топология / Tom Dieck T. Algebraic topology (EMS, 2008)

280 Chapter 11. Homological Algebra

The morphisms named a; b; a0; b0 are induced by the original morphisms with the same name by applying them to representatives. (1) and (2) are inclusions, (3) and

(4) are quotients.

(11.2.1) Proposition. Let .a; b/ and .a0; b0/ be exact. Then a connecting morphism

A0

ı W Ke. / \ Im.b/ ! Im.˛/ C Ke.a0/

is defined by the correspondence .a0/ 1ˇb 1.

Proof. For z 2 Ke. / \ Im.b/ there exists y 2 B such that b.y/ D z; since z 2 Ke. / and b D b0ˇ we have ˇ.y/ 2 Ke.b0/; since Ke.b0/ Im.a0/, there exists x0 2 A0 such that a0.x0/ D ˇ.y/. We set ı.z/ D x0 and show that this assignment is well-defined. If yQ 2 B, b.y/Q D z, then b.y y/Q D 0; since Ke.b/ Im.a/,

We add further hypotheses to the original diagram and list the consequences for the derived diagrams. We leave the verification of (11.2.2), (11.2.3), (11.2.4), (11.2.5) to the reader.

11.2.2 If a0 is injective, then (1) and (3) are bijective. If b is surjective, then (2)

11.2.5 Let .a; b/ and .a0; b0/ be exact. Then, as we have seen, ı is defined. Under these assumptions the bottom lines of the derived diagrams together with ı yield an exact sequence. (See the special case (11.2.6).) Þ

(11.2.6) Kernel–Cokernel Lemma. If in the original diagram a0 is injective and b surjective, then .1/; .2/; .3/, and .4/ are bijective and the kernel-cokernel-sequence

a b ı a0 b0

Ke.˛/ ! Ke.ˇ/ ! Ke. / ! Ko.˛/ ! Ko.ˇ/ ! Ko. /

is exact.

Proof. We show the exactness at places involving ı; the other cases have already been dealt with. The relations ıb D 0 and a0ı D 0 hold by construction.

If the class of x0 is contained in the kernel of a0, then there exists y such that a0.x0/ D ˇ.y/. Hence z D b.y/ 2 Ke. / by commutativity, and ı.z/ D x0.

Suppose z 2 Ke. / is contained in the kernel of ı. Then there exists y such that z D b.y/; ˇ.y/ D a0.x0/ and ˇ.z/ D ˛.x/ 2 Im.˛/. Then b.y a.x// D z and ˇ.y a.x// D ˇ.y/ ˇa.x/ D ˇ.y/ a0˛.x/ D ˇ.y/ a0.x0/ D 0. Hence y a.x/ is a pre-image of z.

We now relate the Kernel–Cokernel Lemma to the Five Lemma (11.2.7); see also (11.1.4). Given a commutative five-term diagram of modules and homomorphisms with exact rows.

We have three derived diagrams.

282 Chapter 11. Homological Algebra

The rows of the first two diagrams are exact for trivial reasons. The exactness of the rows of the original diagram implies that the third diagram has exact rows. From the considerations so far we obtain the exact sequences

This yields:

(11.2.7) Five Lemma. Given a five-term diagram as above. Then the following

Problems

1. Given homomorphisms f W A ! B and g W B ! C between R-modules. Then there is a natural exact sequence

0 ! Ke.f / ! Ke.gf / ! Ke.g/ ! Ko.f / ! Ko.gf / ! Ko.g/ ! 0:

The connection to the previous considerations: The commutative diagram with exact rows

can be viewed as an exact sequence of chain complexes. Its homology sequence (11.3.2) is the desired sequence, if we identify the kernel and cokernel of gf ˚1 with the corresponding modules for gf .

Describe the morphisms of the sequence and give also a direct proof.

11.3 Chain Complexes

The algebraic terminology of chain complexes arose from the definition of homology groups. Since then it also has become of independent interest in algebra (homological algebra). The construction of (singular) homology proceeds in two stages: First one associates to a space a so-called chain complex. Then the chain complex yields, by algebra, the homology groups. The category of chain complexes and chain maps has an associated homotopy structure.

We work in this section with the category R- MOD of left modules over some fixed ring R. A family A D .An j n 2 Z/ of modules An is called a Z-graded

module. We call An the component of degree or dimension n. One sometimes

L

considers the direct sum n2Z An; then the elements in An are said to be homogeneous of degree n. Typical examples are polynomial rings; if kŒx; y is the polynomial ring in two indeterminates x; y of degree 1 say, then the homogeneous polynomials of degree n are spanned by xi yn i for 0 i n, and in this manner we consider kŒx; y as a graded k-module (actually a graded algebra, as defined later). One can also consider formal power series; this would correspond to taking

A sequence C D .Cn; @n j n 2 Z/ of modules Cn and homomorphisms

@n W Cn ! Cn 1, called boundary operators or differentials, is said to be a chain complex, if for each n 2 Z the boundary relation @n 1 ı@n D 0 holds. We associate

to a chain complex C the modules

Zn D Zn.C / D Ker.@n W Cn ! Cn 1/;

Bn D Bn.C / D Im.@nC1 W CnC1 ! Cn/; Hn D Hn.C / D Zn=Bn:

We call Cn (Zn, Bn) the module of n-chains (n-cycles, n-boundaries) and Hn the n-th homology module of the chain complex. (The boundary relation @@ D 0

implies Bn Zn, and therefore Hn is defined.) Two n-chains whose difference is a boundary are said to be homologous. Often, in particular in the case R D Z, we talk about homology groups.

Let C D .Cn; cn/ and D D .Dn; dn/ be chain complexes. A chain map f W C ! D is a sequence of homomorphisms fn W Cn ! Dn which satisfy the commutation rules dn ı fn D fn 1 ı cn. A chain map induces (by restriction and passage to the factor groups) homomorphisms of the cycles, boundaries, and homology groups

Zn.f / W Zn.C / ! Zn.D /;

Bn.f / W Bn.C / ! Bn.D /;

f D Hn.f / W Hn.C / ! Hn.D /:

We certainly have the induced morphisms Hn.f / and Hn.g/. Moreover, there exists a connecting morphism @n W Hn.C 00/ ! Hn 1.C 0/, also called boundary

then we have to replace z0 by z0 C dn0 w0, so that the homology class of z0 is welldefined. Finally, if we change z00 by a boundary, we can replace z by the addition

is exact.

Proof. The boundary operator dn W Cn ! Cn 1 induces a homomorphism dn W Kn D Cn=Bn ! Zn 1;

and its kernel and cokernel are Hn and Hn 1. By (11.2.3) and (11.2.4) the rows of the next diagram are exact.

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11.5 Natural Chain Maps and Homotopies

Let C be an arbitrary category and CHC the category of chain complexes .Cn; cn/ of abelian groups with Cn D 0 for n < 0 and chain maps. A functor F W C ! CHC

a family sn W Fn ! GnC1 of natural transformations such that

dnGC1 ı sn C sn 1 ı dnF D n 'n:

A functor Fn W C ! Z- MOD is called free if there exists a family ..Bn;j ; bn;j / j j 2 J.n// of objects Bn;j of C (called models) and elements bn;j 2 Fn.Bn;j / such that

Fn.f /.bn;j /; j 2 J.n/; f 2 HomC .Bn;j ; X/

is for each object X of C a Z-basis of Fn.X/. A natural transformation 'n W Fn ! Gn from a free functor Fn into another functor Gn is then determined by the values 'n.bn;j / and the family of these values can be fixed arbitrarily in order to obtain a natural transformation. We call F free if each Fn is free. We call G acyclic (with respect to the families of models for F ) if the homology groups Hn.G .Bn;j // D 0 for n > 0 and each model Bn;j .

(11.5.1) Theorem. Let F be a free and G be an acyclic functor. For each natural transformation 'x W H0 ı F0 ! H0 ı G0 there exists a natural transformation ' W F ! G which induces 'x. Any two natural transformations ' and with this property are naturally chain homotopic ([57]).

Proof. We specify a natural transformation '0 by the condition that '.b0;j / rep-

resents the homology class 'Œbx 0;j . Let now 'i W Fi ! Gi be natural transformations .0 i < n/ such that diG 'i D 'i 1diF for 0 < i < n. Consider the

elements 'n 1dnF bn;j 2 Gn 1.Bn;j /. For n D 1 this element represents 0 in H0, by the construction of '0. For n > 1 we see from the induction hypothesis

exact sequence of R-modules. A chain map .' W Fi ! Di j i 2 N0/ induces a homomorphism H0.' / W H0.F / ! H0.D /. Given a homomorphism ˛ W H0.F / ! H0.D0/ there exists up to chain homotopy a unique chain map .'i / such that H0.' / D ˛. This can be obtained as a special case of (11.5.1).

The reader should now study the notion of a projective module (one definition is: direct summand of a free module) and then show that a similar result holds if the Fi are only

Pi is called a projective resolution of the module A. The result stated at the beginning says that projective resolutions are unique up to chain equivalence. (Fundamental Lemma of

homological algebra). Each module has a free resolution.

11.6 Chain Equivalences

A chain map which induces an isomorphism of homology groups is under certain circumstances a chain equivalence. This is one of the results of this section.

The notion of a chain homotopy can be used to develop a homotopy theory of chain complexes in analogy to the topological homotopy theory.

We have the null complex; the chain groups are zero in each dimension. A chain complex is called contractible if it is chain equivalent to the null complex, or

equivalently, if the identity is chain homotopic to the null map. A chain complex is said to be acyclic if its homology groups are zero.

Let f W .K; d K / ! .L; d L/ be a chain map. We construct a new chain complex

Cf , the mapping cone of f , by