Commutative Algebra/Diagram chasing within Abelian categories

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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. 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

Short 5 lemma.svg,

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 :

Factorisation.svg

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:

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):