The Report of Homological Algebra

Abstract

The current article intends to introduce the reader to the concept of injective and projective modules and to describe the CFT. We present a clear view to show the homological algebra and injective and projective modules.

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Zhang, J. (2024) The Report of Homological Algebra. Open Journal of Applied Sciences, 14, 2749-2767. doi: 10.4236/ojapps.2024.149179.

1. Introduction

Before we get into the main text, let’s talk about homological algebra. This work originated from the generalization of some simple topological relations.

This framework is sketched in Chapter 2, we’ll start with simplicial sets, while discussing simple categories. And in this chapter we would like to talk more about simplicial complexes.

We’ll start with module in Chapter 3, and from an algebraic point of view, discuss the projective and injective module with the free module, whose theme is the same as that of [1].

In Chapter 4, we’ll show more detail about Projective and Injective, and knowledge about Yoneda Lemma. Here we can refer to [1] [2] more details can be referred to [3]-[5].

We will abstract the concept of simple sets from algebraic topology, and give a more fundamental understanding and thinking of homology theory. This will provide a transcendental reflection on many of our subsequent study directions which is to obtain global information from local information (Although the method may still be categorical). For example, we can think of sheaf theory as a kind of singular homology which I’ll talk a lot about that in Chapter 1. The article can be thought as a simple one of Sur quelques points d’algèbre homologique [TOHOKU] [1957] without some derived category and with some abstract algebraic topology. More detail can be referred to [6]-[8].

2. Simplicial Sets and Cohomology

2.1. Simplicial Sets

In this section, we will present some brief observations of simplicial sets.

Definition 1.1. Before defining other definitions, we need a bit of the analytic geometry of Euclidean space. Given a set { a 0 ,, a n } of points of , this set is said to be geometrically independent if for any scalars t i , the equations

i=0 n t i =0 and i=0 n t i a i =0 (1)

imply that t 0 = t 1 == t n =0 .

Definition 1.2. Let { a 0 ,, a n } be a geometrically independent set in . We define the n-simplex Δ n spanned by a 0 ,, a n to be the set of all points x of such that

x= i=0 n t i a i , where i=0 n t i =1 (2)

and t i 0 for all i.

Definition 1.3. The points a 0 ,, a n that span Δ n are called the vertices of Δ n ; the number n is called the dimension of Δ n . Any simplex spanned by a subset of { a 0 ,, a n } is called a face of Δ n . The face of Δ n spanned by a 0 ,, a n is called the face opposite a 0 . Their union is called the boundary of Δ n , which we denote as Bd Δ n . The interior of Δ n is defined by the equation Int Δ n = Δ n Bd Δ n .

Definition 1.4. A simplicial set is a family of sets X=( X n ),n=0,1, , elements of X n are n-simplices, and of map X( f ): X n X m , one for each nondecreasing map f:[ m ][ n ] such the following conditions are satisfied:

X( id )=id,X( fg )=X( g )X( f ) (3)

For any nondecreasiong map f:[ m ][ n ] we define the “f-th face” Δ f as the linear map Δ m Δ n that maps any vertex e i Δ m into the vertex e f ( i ) Δ n ,i=0,,m . The geometric realization of the simplicial set Δ[ p ] is the p-dimensional simplex Δ p . We can describe Δ[ p ] as the simplicial set of all singular simplices of Δ p that are compatible with the standard triangulation of Δ p .

A topological space | X | with the underlying set n=0 ( Δ n × X ( n ) )/R , where R is the weakest equivalence relation that identified ( s,x ) Δ n × X ( n ) and ( t,y ) Δ m × X ( m ) with

y=X( f )x,s= Δ f ( t ) (4)

for some increasing mapping f:[ m ][ n ] . We denote the situation as in (4) by ( t,y )( s,x ) . The canonical topology on | X | is the weakest topology for which the canonical mapping n=0 ( Δ n × X ( n ) )/R | X | is continuous.

Definition 1.5. Let Δ p be a simplex. Define two orderings of its vertex set to be equivalent if they differ from one another by an even permutation. Each of these classes is called an orientation of Δ p . The oriented simplex [ v 0 , v 1 , v 2 ] is indicated in the figure by drawing an arrow in the direction from v 0 to v 1 to v 2 . We can check that [ v 1 , v 2 , v 0 ] and [ v 2 , v 0 , v 1 ] are indicated by the same arrow. In particular, 0-simplex has only one element and only one orientation.

Definition 1.6. Let X be a simplicial set. An n-simplex x X n is said to be degenerate if there exists a surjective nondecreasing map f:[ n ][ m ] , m<n , and an element y X m such that x=X( f )y . One can easily check that if x is nondegenerate and x=X( f )y for some f and y, then f is an injection.

Example 1.7. (Triangulation of the Product of Two Simplices) The product of two segments [ 0,1 ]×[ 0,1 ] is not a triangle but a square; it can be naturally divided into two triangles by a diagonal, of two diagonals one is singled out by the fact that its vertices are naturally ordered: [ 00,11 ] .

Generalizing this construction, we define the canonical triangulation ( X ( n ) ,X( f ) ) of the product Δ p × Δ q . One element of X ( n ) is a set of n+1 different pairs of integers ( i 0 , j 0 ),,( i n , j n ) , where they are a directed set. Any increasing mapping f:[ m ][ n ] define X( f ) as follows:

X( f ){ ( i 0 , j 0 ),,( i n , j n ) }={ ( i 0 , j 0 ),,( i n , j n ) } (5)

where i k = i f( k ) , j k = j f( k ) .

Define a mapping

θ n : Δ n × X ( n ) Δ p × Δ q (6)

as follows: with the x-th simplex,

x={ ( i 0 , j 0 ),,( i n , j n ) } X n (7)

θ n associates the simplex Δ n in Δ p × Δ q p+q+2 spanned by the points ( e i a , e j a ) , 0an , where e i (resp. e i ) is the i-th vertex of Δ p (resp. Δ q ). Or, more formally, θ n ( ,x ): Δ n Δ p × Δ q is a linear order-preserving mapping with the image Δ n .

We claim that there exists a commutative diagram.

where φ is a bijection.

Definition 1.8. Let G be a group. Let

( BG ) n = G n (8)

and for f:[ m ][ n ] let

BG( f )( g 1 ,, g n )=( h 1 ,, h m ) (9)

where

h i = j=f( i1 )+1 f( i ) g j , h i =eiff( i1 )=f( i ) (10)

Taking a example: the mapping f:[ 3 ][ 4 ] with f( 0 )=0 , f( 1 )=f( 2 )=2 , f( 3 )=4 , we can see h 1 = g 1 g 2 ; h 2 =e ; h 3 = g 3 g 4 The geometric realization | BG | of BG is called the classifying space of G.

The structure of the geometric realization of a simplicial set X can be clarified like that:

τ : n=0 ( Ind Δ n × X ( n ) )| X | (11)

where X(n) = the set of nondegenerate n-simplices of X.

2.2. Category I

In this section we will briefly introduce some category knowledge, the main of which is the addition category and abelian categories. More details can be referred to [7] [8]

Definition 1.9. The aim is to fix some notations and to recall the axioms of universes. We do not intend neither to enter Set Theory, nor to say more about universes than what we need.

For a set u, we denote by P( u ) the set of subsets of u:P( u )={ x;xu } . For x 1 ,, x n , we denote by x 1 ,, x n the set whose elements are x 1 ,, x n .

A universe U is a set satisfying the following properties:

1) ϕU ,

2) uU implies uU , (equivalently, xU and yx implies yU , or else UP( U ) ),

3) uU implies uU ,

4) uU implies P( u )U ,

5) if IU and u i U for all iI , then iI , u i U ,

6) U

Following Grothendieck, we shall add an axiom to the Zermelo-Fraenkel theory, asking that for any set X there exists a universe U such that XU .

Let U be a universe.

1) A set is called a U-set if it belongs to U.

2) A set is called U-small if it is isomorphic to a set belonging to U.

We can also define order. But that’s enough for what we’ll talk about below.

Definition 1.10. A category C consists of the following data.

1) A family Ob( C ) , whose members are called the objects of C .

2) for all pairs ( X,Y ) of Ob( C ) , a set Ho m C ( X,Y ) , whose elements are called morphisms from X to Y.

3) for any triple ( X,Y,Z ) of Ob( C ) , a map from Ho m C ( X,Y )×Ho m C ( Y,Z ) to Ho m C ( X,Z ) , called the composition map, and denoted ( f,g )gf .

These data satisfying: the composition of morphisms is associative, for any XOb( C ) there exists i d X Ho m C ( X,X ) such that fi d X =f and i d X g=g for an fHo m C ( X,Y ) and any gHo m C ( Y,X ) .

Definition 1.11 A subcategory C of C is a category C such that Ob( C )Ob( ) and for any pair ( X,Y ) of Ob( C ) , Ho m C ( X,Y )Ho m C ( X,Y ) , with the induced composition law, and i d X Ho m C ( X,X ) .

If moreover Ho m C ( X,Y )=Ho m C ( X,Y ) , then C is called a full subcategory of C .

Definition 1.12. Let C be a category. The opposite category, denoted C , is defined by:

Ob( C )=Ob( C )

Ho m C ( X,Y )=Ho m C ( Y,X )

Definition 1.13. Let f:XY be a morphism in C . One says that f is a monomorphism if for any WOb( C ) and any pair ( g, g ) of Ho m C ( W,X ) such that fg=f g , one has g= g . In the dual way, we can define epimorphism.

In a category C an object P is called initial if Ho m C ( P,Y ) has exactly one element for any YOb( C ) . Similarly an object Q is called final if Ho m C ( X,Q ) has only one element, that is, if Q is initial in C . Note that two initial (resp.final) object are naturally isomorphic.

Definition 1.14. Let C and C be two categories. A functor F:C C consists of a map F:Ob( C )Ob( C ) and of maps F:Ho m C ( X,Y )Ho m C ( F( X ),F( Y ) ) for all X,YC , such that

F( i d X )=i d F( X ) ,XC,

F( gf )=F( g )F( f ),f:XY,g:YZ

A contravariant functor from C to C is a functor from C to C . In other words, it satisfies F( gf )=F( f )F( g ) .

Definition 1.15. A diagram in a category C is a family of symbols representing objects of C and a family of arrows between these symbols representing morphisms of these objects. One defines in an obvious way the notion of a commutative diagram.

Definition 1.16. Let F:C C be a functor. We say that F is faithful (resp. full, fully faithful) if Ho m C ( X,Y )Ho m C ( F( X ),F( Y ) ) , is injective (resp. surjective, bijective) for any X,Y in C .

Let F1 and F2 be two functors from C to C . A morphism θ from F1 to F2 consists of the follow data.

XOb( C ),θ( X )Ho m C ( F 1 ( X ), F 2 ( X ) ) (12)

These data satisfying: for any fHo m C ( X,Y ) the following diagram commutes:

We gets a new category whose objects are functors from C to C and morphisms are morphism of such functors.

Definition 1.17. An additive category C is a category C such that:

1) for any pair ( X,Y ) of Ob( C ) , Ho m C ( X,Y ) has a structure of additive (i.e. abelian) group, and the composition law is bilinear.

2) there exists an object 0 such that Ho m C ( 0,0 )=0 .

An additive functor F:A between Ab-category and A is a functor such that each Ho m ( B ,B )Ho m A ( F B ,FB ) is a group homomorphism.

In any additive category A , a kernel of a morphism f:BC is defined to be a map i:AB such that fi=0 and that is universal with respect to this property. Dually, a cokernel of f is a map e:CD , which is universal with respect to having ef=0 . In A , am map i:AB is monic if ig=0 implies g=0 for every map g: A A , and a map e:CD is an epi if he=0 implies h=0 for every map h:D D (The definition of monic and epi in a non-abelian category is slightly different). It’s easy to see that every kernel is monic and that every cokernel is an epi.

2.3. Homology and Cohomology I

The boundary of the geometric simplex Δ 1 is the difference of its vertices (1)-(0). In such a form the boundary appears in the Leibniz formula 0 1 f( x )dx =f( 1 )f( 0 ) . Similarly, the boundary of Δ n is the alternating sum of its faces.

To make these definitions precise we need the following notions.

An n-dimensional chain (or simply n-chain) of a simplicial set X is an element of the free abelian group C n ( X ) generated by all n-simplices of X. So an n-dimensional chain is a formal linear combination x X n a( x )x , where a( x )Z and a( x )=0 for a finite number of simplices x. Let n i :[ n1 ][ n ] be the unique strictly increasing mapping whose image does not contain i[ n ] .

Definition 1.18. The boundary of an n-chain c C n ( x ) is the (n-1)-chain d n c defined by the following formula:

d n ( x X n ( a( x )x ) )= x X n a( x ) i=1 n ( 1 ) i X( n i )( x ) (13)

The so defined boundary operator d n : C n ( X ) C n1 ( X ) is clearly a group homomorphism. For n=0 , we set d 0 =0 . There exists an obvious generalization of this construction, namely, chains with coefficients in an abelian group A. Such a chain is a formal linear combination x X n a( x )x , a( x )A , we set C n ( X )= C n ( X,Z ) . The boundary operator d n : C n ( X,A ) C n+1 ( X,A ) is again defined by before. Dually, one can define cochains with coefficients in A: C n ( X,A ) is the group of functions on X n with values in A. The coboundary d n : C n ( X,A ) C n+1 ( X,A ) is given by the formula:

( d n f )( x )= i=0 n+1 ( 1 ) i f( X( n+1 i )( x ) ) (14)

Formally, chains can be considered as special cases of cochains: there exists an inclusion C n ( X,A ) C n ( X,A ) that maps a chain x X n a( x )x into the function a: X n A . However, this inclusion is incompatible with the action of d n and d n (they act in opposite directions) and, even more important, it is incompatible with the behaviour of C n and C n under simplicial maps XY .

Lemma 1.19. We have d n1 d n =0 for n1 , d n+1 d n =0 for n0 .

Proof: Note first that for any 0j<in1 we have

n i n1 j = n j n1 j1

indeed, both sides of the equality give unique increasing mapping of [ n2 ] into [ n ] not taking values i and j.

To prove the lemma it suffices to check that d n1 d n ( x )=0 for any x X n . Compositions n i n1 j for different i,j all yield increasing maps of [ n2 ] into [ n ] , and the map whose image does not contain i and j appears exactly twice: the first time as n i n1 j with the sign ( 1 ) i+j and the second time as n j n1 i1 with the opposite sign ( 1 ) i+j1 . Hence d n1 d n =0 .

Similarly one proves another.

Definition 1.20. Let us define several algebraic notions. A chain complex is a sequence of abelian groups and homomorphisms.

C : d n+1 C n d n C n1 d n1 (15)

with the property d n1 d n =0 for all n. Homomorphisms d n are called boundary maps or boundary operators.

A cochain complex is a similar sequence

C : d n1 C n d n C n+1 d n+1 (16)

d n+1 d n =0 Any chain complex can be transformed into a cochain complex by setting D n = C n , d n = d n1 . So we will usually consider only cochain complexes.

Definition 1.21. Homology groups of a chain complex C are

H n ( C )= Ker d n / Im d d+1

Cohomology groups of a cochain complex C are

H n ( C )= Ker d n / Im d d+1

A substantial part of homological algebra can be considered as a collection of methods for computing (co)homology of various complexes, which we’ll talk about that later. For a simplicial set X we will use the following notations: H n ( X,A )= H n ( C ( X,A ) ) , H n ( X,A )= H n ( C ( X,A ) ) .

Elements of the group H n ( X,A ) are called homology classes, and those of H n ( X,A ) are called cohomology classes (of the simplicial set X with coefficients)

Definition 1.22. Each homology (resp. cohomology) class is represented by an n-chain c (resp. cochain f) such that d n c=0 (resp. d n f=0 ). Such chains (resp. cochains) are called cycles (resp. cocycles). A cycle c in a given homology class is defined up to a summand of the form b= d n+1 c ; such chains are called boundaries. Similarly, cochains of the form d n1 c are called coboundaries. Two chains whose difference is a boundary are said to be homological.

2.4. Geometry of Chains and Cochains with Coefficient System

Example 1.23. Look at the picture of a tetrahedron Δ 3 .

On any edge two adjacent faces induce opposite orientation. Therefore the corresponding terms in d 2 d 3 ( Δ 3 ) have opposite signs.

Example 1.24. Zero-dimensional homology yields somewhat different information. Since, d 0 =0 , any 0-chain is a cycle. Let us show that there exists a natural isomorphism. H 0 ( X, ) = free abelian group generated by piecewise connected components of | X | . Denote for a moment the group on the right-hand side by π 0 ( X ) . Define a map π 0 ( X ) H 0 ( X, ) by associating with a component of | X | the class of a chain consisting of one (arbitrary) point in this component.

Example 1.25. An important role in geometry is played by various modifications of topological spaces that eliminate some (co)homology classes or generate new ones.

Let X be a triangulated space. The cone CX over X is a triangulated space obtained from X in the following manner:

{ VerticesofCX }={ verticesofX }{ }

and for n1

{ n-simplices ofCX } ={ n-simplices ofX }{ conesof( n1 )-simplices ofXwith the vertex }

so that is the vertex of the cone.

We claim that any hole in X is filled in CX (by the cone over the boundary of the hole) and no new holes appear.

Indeed, let us define the complex of chains C ^ ( X ) of a triangulated space X as follows:

C ^ ( X ) = the free abelian group generated by X ( n )

the boundary operator on C ^ is same as before, we claim that H n ( C ^ ( CX ) )=0 , n>0 and = , n=0 . We have C ^ n ( CX ) C n ( X ) C n1 ( X ) . Later we define the cone of any complex.

Definition 1.26. We can construct chains and cochains of a simplicial set using as coefficients something more involved than just abelian groups. There are two types of coefficient systems: for homology and for cohomology. A homological coefficient system A on a simplicial set X is a family of abelian groups { Ax } , one for each simplex x X n , and a family of homomorphisms A( f,x ): A x A X( f )x , one for each pair x X n , f:[ m ][ n ] , such that the following conditions are satisfied:

A( id,x )=id

A( fg,x )=A( g,X( f )x )A( f,x )

The second equality means that the following diagram is commutative:

A cohomological coefficient system B on a simplicial set X is a family of abelian groups { B x } , one for each simplex x X n , and a family of ho momorphisms B( f,x ): B X( f )x β B x , one for each pair ( x,f ) , such that the following conditions are satisfied:

B( id,x )=id

B( fg,x )=B( f,x )B( g,X( f )x )

The second equality is equivalent to the commutativity of a diagram similar to the one above.

Definition 1.27. Let A be a homological coefficient system on a simplicial set X. An n-dimensional chain of X with coefficients in A is a formal linear combination x X n a( x )x , a( x ) A x . Such chains form an abelian group (under addition) which is denoted by C n ( X,A ) . The boundary of an n-dimensional chain c=a( x )x C n ( X,A ) is an ( n1 ) -dimensional chain d n c C n1 ( X,A ) defined by:

d n c= x X n i=0 n A( n i ,x )( a( x ) ) ( 1 ) i X( n i )( x )

d 0 =0

As before we can easily see that

C ( X,A ): d n+1 C n ( X,A ) d n C n1 ( X,A ) d n1

is a chain complex, i.e. d n1 d n =0 .

Homology groups of the complex C ( X,A ) are called the homology groups of the simplicial set X with coefficients in A; they are denoted by H n ( X,A ) .

Similarly and dually, we can define cohomology groups.

2.5. The Exact Sequence I

We defined groups H n ( X,A ) and H n ( X,B ) , where X is a simplicial set, and A and B are coefficient systems. In some simple cases these groups can be computed directly. But the main technique consists in the study of the behaviour of these groups under the change of X or the change of A.

In this subsection we fix X and study the dependence of homology and cohomology on coefficients. The main tool here is the theorem about the exact sequence.

Definition 1.28. An exact sequence of abelian groups is a complex C with all cohomology groups vanishing (for chain complexes the definition is the same). This means that Ker d n =Im d n1 for all n. Usually such a sequence is written as

0A i B π C0 (17)

To give such a triple is the same as to give an abelian group B and its subgroup A. The homomorphism theorem says that

Imi=Kerπ implies C=B/A (18)

Theorem 1.29. Let X be a simplicial set. Any exact triple of abelian groups canonically determines a cohomology exact sequence:

0 H 0 ( X,A ) H 0 ( X,B ) H 0 ( X,C ) H 1 ( X,A ) H 1 ( X,B ) H n ( X,A ) (19)

and a similar homology sequence.

To proof the theorem we must define the morphisms of complexes and construct the boundary homomorphism.

Theorem 1.30. Let B , C be two complexes. A morphism f : B C is a family of homomorphisms f n : B n C n commuting with differentials:

d n f n = f n+1 d n (20)

Given f n : B n C n , let us construct a family of homomorphisms

H n ( f ): H n ( B ) H n ( C ) (21)

as follows. Let b H n ( B ) be represented by a cocycle b ^ Ker d n B n , Then f n ( b ^ )Ker d n C n , and we define H n ( f )( b ) to be the class of f n ( b ^ ) in H n ( C ) . this class does not depend on the choice of a representative of b modulo Im d n1 .

It is clear also that if g : B C is another morphism of complexes then H n ( fg )= H n ( f ) H n ( g ) . Let Ker f =( Ker f n ) , Coker f =( Coker f n ) , so Ker f and Coker f are complexes.

Lemma 1.31. Let 0A i B π C0 be an exact triple of abelian groups. Then the sequences of groups of chains and of cochains

0 C ( X,A ) i C ( X,B ) π C ( X,C )0

0 C ( X,A ) i C ( X,B ) π C ( X,C )0

are exact.

Proof: An element of C n ( X,A ) is a formal linear combination x X n a( x )x , a( x ) A x . The image of this element under the mapping i n : C n ( X,A ) C n ( X,B ) is x X n i( a( x ) )x and since i is an injection, i n is also an injection. Similarly one proves that π n is a surjection. Further, π n i n ( x X n a( x )x )= ( πi )( a( x ) )x=0 for a( x )x C n ( X,A ) . Let now β= b( x )x C n ( X,B ) and π n ( β )=0 . Then π( b( x ) )=0 for all x X n , i.e., b( x )=i( a( x ) ) , a( x )A and β= i n ( α ) for α= a( x )x C n ( X,A ) . The second sequence is treated similarly.

Theorem 1.32 Proof of 1.29: 1) Exactness at H n ( B ) . First of all, H n ( π ) H n ( i )= H n ( π i )=0 , because π i =0 . Next, let b H n ( B ) and H n ( π )( b )=0 . We construct a H n ( A ) with b= H n ( i )( a ) as follows. Let b ^ B n be a representative of b so that d b ^ =0 . Since H n ( π )( b )=0 , we have π n ( b ^ )=d c ^ for some c ^ C n1 and, p n1 : B n1 C n1 being a surjection, c ^ = p n1 ( b ^ 1 ) for some b ^ 1 B n1 . It is clear that p n ( b ^ d b ^ 1 )=0 so that by the exactness of before, bd b ^ 1 = i n ( a ^ ) for some a ^ A n ; moreover, i n+1 ( d a ^ )=d i n ( a ^ )=d b ^ =0 . As i n+1 is injective, d a ^ =0 . Now one can easily check that a= a ^ mod Im d n1 H n ( B ) satisfies the required property.

2) n ( i , π ) H n ( π )=0 . Let c= H n ( π )( b ) for some b H n ( B ) and let b ^ B n , c ^ C n be representatives of b,c respectively, so that d b ^ =0 . Then the definition of n ( i , π ) shows that n ( i , π )( c )=0 .

3) Ker n ( i , π )Im H n ( π ) . Let n ( i , π )( c )=0 and let c ^ C n be a cocycle representing the cohomology class c. Let c ^ = π n ( b ^ ) and d b ^ = i n+1 ( a ^ ) for b ^ B n , a ^ A n+1 . Then n ( i , π )( c )=0 implies a ^ =d a ^ 1 for some a ^ 1 A n . Let b ^ 1 = b ^ i n ( a ^ 1 ) . Then d b ^ 1 =0 and π n ( b ^ 1 )= p n ( b ) p n i n ( a ^ 1 )= c ^ . Hence c= H n ( π )( b 1 ) , where b 1 = b ^ mod Im d n H n ( B ) .

It’s the same on the other side.

3. Module: With View of Free Module in Projective and Injective

Modules over a ring are a generalization of abelian groups (which are modules over ). In the section, we’ll cover projective and injective with view of the free module.

3.1. Module and the Exact Sequence II

Definition 2.1. Let R be a ring. A (left) R-module is an additive abelian group A together with a function R×AA (the image of ( r,a ) being denoted by ra ) such that for all r,sR and a,bA :

1) r( a+b )=ra+rb

2) ( r+s )a=ra+sa

3) r( sa )=( rs )a

If R has an identity element 1 R and

4) 1 R a=a for all aA

then A is said to be a unitary R-module. If R is a division ring, then a unitary R-module is called a vector space.

Definition 2.2. Let A and B be modules over a ring R. A function f:AB is an R-module homomorphism provided that for all a,cA and rR :

f( a+c )=f( a )+f( c ) and f( ra )=rf( a ) (22)

If R is a division ring, then an R-module homomorphism is called a linear transformation.

Definition 2.3. Let R be a ring, A an R-module and B a nonempty subset of A. B is a submodule of A provided that B is an additive subgroup of A and rbB for all rR , bB . A submodule of a vector space over a division ring is called a subspace.

If X is a subset of a module A over a ring R, then the intersection of all submodules of A containing X is called the submodule generated by X (or. spanned by X). If X is finite, and X generates the module B, B is said to be finitely generated. Let B be a submodule of a module A over a ring R. Then the quotient group A/B is an R-module with the action of R on A/B given by:

r( a+B )=ra+B for all rR,aA

Definition 2.4. A pair of module homomorphisms, A f B g C , is said to be exact at B provided Imf=Kerg . A finite sequence of module homomorphisms, A 0 f 1 A l f 2 A 2 f 3 f n1 A n1 f n A n , is exact provided Im f i =Ker f i+1 for i=1,2,,n1 . At the same time we can define the exact of an infinite sequence.

Lemma 2.5. Let R be a ring and

a commutative diagram of R-modules and R-module homomorphisms such that each row is a short exact sequence. Then

1) α,γ monomorphisms β is a monomorphisms;

2) α,γ epimorphisms β is a epimorphisms;

3) α,γ isomorphisms β is a isomorphisms.

Proof: 1) Let bB and suppose β( b )=0 ; we must show that b=0 . By commutativity we have γg( b )= g β( b )= g ( 0 )=0 . This implies g( b )=0 , since γ is a monomorphism. By exactness of the top row at B, we have bKerg=Imf , say aA . By commutativity, f α( a )=βf( a )=β( b )=0 . By exactness of the bottom row at A , f is a monomorphism, hence α( a )=0 . But α is a monomorphism; therefore a=0 and hence b=f( a )=f( 0 )=0 . Thus β is a monomorphism.

2) Let b B . Then g ( b ) C ; since γ is an epimorphism g ( b )=γ( c ) for some cC . By exactness of the top row at C, g is an epimorphism; hence c=g( b ) for some bB . By commutativity, g β( b )=γg( b )=γ( c )= g ( b ) . Thus g [ β( b ) b ]=0 and β( b ) b Ker g =Im f by exactness, say f ( a )=β( b ) b , a A . Since α is an epimorphism, a =α( a ) for some aA . Consider bf( a )B : β[ bf( a ) ]=β( b )βf( a ) , by commutativity, βf( a )= f α( a )= f ( a )=β( b ) b , hence β[ bf( a ) ]=β( b )βf( a )=β( b )( β( b ) b )= b and β is an epimorphism.

3) is an immediate consequence of 1) and 2).

Definition 2.6 Let R be a ring and 0 A 1 f B g A 2 0 . a short exact sequence of R-module homomorphisms. Then the following conditions are equivalent.

1) There is an R-module homomorphism h: A 2 B with gh= 1 A 2 ;

2) There is an R-module homomorphism k:B A 1 with kf= 1 A 1 ;

3) the given sequence is isomorphic (with identity maps on A 1 and A 2 ) to the direct sum short exact sequence 0 A 1 f A 1 A 2 g A 2 0 ; in particular B A 1 A 2 .

A short exact sequence that satisfies the equivalent conditions is said to be split or a split exact sequence.

3.2. Free Module

In this subsection we show free modules, the most important examples of which are vector spaces over a division ring. We’ll show the objects without proof.

Definition 2.7. A subset X of an R-module A is said to be linearly independent provided that for distinct x 1 ,, x n X and r i R . r 1 x 1 + r 2 x 2 ++ r n x n =0 r i =0 for every i. A set that is not linearly independent is said to be linearly dependent. If A is generated as an R-module by a set Y, then we say that Y spans A. If R has an identity and A is unitary, Y spans A if and only if every element of A may be written as a linear combination: r 1 y 1 + r 2 y 2 ++ r n y n =0 ( r i R, y i Y ) ; A linearly independent subset of A that spans A is called a basis of A. Observe that the empty set is linearly independent and is a basis of the zero module.

Definition 2.8. A unitary module F over a ring R with identity, which satisfies F has a nonempty basis is called a free R-module.

Definition 2.9. A maximal linearly independent subset X of a vector space V over a division ring D is a basis of V. Every vector space V over a division ring D has a basis and is therefore a free D-module. More generally every linearly independent subset of V is contained in a basis of V. If V is a vector space over a division ring D and X is a subset that spans V, then X contains a basis of V.

Definition 2.10. Let R be a ring with identity and F a free R-module with an infinite basis X. Then every basis of F has the same cardinality as X. If V is a vector space over a division ring D, then any two bases of V have the same cardinality.

3.3. Projective and Injective Module

Every free module is projective and arbitrary projective modules (which need not be free) have some of the same properties as free modules. Injectivity, which is also studied here, is the dual notion to projectivity. Before we start our section we consider such example when expressed in modern language, the Riemann-Roch theorem give a formula for the difference of the dimensions of two vector spaces attached to algebraic line bundle over a non-singular projective curve. Thus, we can see easily where the projective or injective module come from (we don’t expect its historic origin which comes from Homological Algebra written by Cartan). At the same time, in order to fit in with the category we mentioned above, we will make sacrifices to use its abstract algebraic language.

Definition 2.11. A module P over a ring R is said to be projective if given any diagram of R-module homomorphisms

with bottom row exact (that is, g an epimorphism), there exists an R-module homomorphism h:PA such that the diagram

is commutative (that is, gh=f ).

Theorem 2.12 Every free module F over a ring R with identity is projective.

Proof: We are given a diagram of homomorphisms of unitary R-modules:

with g an epimorphism and F a free R-module on the set X ( π:XF ). For each xX , f( π( x ) )B . Since g is an epimorphism, there exists a x A with g( a x )=f( π( x ) ) . Since F is free, the map XA given by x a x induces an R-module homomorphism h:FA such that h( π( x ) )= a x for all xX . Consequently, ghπ( x )=g( a x )=fπ( x ) for all xX so that ghπ=fπ:XB , we have gh=f . Therefore F is projective. We also can see every module A over a ring R is the homomorphic image of a projective R-module.

Theorem 2.13 Let R be a ring. The following conditions on an R-module P are equivalent.

1) P is projective;

2) there is a free module F and an R-module K such that FKP ;

3) every short exact sequence 0A f B g P0 is split exact.

Proof: 1) 3) Consider the diagram

with bottom row exact by hypothesis. Since P is projective there is an R-module homomorphism h:PB such that gh= 1 p . Therefore, the short exact sequence 0A f B g P0 is split exact.

3) 2) There is a free R-module F and an epimorphism g:FP . If K=Kerg , then 0K F g P0 . By hypothesis the sequence splits.

2) 1) Let π be the composition FKPP where the second map is the canonical projection. Similarly let τ be the composition PKPF with the first map the canonical injection. Given a diagram of R-module homomorphisms

with exact bottom row, consider the diagram

Since F is projective, there is an R-module homomorphism h 1 :FA such that g h 1 =fπ . Let h= h 1 τ:PA . Then gh=g h 1 τ=( fπ )τ=f( πτ )=f 1 P =f . Therefore, P is projective.

Definition 2.14 A module J over a ring R is said to be injective if given any diagram of R-module homomorphisms

with top row exact (that is, g a monomorphism), there exists an R-module homomorphism h:BJ such that the diagram

is commutative (that is, hg=f ).

Theorem 2.15. A right R-module E is injective if and only if for every right ideal J of R, every map JE can be extended to a map RE .

Proof: The “only if” direction is a special case of the definition of injective. Conversely, suppose given an R-module B, a submodule A and a map α:AE . Let be the poset of all extensions α : A E of α to an intermediate submodule A A B ; the partial order is that α α if α extends α . By Zorn’s lemma there is a maximal extension α : A E in ; we have to show that A =B . Suppose there is some bB not in A . The set J={ rR:br A } is a right ideal of R. By assumption, the map J b A α E extends to a map f:RE . Let A be the submodule A +bR of B and define α : A E by a ( a+br )= a ( a )+f( r ) , a A and rR . This is well defined because α ( br )=f( r ) for br in A bR , and α extends α , contradicting the existence of b. Hence A =B .

Definition 2.16. An abelian group D is said to be divisible if given any yD and 0nZ , there exists xD such that nx=y . For example, the additive group is divisible, but is not. An abelian group D is divisible if and only if D is an injective (unitary) Z-module.

Theorem 2.17. Let R be a ring. The following conditions on an R-module J are equivalent.

1) J is injective;

2) J is a direct summand of any module B of which it is a submodule;

3) every short exact sequence 0J f B g C0 is split exact.

Proof like before.

Example 2.18. The divisible abelian groups and p =[ 1 p ]/ are injective. Every injective abelian group is direct sum of these. In particular, the injective abelian group / is isomorphic to p .

If A is an abelian group, let I( A ) be the product of copies of the injective group / , indexed by the set Ho m Ab ( A,/ ) , then I( A ) is injective.

Example 2.19. Nice rings every projective module is a free module like , fields, division rings

4. Homological Algebra

Homological algebra is a tool used to prove nonconstructive existence theorems in algebra. It also provides obstructions to carrying out various kinds of constructions; when the obstructions are zero, the construction is possible. In the section, we will show many theorems without proof because of space. And we’ll skip content like δ-function that is interesting.

Definition 3.1. Let A be an abelian category. Then Ho m A ( M, ) is a left exact functor from A to Ab for every M in A. That is, given an exact sequence 0A f B g C0 , the following sequence of abelian groups is also exact:

Hom( M,A )Hom( M,B ) g * Hom( M,C )0 (23)

Ho m A ( ,M ) is a left exact contravariant functor.

Theorem 3.2. Yoneda Embedding Every additive category A can be embedded in the abelian category A b A op by the functor h sending A to h A =Ho m A ( ,A ) . Since each Ho m A ( M, ) is left exact, h is a left exact functor. Since the functors h A are left exact, the Yoneda embedding actually lands in the abelian subcategory of all left exact contravariant functors from A to Ab whenever A is an abelian category.

Lemma 3.3. The Yoneda embedding h reflects exactness. That is, a sequence A f B g C in A is exact, provided that for every M in A the following sequence is exact:

Hom( M,A ) f * Hom( M,B ) g * Hom( M,C ) (24)

Theorem 3.4. M is projective if and only if Ho m A ( M, ) is an exact functor. That is, the sequence of groups:

0Hom( M,A ) f * Hom( M,B ) g * Hom( M,C )0 (25)

is exact for every exact sequence A f B g C in A.

Definition 3.5. Let M be an object of A. A left resolution of M is a complex P with P i =0 for i<0 ,together with a map ϵ: P 0 M so that the augmented complex:

P 2 d P 1 d P 0 ϵ M0 (26)

is exact. It is a projective resolution if each P i is projective.

Theorem 3.6. Every R-module M has a projective resolution. More generally, if an abelian category A has enough projectives, then every object M in A has a projective resolution.

Forming a resolution by splicing.

Theorem 3.7. Let P ϵ M be a projective resolution of M and f:MN a map in A. Then for every resolution Q η N of N there is a chain map f: P Q lifting f in the sense that η f 0 = f ϵ . The chain map f is unique up to chain homotopy equivalence.

Theorem 3.8. Suppose given a commutative diagram

where the column is exact and the rows are projective resolutions. Set P n = P n P n Then the P n assemble to form a projective resolution P of A, and the right-hand column lifts to an exact sequence of complexes 0 P P P 0 where are the natural inclusion and projection, respectively.

Definition 3.9 Let M be an object of A. A right resolution of M is a cochain complex I with I i =0 for i<0 and a map M I 0 such that the augmented complex:

0M I 0 d I 1 d I 2 d (27)

is exact. This is the same as a cochain map M I , where M is considered as a complex concentrated in degree 0. It is called an injective resolution if each I i is injective.

The other theorems mentioned above are all in dual form in injective.

What Is Homological Algebra?

Now we have met some injective and projective module and also found the continuity with algebraic topology. It’s appropriate to ask, what is homological algebra?

I’d like to the view that homological algerbra is a tool used to prove some theorems in algebra and it can show us how far away from the good structure. At the same time, it can also provide enough details in arithmetic. From a historical point of view, homological algebra is actually a metaphysical weapon. Kant used to say: There can be no doubt that all our knowledge begins with experience. Whether or not there is such knowledge, which does not rely on experience, or even on any sensory impressions, is at least a question that needs to be examined more carefully, and one that cannot be answered immediately and lightly. Therefore, it is easy to see that the weapon we use is actually derived from algebraic topology, but it is beyond the category of algebraic topology and has become a kind of prior knowledge. This provides us with a philosophical reflection on how to understand mathematics, Copernicus, a philosopher born 300 years ago, said that the intellectual category can only be used empirically, not transcendentally. Empirical know- ledge such as arithmetic, is difficult to be as transcendent as such knowledge. However, this kind of prior knowledge is difficult to understand and acquire, and it still requires a lot of empirical knowledge as a background refinement and the field is still breeding lilacs out of the dead land.

Conflicts of Interest

The author declares no conflicts of interest regarding the publication of this paper.

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