas edit

Theorem 4.21 (the long five lemma):

Theorem 4.22 (the snake lemma):