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

288 Chapter 11. Homological Algebra

The suspension †K of K is defined by .†K/n D Kn 1 and d †K D d K . The canonical injection and projection yield an exact sequence of chain complexes 0 ! L ! Cf ! †K ! 0. Associated is an exact sequence (11.3.2), and the boundary morphism @ W HnC1.†K/ ! Hn.L/ equals Hn.f /, if we use the canonical identifications HnC1.†K/ Š Hn.K/. The next result shows a typical difference between the topological and the algebraic homotopy theory.

(11.6.1) Theorem. Let Cf be contractible. Then f is a chain equivalence.

Proof. The inclusion W L ! Cf , y 7!.y; 0/ is null homotopic, since Cf is contractible. Let s W ' 0 be a null homotopy. We write s.y/ D . .y/; g.y// 2 L ˚ K (without notation for the dimensions). The condition @s C s@ D then reads

.@ y C fgy C @y; @gy C g@y/ D .y; 0/;

i.e., @g D g@ and @ C @ D id fg. Hence g is a chain map, and because of the-relation, a right homotopy inverse of f .

The projection W Cf ! †K is likewise null homotopic. Let t W ' 0 be a null homotopy. We write t.y; x/ D h.y/ C .x/. The equality @t C t@ D then means

@hy C h@y @ x C @x C hf x D x;

(11.6.2) Proposition. Let K be acyclic and suppose that Zn Kn is always a direct summand. Then K is contractible.

@

Proof. We have the exact sequence 0 ! Zn ! Kn ! Bn 1 ! 0, and since K is acyclic we conclude Zn D Bn. Moreover there exists tn 1 W Bn 1 ! Kn with @tn 1 D id, since Zn is a direct summand of Kn. We therefore have a direct decomposition Kn D Bn ˚ tn 1Bn 1. We define s W Kn ! KnC1 by sjBn D tn and snjtn 1Bn 1 D 0. With these definitions one verifies separately on Bn as well

as on tn 1Bn 1 that @s C s@ is the identity, i.e., s is a null homotopy of the identity.

(11.6.3) Theorem. Let f W K ! L be a chain map between chain complexes which consist of free modules over a principal ideal domain R. If f induces isomorphisms f W H .K/ Š H .L/, then f is a chain equivalence.

Proof. The exact homology sequence and the hypothesis imply that Cf is acyclic. A submodule of a free R-module is free. Hence the boundary groups of the complex Cf are free, and therefore the exact sequence 0 ! Zn ! Cfn ! Bn 1 ! 0 splits.

Now we apply (11.6.1) and (11.6.2), in order to see that f is a chain equivalence.

In the topological applications we often have to work with large chain complexes. In some situations it is useful to replace them by smaller chain equivalent complexes. A graded R-module A D .An/ is said to be of finite type if the modules An are finitely generated R-modules.

(11.6.4) Proposition. Let R be a principal ideal domain. Let C D .Cn/ be a chain complex of free R-modules such that its homology groups are finitely generated. Then there exists a free chain complex D of finite type which is chain equivalent to C .

Proof. Let Fn be a finitely generated submodule of Zn.C / which is mapped onto Hn.C / under the quotient map Zn.C / ! Hn.C /, and denote by Gn the kernel of the epimorphism Fn ! Hn.C /. Define a chain complex D D .Dn; dn/ by Dn D Fn ˚ Gn 1 and dn.x; y/ D .y; 0/. Then D is a free chain complex of finite type and Hn.D/ D Fn=Gn Š Hn.C /. Since Gn is a free submodule of Bn.C / we can choose for each n a homomorphism 'n W Gn ! CnC1 such that cnC1'n.y/ D y

11.7 Linear Algebra of Chain Complexes

We work in the category R- MOD for a commutative ring R.

11.7.1 Graded modules. Let A D .An/ and B D .Bn/ be Z-graded left R-

.f ˝ g/.a ˝ b/ D . 1/jgjjajf .a/ ˝ g.b/:

Here jaj denotes the degree of a. The formula for the tensor product obeys the (heuristic) “graded sign rule”: Whenever entities of degree x and y are interchanged, then the sign . 1/xy appears. The tensor product of objects and of morphisms is associative and compatible with composition (in the graded sense)

.f ˝ g/ ı .f 0 ˝ g0/ D . 1/jgjjf 0 jff 0 ˝ gg0

(sign rule). This composition is associative, as it should be. When we use the degree as upper index (e.g., in cohomology), then the agreement Ak D A k is sometimes

290 Chapter 11. Homological Algebra

The algebra is associative, if always x .y z/ D .x y/ z holds, and commutative, if always x y D . 1/jxjjyjy x holds (sign rule). A unit element 1 2 A0 of the

algebra satisfies 1 x D x D x 1. Let M D .M n/ be a Z-graded R-module. A

of R-linear maps is the structure of an A -module on M , provided the associativity a .b x/ D .a b/ x holds for a; b 2 A and x 2 M . If A has a unit element, then the module is unital, provided 1 x D x always holds. Let A and B be Z-

then A ˝ B is associative. If both have a unit element 1, then 1 ˝ 1 is a unit element for the tensor product. If both algebras are commutative, then their tensor product is commutative. The tensor product of graded algebras is an associative functor.Þ

11.7.3 Tensor product of chain complexes. Let .A ; dA/ and .B ; dB / be chain complexes. Then the graded module A ˝ B is a chain complex with boundary operator d D dA ˝ 1 C 1 ˝ dB . Here we have to take the sign rule into account,

i.e.,

d.a ˝ b/ D dAa ˝ b C . 1/jaja ˝ dB b:

11.7.4 Dual chain complex. We regard the ground ring R as a trivial chain complex with R in degree 0 and zero modules otherwise. Let .An; @/ be a chain complex.

" W A n ˝ An ! R; ' ˝ a 7!'.a/;

" D 0 otherwise, becomes a chain map. This condition, ".' ˝ a/ D 0 and 11.7.3 yield for ' ˝ a 2 A n 1 ˝ An

0 D d ".' ˝ a/ D ".d.' ˝ a/

D".ı' ˝ a C . 1/j'j' ˝ @a

D.ı'/.a/ C . 1/j'j'.@a/;

11.7.5 Hom-Complex. For graded modules A and B we let Hom.A ; B / be

d.fi / D .@ ı fi / .. 1/nfi ı @/

for .fi W Ai ! BiCn/ , i.e., the a-component pra.df / 2 Hom.Aa; BaCn 1/ for f D .fa/ 2 Hom.A ; B /n is defined to be

pra.df / D @ ı fa . 1/nga 1 ı @:

One verifies dd D 0. This definition generalizes our convention about the dual module. Þ

11.7.6 Canonical maps. The following canonical maps from linear algebra are

Problems

1. Tensor product is compatible with chain homotopy. Let s W f ' g W C ! C 0 be a chain homotopy. Then s ˝ id W f ˝ id ' g ˝ id W C ˝ D ! C 0 ˝ D is a chain homotopy.

2. A chain complex model of the unit interval is the chain complex I with two non-zero groups I1 Š R with basis e, I0 Š R ˚ R with basis e0; e1 and boundary operator d.e/ D e1 e0 (in the topological context: the cellular chain complex of the unit interval). We use this model to define chain homotopies with the cylinder I ˝ C . Note

Cn ˚ Cn ˚ Cn 1 Š .I ˝ C /n; .x1; x0; y/ 7!e1 ˝ x1 C e0 ˝ x0 C e ˝ y:

A chain map h W I ˝ C ! D consists, via these isomorphisms, of homomorphisms htn W Cn ! Dn and sn W Cn ! DnC1. The ht are chain maps (t D 0; 1) and dsn.y/ D h1n.y/ h0n.y/ sn 1cy, i.e., s W h1 ' h0 is a chain homotopy in our previous definition. 3. Imitate the topological definition of the mapping cone and define the mapping cone of a chain map f W C ! D as a quotient of I ˝C ˚D. The n-th chain group is then canonically isomorphic to Cn 1 ˚ Dn and the resulting boundary operator is the one we defined in the section on chain equivalences. Consider also the mapping cylinder from this view-point.

292 Chapter 11. Homological Algebra

11.8 The Functors Tor and Ext

Let R be a principal ideal ring. We work in the category R- MOD; this comprises

the category of abelian groups (Z-modules). An exact sequence 0 ! F1 ! F0 ! A ! 0 with free modules F1; F0 is a free resolution of A. Since submodules of

free modules are free, it suffices to require that F0 is free. Let F .A/ denote the free R-module generated by the set A. Denote the basis element of F .A/ which

0 ! K.A/ ! F .A/ ! A ! 0

will be called the standard resolution of A. We take the tensor product (over R)

of this sequence with a module G, denote the kernel of i ˝ 1 by TorR.A; G/ D Tor.A; G/ and call it the torsion product of A; G.

We now derive some elementary properties of torsion products. We show that Tor.A; G/ can be determined from any free resolution, and we make Tor. ; / into a functor in two variables. In the next lemma we compare free resolutions.

(11.8.1) Lemma. Given a homomorphism f W A ! A0 and free resolution F and F 0 of A and A0, there exists a commutative diagram

shows that this homomorphism does not depend on the choice of .f1; f0/. Let us denote this homomorphism by T .f I F ; F 0/. If g W A0 ! A00 is given and F 00 a

free resolution of A00, then T .gI F 0; F 00/ ı T .f I F ; F 0/ D T .gf I F ; F 00/. This implies that an isomorphism f induces an isomorphism T .f I F ; F 0/. In particular each free resolution yields a unique isomorphism Ker.i ˝ 1/ Š Tor.AI G/, if we compare F with the standard resolution. The standard resolution is functorial in A. This fact is used to make Tor. ; G/ into a functor. It is clear that a homomorphism G ! G0 induces a homomorphisms Tor.A; G/ ! Tor.A; G0/. Hence Tor is also a functor in the variable G (and the two functor structures commute).

If we view 0 ! F1 ! F0 ! 0 in (11.8.1) as a chain complex, then .f1; f0/ is a chain map and s yields a chain homotopy between .f1; f0/ and .f10; f00/.

(11.8.2) Proposition. Elementary properties of torsion groups in the category of abelian groups are:

(1)Let A be a free abelian group. Then Tor.A; G/ D 0.

(2)Tor.Z=n; G/ Š fg 2 G j ng D 0g G.

(3)If G is torsion free, then Tor.Z=n; G/ D 0.

(4)Tor.Z=m; Z=n/ Š Z=d with d the greatest common divisor of m; n.

(5)A direct sum decomposition A Š A1 ˚A2 induces a direct sum decomposition

Tor.A; G/ Š Tor.A1; G/ ˚ Tor.A2; G/.

Proof. (1) 0 ! 0 ! A ! A ! 0 is a free resolution. (2) Use the free resolution

n

0 ! Z ! Z ! Z=n ! 0. (3) and (4) are consequences of (2). In order to verify (5), use the direct sum of free resolutions.

We can also work with a resolution of the other variable. Let Q1 Q0 B be a free resolution and define Tor0.A; B/ D Ker.A ˝ Q1 ! A ˝ Q0/.

(11.8.3) Proposition. There exists a canonical isomorphism

Tor.A; B/ Š Tor0.A; B/:

Proof. Let P1 ! P0 ! A be a free resolution. From the resolutions of A and B we obtain a commutative diagram:

294 Chapter 11. Homological Algebra

The Kernel–Cokernel Lemma (11.2.6) yields an isomorphism ı of Tor.A; B/ D Ker. / with the submodule Tor0.A; B/ of Coker ˛.

Interchanging the tensor factors yields an isomorphism Tor.B; A/ Š Tor0.A; B/. We combine this with (11.8.3) and see that the isomorphisms (11.8.2) also hold if we interchange the variables. It is now no longer necessary to use the notation Tor0.

The functor Ext is defined in analogy to the functor Tor, the tensor product is replaced by the Hom-functor.

i p

Let R be a principal ideal domain and 0 ! K.A/ ! F .A/ ! A ! 0 the standard free resolution of A as above. We apply the functor HomR. ; B/ to this sequence. The cokernel of i W Hom.F .A/; B/ ! Hom.K.A/; B/ is defined to be ExtR.A; B/ D Ext.A; B/. We show that Ext.A; B/ can be determined from any free resolution. We start with a diagram as in (11.8.1) and obtain a well-defined homomorphism Coker.Hom.i; B// ! Coker.Hom.i0; B//; in particular we obtain an isomorphism Ext.A; B/ Š Coker.Hom.i; B//.

(11.8.4) Proposition. Elementary properties of Ext in the category of abelian groups are:

The foregoing develops what we need in this text. We should at least mention the general case. Let 0 C P0 P1 be a projective resolution of the R-module C and let A be another R-module. We apply Hom. ; A/ to the chain complex P and obtain a cochain complex Hom.P ; A/; its i-th cohomology group (i 1) is denoted ExtiR.C; A/. Since projective resolutions are unique up to chain equivalence, the ExtiR-groups are unique up to isomorphism. For principal ideal domains only Ext1 occurs, since we have resolution of length 1. The notation Ext has its origin in the notion of extensions of modules. An exact sequence

0 ! A ! Bn 1 ! ! B1 ! B0 ! C ! 0

is called an n-fold extension of A by C . One can obtain ExtnR.C; A/ as certain congruence classes of n-fold extension of A by C , see [120, Chapter III]. Write

EE0 if there exists a commutative diagram

The congruence relation is generated by .

Problems

1.Suppose Tor.A; Z=p/ D 0 for each prime p. Then the abelian group A is torsion free.

2.The kernel of A ! A ˝Z Q, a 7!a ˝ 1 is the torsion subgroup of A.

3.Does there exist a non-trivial abelian group A such that A ˝ F D 0 for each field F ?

11.9 Universal Coefficients

We still work in R- MOD for a principal ideal domain R. Let C D .Cn; cn/ be a chain complex of modules. Then C ˝ G D .Cn ˝ G; cn ˝ 1/ is again a chain complex.

(11.9.1) Proposition (Universal Coefficients). Let C be a chain complex of free modules. Then there exists an exact sequence

˛ ˇ

0 ! Hq .C / ˝ G ! Hq .C ˝ G/ ! Tor.Hq 1.C /; G/ ! 0:

The sequence is natural in C and G and splits. The homomorphism ˛ sends Œz ˝ g for a cycle z to the homology class Œz ˝ g .

Proof. The sequence n n cn Bn 1 ! 0 is exact; Bn 1 is a submodule

!

C

!

0 ! Z

of Cn 1 and hence free. Therefore the sequence splits and the induced sequence

0 ! Zn ˝ G ! Cn ˝ G ! Bn 1 ˝ G ! 0

is again a split exact sequence. We consider the totality of these sequences as an exact sequence of chain complexes, the Z- and the B-complex have trivial boundary operator. Associated to this short exact sequence of chain complexes is a long exact homology sequence of the form

One verifies that the boundary operator (11.3.1) of the homology sequence is i ˝ 1, where i W Bn Zn. The sequence Bn ˝ G ! Zn ˝ G ! Hn ˝ G ! 0 is exact, hence the cokernel of i ˝ 1 is Hn.C / ˝ G, and the resulting map Hn.C / ˝ G ! Hn.C ˝ G/ is ˛. The kernel of Bn 1 ˝ G ! Zn 1 ˝ G is Tor.Hn 1.C /; G/, because 0 ! Bn 1 ! Zn 1 ! Hn 1.C / ! 0 is a free resolution. Let r W Cn ! Zn be a splitting of Zn Cn. Then

r˝1

Zn.C ˝ G/ Cn ˝ G ! Zn ˝ G ! Hn.C / ˝ G

maps Bn.C ˝G/ to zero and induces W Hn.C ˝G/ ! Hn.C /˝G with ˛ D id,

296 Chapter 11. Homological Algebra

Let again C D .Cn; cn/ be a chain complex with free R-modules Cn. We obtain the cochain complex with cochain groups Hom.Cn; G/ and cohomology groups H n.C I G/.

(11.9.2) Proposition (Universal Coefficients). There exists an exact sequence

0 Ext.H .C /; G/ H n.C G/ ˛ Hom.H .C /; G/ 0:

! n 1 ! I ! n !

The map ˛ sends the cohomology class of the cocycle ' W Cn ! G to the homomorphism Hn.C / ! G, Œc 7!'.c/. The sequence is natural with respect to chain maps (variable C ) and module homomorphisms (variable G). The sequence splits, and the splitting is natural in G but not in C .

Proof. Again we start with the split exact sequence 0 ! Zn ! Cn ! Bn 1 ! 0 and the induced exact sequence

0 Hom.Zn; G/ Hom.Cn; G/ Hom.Bn 1; G/ 0:

We consider the totality of these sequences as an exact sequence of cochain complexes, the Z- and the B-complex have trivial coboundary operator. Associated to this short exact sequence of cochain complexes is a long exact cohomology sequence of the form

We need:

(11.9.3) Lemma. The formal coboundary operator d n (without the additional sign introduced earlier!) is the homomorphism induced by i W Bn ! Zn.

Proof. Let ' W Zn ! G be given. Then d n.'/ is obtained as follows: Extend ' to

From the exact sequence 0 ! Bn ! Zn ! Hn.C / ! 0 we obtain the exact sequence

i

Hom.Bn; G/ Hom.Zn; G/ Hom.Hn.C /; G/ 0:

We use it to identify the Ker i with Hom.Hn.C /; G/. One verifies that ˛ is as claimed in the statement (11.9.2). From the free presentation and the definition

of Ext we thus obtain the exact sequence of the theorem. The naturality of this sequence is a consequence of the construction. It remains to verify the splitting. We choose a splitting r W Cn ! Zn of the inclusion Zn Cn. Now consider the diagram

Without going into the definition of Ext we see from the discussion:

(11.9.4) Proposition. Suppose Hn 1.C / is a free R-module. Then the homomorphism ˛ W H n.C I G/ ! Hom.Hn.C /; G/ in (11.9.2) is an isomorphism.

Proof. The sequence 0 ! Bn 1 ! Zn 1 ! Hn 1 ! 0 splits and therefore the cokernel of d n 1 is zero.

Given a cochain complex C D .C q ; ıq / we can view it as a chain complex C D .Cq ; @q / by a shift of indices: We set Cq D C q and we define @q W Cq ! Cq 1 as ı q W C q ! C qC1. We can now rewrite (11.9.1):

(11.9.5) Proposition. Let C be a cochain complex of free R-modules. Then we have a split exact sequence

0 ! H q .C / ˝ G ! H q .C ˝ G/ ! Tor.H qC1.C /; G/ ! 0:

Let now C be a chain complex of free modules. We apply (11.9.5) to the dual cochain complex with C q D Hom.Cq ; R/ and cohomology groups H q .C I R/.

(11.9.6) Proposition. Let C be a free chain complex and G be a module such that either H .C / is of finite type or G is finitely generated. Then there exists a natural exact sequence

0 ! H p.C / ˝ G ! H q .C I G/ ! Tor.H qC1.C /; G/ ! 0

and this sequence splits.

Proof. If G is finitely generated we have a canonical isomorphism of the form Hom.C; R/ ˝ G Š Hom.C; G/; we use this isomorphism in (11.9.5). If H .C / is of finite type we replace C by a chain equivalent complex C 0 of finite type (see (11.6.4)). In that case we have again a canonical isomorphism Hom.C 0; R/ ˝ G Š Hom.C 0; G/. We apply now (11.9.5) to C 0.