A COURSE IN MODERN ALGEBRAIC TOPOLOGY...A COURSE IN MODERN ALGEBRAIC TOPOLOGY IGOR KRIZ Abstract. This is a course in algebraic topology for anyone who has seen the fundamental group

A COURSE IN MODERN ALGEBRAIC TOPOLOGY

IGOR KRIZ

Abstract. This is a course in algebraic topology for anyone who has seen the fundamentalgroup and homology. Traditionally (20 years ago), the syllabus included cohomology, theuniversal coefficient theorem, products, Tor and Ext and duality. We will cover all thosetopics, while also building a modern framework of derived categories and derived functorsusing the Cartan–Eilenberg method. We will apply these methods to modules, topologicalspaces, simplicial sets, sheaves, and spectra (=generalized cohomology theories).

Contents

0. Notations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31. Homology and Cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.1. Singular homology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31.2. Singular cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41.3. Universal coefficient theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51.4. The broader context of Tor and Ext . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 71.5. Example: group homology and cohomology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.5.1. Connection with topology. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 91.6. Kunneth theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 101.7. Eilenberg–Zilber theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.8. The cup product. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2. Cell structure on R-chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162.1. Mapping cone . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1. Chain homotopies as chain maps. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182.2. Cell R-chain complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 192.3. Derived category of R-chain complexes. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

3. Derived categories and derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243.1. Derived categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

3.1.1. Derived categories in the 2-categorical sense . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.2. Kan extensions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 253.3. Derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 283.4. Co-localization and localization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

3.4.1. Example: Tor and Ext as left derived functors . . . . . . . . . . . . . . . . . . . . . . . . . . 323.5. Co-localization in abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333.6. Localization in abelian categories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.6.1. Digression: lim1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 383.7. Total Tor and Ext functors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40

Appendix A. Solutions to Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

Date: October 24, 2017.Notes taken by Ruian Chen from Kriz’s Math 695 class taught in University of Michigan in Fall 2017.

1

2 IGOR KRIZ

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

A COURSE IN MODERN ALGEBRAIC TOPOLOGY 3

0. Notations

Some common categories are denoted as follows:Set sets, functionsTop (topological) spaces, (continuous) mapsAb abelian groups, group morphisms

R-Mod (left) R-modules, R-module (R-linear) morphismsUnless otherwise specified, all rings considered here are unital, associative and commuta-

tive.If A is an abelian category, the category of chain complexes in A and chain maps are

denoted Ch(A ). In the special case A = R-Mod, we abbreviate Ch(A ) = Ch(R).We use ∼=, ' and ∼ to denote isomorphisms, (strong) homotopy (equivalences), and weak

equivalences, respectively.

1. Homology and Cohomology

In this section, we define (singular) homology and cohomology with coefficients. Onereason we are interested in them is that, if A is an R-module, where R is a (commutative)ring, the homology and cohomology are functors from the homotopy category of spaces toR-modules. For example, if R is a field F , then the homology and cohomology groups withcoefficients in F is an F -vector space.

1.1. Singular homology. The standard nsimplex is the topological space

∆n = {(t0, ..., tn) ∈ Rn+1 :∑

ti = 1, ti ≥ 0}.

For each i = 0, ..., n, the i-th face map is defined by

∂i : ∆n−1 −→ ∆n

(t0, ..., tn−1) 7−→ (t0, ..., ti−1, 0, ti, ..., tn−1)

A singular n-simplex in a space X is a map σ : ∆n → X. Let Sn be the set of all singularn-simplexes, and CnX = ZSnX the free abelian group on Sn. Define a differential d = dn :CnX → Cn−1X by

d(σ) =n∑i=0

(−1)iσ ◦ ∂i

for σ ∈ SnX. One can check that d2 = 0, so we obtain a chain complex

C∗X = · · · −→ CnXd−→ Cn−1X

d−→ Cn−2X −→ · · · ,

called the singular chain of X.

Theorem 1.1.1. The singular chain forms a functor

C∗ : Top −→ Ch(Z).

Moreover, C∗X is a chain complex of free abelian groups. �

Definition. The n-th singular homology of a space X with coefficient in an abelian groupA is defined as

Hn(X;A) := Hn(C∗X ⊗ A).

4 IGOR KRIZ

Corollary 1.1.2. For each n, the n-th singular homology forms a functor

Hn(−;A) : Top −→ Ab. �

Exercise 1. Let S be a set, and A an abelian group. Show that

ZS ⊗ A ∼= A⊕S.

For Y ⊆ X, define the singular chain of X relative to Y to be C∗(X, Y ) := C∗(X)/C∗(Y ).In particular, o we have a short exact sequence

0 −→ C∗Y −→ C∗X −→ C∗(X, Y ) −→ 0.

Since CnY ↪→ CnX is induced by the set map SnY ↪→ SnX, we have

CnX ∼= Z⊕SnY ⊕ Z⊕(SnX\SnY ),

so Cn(X/Y )⊗ ∼= Z⊕(SnX\SnY ) is also free, and the short exact sequence splits. Therefore,using Exercise 1, we have a short exact sequence

0 −→ C∗Y ⊗ A −→ C∗X ⊗ A −→ C∗(X, Y )⊗ A −→ 0

for any abelian group A, which again splits.

Definition. The n-th relative singular homology of a space X relative to a subspace Y withcoefficient in an abelian group A is defined as

Hn(X : A) := Hn(C∗(X, Y )⊗ A).

In particular, we have a long exact sequence

· · · −→ Hn(Y ;A) −→ Hn(X;A) −→ Hn(X, Y ;A) −→ Hn−1(Y ;A) −→ · · ·This is the Exactness Axiom of the Eilenberg–Steenrod axioms. In fact, all the otherEilenberg–Steenrod axioms hold, and proof is exactly the same as the case with Z coef-ficient. We recall that the Dimension Axiom reads

Hn(∗;A) =

{A n = 0,

0 n 6= 0.

1.2. Singular cohomology. For a space X and an abelian group A, the singular cochainof X is defined as

C∗(X;A) := Hom(C∗X,A).

This is a cochain complex, with differential

dn = Hom(dn+1, A) : Hom(Cn, A) −→ Hom(Cn+1, A).

Note. A cochain complex C∗ can be made into a chain complex C∗ by setting C−n = Cn andd−n = dn.

Definition. The n-th singular cohomology of a space X with coefficient in an abelian groupA is defined as

Hn(X;A) := Hn(Hom(C∗X,A)).

For a subspace Y ⊆ X, the n-th singular cohomology of X relative to Y with coefficient inA is defined as

Hn(X, Y ;A) := Hn(Hom(C∗(X, Y ), A)).


Theorem 1.2.1. The singular cochain forms a functor

C∗ : Topop −→ Ch(Z).

Therefore, for each n, the n-th singular cohomology forms a functor

Hn(−;A) : Topop −→ Ab. �

Since the short exact sequence for singular chains splits, and that Hom(−, A) takes directsum to direct sum (in fact, direct product), we also have a long exact sequence

· · · −→ Hn(X, Y ;A) −→ Hn(X;A) −→ Hn(Y ;A) −→ Hn+1(X, Y ;A) −→ · · · .

1.3. Universal coefficient theorems. Let C be a chain complex of free abelian groups.We denote

Zn := Ker[Cndn−→ Cn−1], Bn := Im[Cn+1

dn+1−−−→ Cn], Hn := HnC ∼= Zn/Bn.

Then we have short exact sequences

0 −→ Bn −→Zn −→ Hn −→ 0,

0 −→ Zn −→Cnd−−→ Bn−1 −→ 0.

Since a subgroup of a free abelian group is free abelian, both Zn and Bn are free abelian. Inparticular, the second short exact sequence splits, and we have

Cnϕn−→ Zn ⊕Bn−1.

Furthermore, C decomposes as follows:

Cn+1

dn+1 //

∼= ϕn+1

��

Cndn //

∼= ϕn

��

Cn−1

∼= ϕn−1

��Zn+1⊕ Zn⊕ Zn−1⊕Bn

)

77

Bn−1

)

77

Bn−2

Therefore, we obtain:

Theorem 1.3.1 (Structure theorem for chain complex of free abelian groups). If C is achain complex of free abelian groups, we have

C =⊕n∈Z

(Bn ↪−→ Zn)[n],

where Bn ↪→ Zn is regarded as a two-stage chain complex on degrees 1 and 0, and [n] is thefunctor on chain complexes which shifts the degrees up by n, i.e., (C[n])i = Ci−n. �

As a result, to understand the homology of C ⊗ A, where C is a chain complex of freeabelian groups and A any abelian group, it suffices to understand it for the special case of atwo-stage chain complex of the form

(1.3.1) C = · · · −→ 0 −→ B ↪−→ Z −→ 0 −→ · · ·

6 IGOR KRIZ

where B and Z are free abelian groups on degrees 1 and 0. We set H = Z/B, so we have ashort exact sequence

(1.3.2) 0 −→ B −→ Z −→ H −→ 0.

Since −⊗ A is right exact, we have an exact sequence

B ⊗ A −→ Z ⊗ A −→ H ⊗ A −→ 0,

soH0(C ⊗ A) = H ⊗ A.

However, we do not know what H1 is; rather, we give it a name:

TorZ1 (H,A) := H1(C ⊗ A).

Note. TorZ1 , or in fact TorRn for any commutative ring R and any integer n, is symmetric inthe two coordinates; see Section 3.7.

Exercise 2. Prove the following calculations on TorZ1 :

(1) TorZ1 (Z/m,Z/n) = Z/ gcd(m,n), for any m,n ∈ N.(2) TorZ1 (A,B) = 0 if B is torsion-free.

Theorem 1.3.2 (Universal coefficient theorem for homology). Let C be a chain complex offree abelian groups. Then for an abelian group A, there is a split exact sequence

0 −→ HnC ⊗ A −→ Hn(C ⊗ A) −→ TorZ1 (Hn−1C,A) −→ 0.

In particular, there is a (non-canonical) isomorphism

Hn(C ⊗ A) = ((HnC)⊗ A)⊕ TorZ1 (Hn−1C,A). �

Example. If n is even, then

Hi(RPn) =

Z i = 0,

Z/2 0 < i < n odd,

0 else

Using the universal coefficient theorem, we conclude that

Hi(RPn;Z/2) =

{Z/2 0 ≤ i ≤ n,

0 else

Similarly, if n is odd, then

Hi(RPn) =

Z i = 0, n,

Z/2 0 < i < n odd,

0 else

Using the universal coefficient theorem again, we obtain

Hi(RPn;Z/2) =

{Z/2 0 ≤ i ≤ n,

0 else


Similarly, to understand the cohomology of Hom(C,A), it suffices to understand the casewhen C is the two-stage complex (1.3.1). Applying Hom(−, A) to the short exact sequence(1.3.2), we have an exact sequence

0 −→ Hom(H,A) −→ Hom(Z,A) −→ Hom(B,A),

so againH0 Hom(C,A) = Hom(H,A),

and we also give a name to H1:

Ext1Z(C,A) := H1 Hom(C,A).

Remark. Both TorZ1 and Ext1Z are independent of the choice of C; we defer the proof to our

general discussion of Tor and Ext in the next section.

Exercise 3. Prove the following calculations on ExtZ1 :

(1) ExtZ1 (Z/m,Z/n) = Z/ gcd(m,n), for any m,n ∈ N.(2) ExtZ1 (Z, A) = 0 for any abelian group A.

Theorem 1.3.3 (Universal coefficient theorem for cohomology). Let C be a chain complexof free abelian groups. Then for an abelian group A, there is a split exact sequence

0 −→ Ext1Z(Hn−1C,A) −→ Hn Hom(C,A) −→ Hom(HnC,A) −→ 0.

In particular, there is a (non-canonical) isomorphism

Hn Hom(C,A) = Hom(HnC,A)⊕ Ext1Z(Hn−1C,A). �

1.4. The broader context of Tor and Ext. In this subsection, we fix a commutative andunital ring R.

Definition. For R-modules M and N , let C be a free resolution of M . Define

TorRn (M,N) := Hn(C ⊗R N),

andExtnR(M,N) := Hn(HomR(C,N)).

Note. If R is not commutative, we must distinguish left and right R-modules. We usuallyconsider left R-modules, as right R-modules are precisely left Rop-modules, where Rop is the“same” ring as R, except with multiplication reversed. If a module has both left and rightR-module structures, then it is called an R-bimodule. Note that over a commutative ring, amodule is naturally equipped a bimodule structure.

In general, M ⊗R N is defined when M is a right R-module and N is a left R-module, inwhich case it is just an abelian group. If both M and N are R-bimodules, then M ⊗R N isan R-module. TorRn (M,N) is behaved analogously.

Similarly, HomR(M,N) is defined when M and N are both left or right R-modules, inwhich case it is an abelian group. If both M and N are R-bimodules, then HomR(M,N) isan R-module. ExtnR(M,N) behaves analogously.Theorem 1.4.1.

(1) Every R-module M has a free resolution C.(2) If C is a free resolution of M , D is any resolution of N , and f : M → N is an

R-module map, then there exists an R-chain map f : C → D, unique up to R-chain

homotopy, such that H0(f) = f .

8 IGOR KRIZ

Proof of existence of free resolution. Given an R-module M , consider the free R-module RMon M . The universal property gives an R-module map

RM −−� M −→ 0(m) 7−→ m

Define M0 := M . Suppose Mn has been constructed, we have a short exact sequence

0 −→Mn+1 −→ RMn −→Mn −→ 0.

Then· · · −→ RM2 −→ RM1 −→ RM0.

is a free R-resolution of M . �

We postpone the proof of uniqueness (up to homotopy) to §2.2, where we prove a slightmore general result (Corollary 2.2.4).

Exercise 4. Verify that −⊗R N and HomR(−, N) preserve R-chain homotopies.

Corollary 1.4.2. For any R-module M and N , the R-modules TorRn and ExtnR are well-defined. Moreover, they define functors

TorRn (−, N) : Ch(R) −→ R-Mod

andExtnR(−, N) : Ch(R) −→ R-Modop. �

Exercise 5. Let K be a field, and R =∧K [x] ∼= K[x]/(x2), which is K{1, x} as an abelian

group. Let K be an R-module where x acting by 0. Compute TorRn (K,K) and ExtnR(K,K)for all n.

1.5. Example: group homology and cohomology. For a group G, we define group ho-mology and cohomology with coefficients in a Z[G]-module (also referred to as “G-module”)M as

Hn(G;M) := TorZ[G]n (Z,M), Hn(G;M) := ExtnZ[G](Z,M),

where Z is the trivial Z[G]-module, where any g ∈ G acting by identity.

Example (The case G = Z/k). In this example, we compute the group (co)homologyof Z/k with integral coefficients. Choose a generator α for Z/k, and regard Z[Z/k] =Z{1, α, ..., αk−1} as an abelian group. We construct an augmented free Z[Z/k]-resolution ofZ as follows:

· · · −→ Z[Z/k]N−→ Z[Z/k]

T−→ Z[Z/k]N−→ Z[Z/k]

T−→ Z[Z/k]ε−→ Z −→ 0,

where ε : 1 7→ 1, soKer ε = Z{1− α, α− α2, ..., αk−2 − αk−1}.

Set

T =

1 −1

1 −1. . . . . .

1 −1−1 · · · 1

,


so KerT = Z{1 + α + · · ·+ αk−1}. Set

N =

1 · · · 1...

. . ....

1 · · · 1

so KerN = Ker ε. This shows that

C := · · · −→ Z[Z/k]N−→ Z[Z/k]

T−→ Z[Z/k]N−→ Z[Z/k]

is a free Z[Z/k]-resolution of Z. Now we compute

C ⊗Z[Z/k] Z = · · · −→ Z k−→ Z 0−→ Z k−→ Z 0−→ Z,

andHomZ[Z/k](C,Z) = · · · ←− Z k←− Z 0←− Z k←− Z 0←− Z,

so

Hn(Z/k;Z) =

Z n = 0,

Z/k n > 0 odd,

0 else,

Hn(Z/k;Z) =

Z n = 0,

Z/k n > 0 even,

0 else,

Exercise 6. Let Z be Z equipped with a Z[Z/2]-module where the generator α acts by −1.

Compute H∗(Z/2; Z) and H∗(Z/2; Z).

1.5.1. Connection with topology. Let G be a discrete group. Then there exists a connected

CW-complex X where π1X ∼= G, and the universal cover X of X is contractible. Such a

CW-complex is unique up to homotopy equivalence. One way to construct such X is byusing “cells” of the form

Sn−1 ×G ↪−→ Dn ×G.Even more concretely, we start with a connected CW-complex X0 with π1(X0) = G, and let

X0 be its universal cover. Attach a cone

CX0 = X0 × I/(x, 1) ∼ (y, 1)

of X0 for each element in G, we form

X1 := X0 q(G× CX0

)/gx ∼ (g, x, 0) .

Repeat this process, and we can form

Xn+1 := X0 q(G× CXn

)/gx ∼ (g, x, 0) ,

X :=⋃

Xn,

X := X/G.

Since X is contractible, the singular chain C∗X is a Z[G]-chain complex, in fact, a freeZ[G]-resolution of Z. If M is a trivial Z[G]-module, then

Hn(G;M) = Hn

(C∗X ⊗Z[G] M

)= Hn

(C∗X ⊗Z[G] Z⊗Z M

)

10 IGOR KRIZ

= Hn(C∗X ⊗Z M)

= Hn(X;M).

Similarly,

Hn(G;M) = Hn HomZ[G]

(C∗X,M

)= Hn HomZ[G]

(C∗X,HomZ(Z,M)

)= Hn HomZ[G]

(C∗X ⊗Z[G] Z,M

)= Hn(X;M).

At least for a connected CW-complex X with universal cover X, and any Z[π1X]-moduleM , this suggests the following definitions:

Hn(X;M) :− Hn

(C∗X ⊗Z[π1X] M

)and

Hn(X;M) := Hn HomZ[π1X]

(C∗X,M

).

If X is a connected space, a Z[π1X]-module is essentially the same thing as a locally constantabelian sheaf on X. For CW-complexes (or more generally, locally contractible spaces), thesheaf cohomology with coefficient the locally constant abelian sheaf M coincides with thesingular cohomology with coefficient the Z[π1X] module Mx, for any x ∈ X.

1.6. Kunneth theorem. In the next few sections, we investigate the homology and coho-mology of products of spaces. As preparation, we first calculate the homology (and coho-mology) of a suitable product of chain complexes. In light of the Eilenberg–Zilber theorem,the direct (categorical) product which coincides with the direct sum (categorical coproduct)is not the correct notion. Instead, we consider the tensor product that is internal to thecategory of chain complexes.

Definition. Define the tensor product

−⊗R − : Ch(R) −→ Ch(R)

of R-chain complexes by

(A⊗R B)n :=⊕p+q=n

Ap ⊗R Bq,

dC⊗RDn :=

⊕p+q=n

((−1)qdCp ⊗R Id + Id⊗RdDq

).

The tensor-hom adjunction also makes sense in the category of chain complexes.

Definition. Given A -chain complexes X and Y , define an Ab-chain complex HomA (X, Y )by

HomA (X, Y )n =∏m∈Z

A (Xm, Yn+m),

with(df)(x) = df(x)− (−1)deg xfd(x)

for x ∈ X homogeneous, where deg denotes the homogeneous degree.


Lemma 1.6.1. [X, Y ] ∼= H0HomA (X, Y ).

Proof. Cycles on the right-hand side are A -chain maps. A A -chain null-homotopy h corre-sponds via h(x) = (−1)deg xg(x) to an element g ∈ HomA (X, Y ). �

Theorem 1.6.2. For any C,D,E ∈ Ch(R), there is an isomorphism

HomR(C ⊗R D,E) ∼= HomR(C,HomR(D,E)).

Now we return to the main theorem of the section:

Theorem 1.6.3 (Kunneth theorem). Suppose C and D are Z-chain complexes, at least oneof which is Z-cell chain complex, for instance, bounded below free Z-chain complex. Thenthere is a natural split short exact sequence

0 −→⊕p+q=n

Hp(C)⊗Z Hq(D) −→ Hn(C ⊗Z D) −→⊕

p+q=n−1

TorZ1 (Hp(C), Hq(D)) −→ 0.

In particular, there exists a (non-canonical) isomorphism

Hn(C ⊗Z D) ∼=⊕p+q=n

Hp(C)⊗Z Hq(D)⊕⊕

p+q=n−1

TorZ1 (Hp(C), Hq(D)) .

Proof. Since C ⊗ D ∼= D ⊗ C, we assume without loss of generality that C is cell, and inparticular free. Then C is quasi-isomorphic to⊕

p

F (HpC)[p],

where F (−) is the free resolution functor. By cell-approximation, we may also assumewithout loss of generality that D is cell, then C ⊗D is quasi-isomorphic to⊕

p+q

(F (HpC)⊗ F (HqD)) [n].

Then the statement follows from the definition of Tor. �

1.7. Eilenberg–Zilber theorem. For a space X, we have C0(X) ∼= ZS0(X) ∼= ZX. The-refore, for spaces X and Y , we have

C0(X)⊗ C0(Y ) ∼= (ZX)⊗Z (ZY ) ∼= Z(X × Y s) ∼= C0(X × Y ).

We will use this identification throughout.

Theorem 1.7.1 (Eilenberg–Zilber theorem). There exists a natural equivalence between anytwo functors of C(X)⊗ZC(Y ) and C(X×Y ), which is unique up to natural chain homotopy,subject to the condition that η is identity on degree 0.

Remark. The proof uses the method of acyclic models ; the slogan is “it is easy to mapsomething free to something acyclic.” This idea was already used in proving the excisionaxiom for singular homology.

Proof. We extend from ψ0 = Id by induction to a chain map ψ : C(X×Y )→ C(X)⊗ZC(Y ).Suppose we have defined ψk for 0 ≤ k < n, and we want to extend to ψn. Since Cn(X×Y ) ∼=ZSn(X × Y ), it is necessary and sufficient to construct ψn on a single singular simplexσn : ∆n → X × Y . By the universal property of product, σ factors as

∆n ι=∆−−→ ∆n ×∆n f×g−−→ X × Y.

12 IGOR KRIZ

By a Yoneda type of argument, it is necessary and sufficient to construct ψn(ι). Morespecifically, there is a commutative diagram

Cn(∆n ×∆n)ψn //

Cn(f×g)��

(C(∆n)⊗Z C(∆n))n

C(f)⊗C(g)

��Cn(X × Y ) // (C(X)⊗Z C(Y ))n

and since ι ∈ Sn(∆n ×∆n) maps to σn ∈ Sn(X × Y ), ψn(ι) uniquely determines the imageof σn.

To construct ψn(ι) ∈ (C(∆n)⊗Z C(∆n))n, note that it is subject to the condition

d⊗nψn(ι) = ψn−1(d×n ι),

where d⊗ is the differential in C(∆n)⊗Z C(∆n), while d× is the differential in Cn(∆n×∆n).Note that the right-hand side is already defined by inductive hypothesis, and it is in additiona cycle since

d×n−1ψn−1d×n ι = ψn−2d

×n−1d

×n ι = 0.

For ψn(ι) to be defined, we need the right-hand side to be a boundary. For n > 1, this isalready the case since ∆n is contractible, so Hn−1(C(∆n)⊗ZC(∆n)) = 0. In the special casen = 1, we make a concrete choice of ψ1, called the Alexander–Whitney map.

Similarly, we may extend from ϕ0 = Id a chain map ϕ : C(X) ⊗Z C(Y ) → C(X × Y ).Note that

(C(X)⊗Z C(Y ))n =⊕k+`=n

Ck(X)⊗Z C`(Y ).

Assume by induction that ϕk′,`′ is defined for k′ < k and `′ < `. Since

Ck(X)⊗Z C`(Y ) ∼= ZSk(X)⊗Z ZS`(Y ) ∼= Z(Sk(X)× S`(Y )),

it is necessary and sufficient to define for any given pair (k, `)

ϕk,`(σk, τ`)

for any pair of simplices σk : ∆k → X and τ` : ∆` → Y . Again, it is necessary and sufficientto define ϕk,` on the universal element σ0 = Id∆k and τ0 = Id∆` .

Again, the only requirement is

d⊗nϕn(σ0 ⊗ τ0) = ϕn−1d×n−1(σ0 ⊗ τ0).

Still,d⊗n−1ϕn−1d

×n−1(σ0 ⊗ τ0) = 0,

and Hk+`−1(∆k × ∆`) = 0, so ϕ extends except when k + ` = 1. In the special case with(k, `) being (1, 0) or (0, 1), we have identifications

C1(∆1)⊗Z C0(∆0) ∼= C1(∆1 ×∆0)

C0(∆0)⊗Z C1(∆1) ∼= C1(∆0 ×∆1)

since ∆0 = ∗ is a singleton and C0(∆0) = Z. Then we can make a concrete choice ofϕ1 = ϕ1,0 ⊕ ϕ0,1, called the shuffle map.

Finally, to construct the natural chain homotopies, we use a similar procedure, and therewill be no special case since chain homotopies go up by 1. To construct a homotopy h : ψ ' ψ′

between two natural chain maps C(X×Y )→ C(X)⊗ZC(Y ), note that (ψ′)0−ψ0 = Id− Id =


0, so we may set h0 = 0. To lift, it suffices to define hn(ι), where ι = ∆ : ∆n → ∆n ×∆n.Again, the only requirement is

d⊗n+1hn(ι) = (ψ′n − ψn)(ι)− hn−1d×n (ι).

Again, we want the right-hand side to be a boundary, or equivalently a cycle by acyclicityof C(∆k ×∆`); this follows from the computation

d⊗n (ψ′n − ψn)(ι)− d⊗nhn−1d×(ι) = (ψ′n−1 − ψn−1)d×n (ι)− d⊗nhn−1d

×(ι)

= (d⊗nhn−1 + hnd×n−1)d×n (ι) + d⊗nhn−1d

×(ι)

= hnd×n−1d

×n (ι) = 0.

It is similar to treat other homotopies between natural chain maps between any other twofrom C(X × Y ) and C(X)⊗Z C(Y ). �

Exercise 7. Prove that any two natural chain maps ϕ, ϕ′ which are identities in degree zeroare naturally chain homotopic.

Remark. In the proof above, the Alexander–Whitney map is unital and associative, but notcommutative, while the shuffle map is unital, associative and commutative.

Corollary 1.7.2. We have a natural chain-homotopy equivalence C(X × Y ) ' C(X) ⊗ZC(Y ). �

As a common consequence of the Kunneth theorem and the Eilenberg–Zilber theorem, wehave

Corollary 1.7.3. For spaces X and Y , there is a natural split short exact sequence

0 −→⊕p+q=n

Hp(X)⊗Z Hq(Y ) −→ Hn(X × Y ) −→⊕

p+q=n−1

TorZ1 (Hp(X), Hq(Y )) −→ 0.

In particular, there exists a (non-canonical) isomorphism

Hn(X × Y ) ∼=⊕p+q=n

Hp(X)⊗Z Hq(Y )⊕⊕

p+q=n−1

TorZ1 (Hp(X), Hq(Y )) . �

Example. In this example, we use the Kunneth theorem to compute H∗(RP3×RP2). Recallthat

Hn(RP2) =

Z n = 0

Z/2 n = 1

0 else

Hn(RP2) =

Z n = 0, 3

Z/2 n = 1

0 else

therefore we haveq1 Z/2 Z/2 0 Z/20 Z Z/2 0 Z

Hp(RP3)⊗Hq(RP2) 0 1 2 3 p

14 IGOR KRIZ

and the only nontrivial Tor is TorZ1 (Z/2,Z/2) = Z/2. Therefore

Hn(RP3 × RP2) =

Z n = 0

Z/2⊕ Z/2 n = 1

Z/2 n = 2, 4

Z⊕ Z/2 n = 3

0 else

Corollary 1.7.4. For any commutative ring R we have an isomorphism

C(X × Y ;R) ∼= C(X;R)⊗R C(Y ;R)

Proof. By associativity and commutativity of the tensor products, as well as the Kunneththeorem, we have

C(X;R)⊗R C(Y ;R) ∼= (C(X)⊗Z R)⊗R (C(Y )⊗Z R)∼= C(X)⊗Z (R⊗R R)⊗Z C(Y )∼= C(X)⊗Z R⊗Z C(Y ) ∼= C(X × Y )⊗Z R∼= C(X × Y ;R). �

�

Remark. Over a field F , all F -modules are free, so in particular all higher Tor vanishes.Then Kunneth theorem reduces to a natural isomorphism

Hn(C ⊗F D) =⊕k+`=n

Hk(C)⊗F H`(D).

Over general R, however, Kunneth theorem becomes a spectral sequence (see !!):

E2p,q =

⊕k+`=q

TorRq (Hk(C), H`(D))⇒ Hp+q(C ⊗R D).

Exercise 8. Calculate H∗(RP3 × RP2;Z/2).

1.8. The cup product. The recipe of the construction is Eilenberg–Zilber theorem appliedto the diagonal map ∆ : X → X ×X. For simplicity, we work with coefficient Z, althoughthe store is the same for any commutative ring.

By definition of the cochain complex Hom(C,Z) of a chain complex C, there is a naturalevaluation map

ev : Hom(C,Z)⊗ C −→ Z.In fact, the tensor-hom adjunction Hom(C⊗D,Z) ∼= Hom(C,Hom(D,Z)) is given by sendingf ∈ Hom(C,Hom(D,Z)) to

C ⊗D f⊗Id−−−→ Hom(D,Z)⊗D ev−→ Z.

From the Eilenberg–Zilber theorem, we have a natural chain map

ρ : C(X)C(∆)−−−→ C(X ×X)

ψ−→ C(X)⊗ C(X).

We wish to construct a “dual” map

λ : C∗(X)⊗ C∗(X)'−→ C∗(X);


it suffices to construct a natural map

Hom(C(X),Z)⊗ Hom(C(X),Z) −→ Hom(C(X)⊗ C(X),Z)

and then post-compose with Hom(ρ,Z). Indeed, the desired map above can be realized, bythe tensor-hom adjunction, as the chain map

Hom(C(X),Z)⊗ Hom(C(X),Z)⊗ C(X)⊗ C(X)

Id⊗τ⊗Id��

3 f ⊗ g ⊗ x⊗ y_

��Hom(C(X),Z)⊗ C(X)⊗ Hom(C(X),Z)⊗ C(X)

ev⊗ ev

��

3 (−1)deg g deg xf ⊗ x⊗ g ⊗ y_

��Z 3 (−1)deg g deg xf(x)g(y)

where τ switches the factors in the tensor product.On the other hand, fix k + ` = n, then the tensor product on the (co)chain restricts to a

natural mapµ : Ck ⊗D` −→ (C ⊗D)n

x⊗ y 7−→ x⊗ yNote that

d(x⊗ y) = (dx)⊗ y + (−1)deg xx⊗ (dy),

so if x and y are both cocycles, then so is x ⊗ y; this shows the tensor product preservescocycles. Furthermore, and if y is a coboundary, then (dx) ⊗ y = d(x ⊗ y), so the tensorproduct preserves coboundaries in the first coordinate, and by symmetric also in the secondone. This implies that it descends into cohomology:

µ : Hk(C)⊗H`(D) −→ Hn(C ⊗D)[x]⊗ [y] 7−→ [x⊗ y]

where [−] denotes cohomology class. Finally, compose this with H∗(λ), we obtain the cupproduct.

Definition. Fix any k + ` = n, the cup product is defined as the natural map

^ : Hk(X)⊗H`(X)µ−−−→ Hk+`(C∗X ⊗ C∗X)

H∗(λ)−−−→ Hk+`(X)

16 IGOR KRIZ

2. Cell structure on R-chain complexes

2.1. Mapping cone.

Note. In fact, this construction defines a functor R-Mod → Ch(R) since the free R-moduleconstruction is functorial.

Definition. The mapping cone Cf of an R-chain map f : C → D is defined as

(Cf)n := Dn ⊕ Cn−1,

dn :=

(dD (−1)n−1f0 dC .

)There is a sequence of R-chain complexes

Cf−→ D

i−→ Cf.

Lemma 2.1.1. For any R-chain complex E, the sequence of R-modules

[E,C]f∗−→ [E,D]

i∗−→ [E,Cf ],

where [−,−] denotes the R-chain homotopy classes of maps, is exact.

Proof. First of all, we claim that if ' 0, so i∗f∗ = (if)∗ = 0. Define an R-chain homotopyhn : Cn → (Cf)n+1 by hn = (−1)n(0, Id):

Cn+1

��

f // Dn+1

��

⊕Cn

��f{{Cn

��

f //

h77

Dn

��

⊕Cn−1

��f{{Cn−1

f //

h

77

Dn−1⊕

Cn−2

Then

(dn+1hn + hn−1dn)(cn) =((−1)nfn((−1)ncn), dn((−1)ncn)

)+(0, (−1)n−1(dncn)

)= (fn(cn), 0) = (infn)(cn),

so dh+ hd = if as required.Now suppose [α] ∈ Ker i∗ is represented by α : E → D, so there exists an R-chain

homotopy h : iα ' 0. Write hn = (kn, βn) : En → (Cf)n+1:

En+1

��

α // Dn+1

��

⊕Cn

��f{{En

��

α //

k

77

β

44

Dn

��

⊕Cn−1

��f{{En−1

α //

k

77

β

44

Dn−1⊕

Cn−2

Then

(inαn)(en) =(αn(en), 0

)


(dn+1hn + hn−1dn)(en) =((dn+1kn + kn−1dn)(en), 0

)+((−1)nfn(βnen), dn(βnen)

)+(0, βn−1(dnen)

)Comparing coordinates gives{

αn = dn+1kn + kn−1dn + (−1)n(fnβn) = αn,

0 = dnβn + βn−1dn.

The second identity implies that γn = (−1)nβn defines an R-chain map γ : E → C, and thefirst identity implies that kn : En → Dn+1 defines an R-chain homotopy k : α ' fγ. Thismeans [α] = f∗[γ] ∈ Im f∗. �

We could extend the sequence into

Cf−→ D

i−→ Cf −→ Ci.

Define maps p : Ci→ C[1] and q : C[1]→ Ci by

Dn ⊕ Cn−1 ⊕Dn−1pn−→ Cn−1

pn−→ Dn ⊕ Cn−1 ⊕Dn−1

(bn, cn−1, bn−1) 7−→ cn−1 7−→ (0, cn−1,−fn−1cn−1)

Lemma 2.1.2. p and q define R-chain maps.

Proof. Since

pn =(0 1 0

), dCi

n =

dDn (−1)n−1fn (−1)n−1

0 dCn−1 00 0 dDn−1

, qn =

01−fn

,

we computepn−1d

Cin =

(0 dCn−1 0

)= dC[1]

n pn,

and

dCin qn =

(−1)n−1f − (−1)n−1fdCn−1

−dDn−1fn−1

=

0dCn−1

−fn−1dDn−1

= qn−1dC[1]n . �

Clearly pq = Id; although qp 6= Id, the following is true:

Exercise 9. Show that qp ' Id, so p and q form an R-chain homotopy equivalence.

Therefore, we have a commutative diagram

Cf // D

i // Cfj //

!!

Ci

'��

C[1]

which induces, by Lemma 2.1.1, an exact sequence

[E,C] −→ [E,D] −→ [E,Cf ] −→ [E,C[1]]

of R-modules for any R-chain complex E.

18 IGOR KRIZ

We may extend the commutative diagram one step further to

Cf // D

i // Cfj //

!!

Cik //

' p

�� ""

Cj

' p′

��C[1]

−f [1]// D[1]

and investigate Cj. Recall that

(Cj)n = (Ci)n ⊕ (Cf)n−1 = Dn ⊕ Cn−1 ⊕Dn−1 ⊕Dn−1 ⊕ Cn−2,

so the map p′ : Cj → D[1] is the projection onto the third factor. It follows that thecomposite p′k sends (dn, cn−1, dn−1) to −fn−1cn−1, which makes the square commute.

Corollary 2.1.3 (Cofiber sequence). For any R-chain complex E, we have a long exactsequence

· · · −→ [E,C[n]] −→ [E,D[n]] −→ [E, (Cf)[n]] −→ [E,C[n+ 1]] −→ · · · .Proof. To extend to the right, we repeat the process above using Lemma 2.1.1, and to extendto the right, we replace E by E[−1]. �

Remark. This phenomenon is called stability.

Note. If Cε−→ M is an R-free resolution, then Cε = C[1], where C is the augmented

resolution.

Exercise 10. Show that HiX ∼= [R[i], X] for any R-chain complex X. Here R is regardedas a chain complex concentrated in degree 0.

Corollary 2.1.4. For any R-chain map f : C → D, we have a long exact sequence

· · · −→ Hn(C) −→ Hn(D) −→ Hn(Cf) −→ Hn−1(C) −→ · · ·in homology. �

2.1.1. Chain homotopies as chain maps. Consider the mapping cone of the “signed diagonal”

IC := C [(Id,− Id) : C −→ C ⊕ C] ;

more explicitly, define

(IC)n := Cn ⊕ Cn ⊕ Cn−1,

dICn :=

dCn 0 (−1)n−1 Id0 dCn (−1)n Id0 0 dCn−1

.

Proposition 2.1.5. The two inclusions i0, i1 : C → IC given by (Id, 0, 0) and (0, Id, 0) arequasi-isomorphisms. Moreover, two R-chain maps f, g : C → D are R-chain homotopic if

and only if there exist an R-chain map h : IC → D such that the diagram

C

∼ i0��

f

''C IC

Id⊕ Idoo h // D

C

∼ i1

OO

g

77


commutes, in which case the R-chain homotopy is given by hn = hn+1(0, 0, (−1)n Id).

Proof. It is clear by definition that pj(Zn(IC)) = Zn(C) and pj(Bn(IC)) = Bn(C) forj = 0, 1, so ij are quasi-isomorphisms. For the second assertion, note that there is a bijectioncorrespondence{

h = {hn : Cn−1 → Dn}}←→

{h = {hn : (IC)n → Dn}

}h 7−→ {hn = fn ⊕ gn ⊕ (−1)n−1hn−1}

{hn = hn+1(0, 0, (−1)n Id)} ←−[ h

between the collections of R-morphisms. Furthermore, h defines a chain map IC → D ifand only if

hn ◦ dICn+1 =(fn gn (−1)n−1hn−1

)dCn+1 0 (−1)n Id0 dCn+1 (−1)n+1 Id0 0 dCn

=(fnd

Cn+1 , gnd

Cn+1 , (−1)nfn − (−1)ngn − (−1)nhn−1d

Cn

)=(dDn+1fn+1 , dDn+1gn+1 , (−1)ndDn hn

)= dDn+1 ◦ hn+1

In the second-to-last equality, the first two coordinates automatically agree since f and gare R-chain maps, while the last coordinates agree if and only if

fn − gn = hn−1dCn + dDn hn,

that is, h defines an R-chain homotopy between f and g. �

Remark. Equivalently, IC may be described as the tensor product IR ⊗R C, where

IR = · · · −−−→ 0 −−−→ R(1,−1)−−−→ R⊕R −−−→ 0 −−−→ · · · ,

is concentrated in degrees 0 and 1, and the tensor product is defined in Section 3.7. Notethat there two natural inclusions R → IR with − ⊗R C applied give the two inclusionsi0, i1 : C → I ⊗R C.

2.2. Cell R-chain complexes. In this section we investigate an analogue of a cell complexof spaces.

Definition. C is a cell R-chain complex if C is the directed union of a sequence of R-chaincomplexes

0 = C(−1) ⊆ C(0) ⊆ C(1) ⊆ · · · ,where C(n+1) is the mapping cone of an R-chain map of the form⊕

i∈Z

(RSi,n)[i] −→ C(n),

where Si,n is a set, and R(−) is the free R-module functor, and the source is regarded as anR-chain complex with zero differentials.

Equivalently, we may choose a set Sn,i of cycles (possibly the 0!) in C(n) of degree i, anddeclare them to be boundaries by attaching a free R-module on each of new elements (cells)whose boundaries are the chosen cycles.

Note that a cell R-chain complex is free.

20 IGOR KRIZ

Lemma 2.2.1. A bounded below R-free chain complex C is cell.

Proof. Suppose Cn = 0 for n < N . We inductively attach CN first, then CN+1, and soon. �

Note that the converse is not true; see Exercise 11.

Proposition 2.2.2. Suppose E is a cell R-chain complex, and suppose X is an exact R-chaincomplex. Then

[E,X] = 0.

Proof. Let f : E → X be an R-chain map. We will construct, by successively extending,chain homotopies h : f |E(n)

' 0. Assuming it has been constructed for n− 1, so

dnhn−1 + hn−2dn−1 = fn−1.

We want to define hn(s) for s ∈ Si,n so that

(dn+1hn + hn−1dn)(s) = fn(s),

sodn+1(hn(s)) = fn(s)− hn−1dn(s) ∈ Xn.

Note that f(s)− hn−1dn(s) is an n-cycle since

dn(fn(s)− hn−1dn(s)) = dnfn(s)− dnhn−1dn(s)

= fn−1(dn(s))− dnhn−1(dn(s))− hn−2 dn−1(dn(s))︸︷︷︸=0

= 0

by the inductive hypothesis. Since HnX = 0, this must also be a boundary of some elementin Xn+1, which we could then define to be hn(s). �

Theorem 2.2.3 (Whitehead theorem in Ch(R)). If f : C → D is a quasi-isomorphism,then for every cell R-chain complex E, the natural map

[E, f ] : [E,C]∼=−→ [E,D]

is an isomorphism.

Remark. More fancily, we say that the cell R-chain complexes are co-local with respect toquasi-isomorphisms in hCh(R); see §3.4 for the definition of co-local objects. In particular,Lemma 3.4.1 implies that quasi-isomorphic cell R-chain complexes are actually R-chainhomotopy equivalent. This is analogous to the classical Whitehead theorem for cell complexesin the topological case.

Proof of the Whitehead theorem. In light of the cofiber sequence (Corollary 2.1.3), it sufficesto show that [E, (Cf)[n]] = 0 for all n. By the previous Proposition, it suffices to show thatHi((Cf)[n]) = 0 for all i. Using the previous Lemma and the cofiber sequence, we have

· · · −→ [R[i], C[n]]∼=−→ [R[i], D[n]] −→ [R[i], (Cf)[n]] −→ · · · .

Therefore, Hi((Cf)[n]) = [R[i], (Cf)[n]] = 0. �


Corollary 2.2.4. If f : M → N is an R-module morphism, C is an R-free resolution of M ,and D is any resolution of N , then the diagram

C

εC��

f // D

εD��

Mf// N

completes uniquely up to homotopy.

Proof. Regard M and N as R-chain complexes concentrated in degree 0, then εD is a quasi-isomorphism. Since C is a cell, the Whitehead theorem implies that [fεC ] ∈ [C,N ] has aunique preimage in [C,D]. �

Exercise 11. In the setting in Exercise 5, show that the R-free chain complex

C = · · · −→ Rx−→ R

x−→ R −→ · · ·

is not cell.

Theorem 2.2.5 (Cell approximation theorem in Ch(R)). Let C be an R-chain complex.Then there exists a cell R-chain complex C ′ and a quasi-isomorphism

γC = γ : C ′ −→ C.

Proof. We construct C ′ explicitly. First, we define

C ′(0) = R(ZC) =⊕i∈Z

(R(ZiC))[i],

and γ(0) the natural projection. Note that H∗γ(0) is onto. Now suppose γ(n) : C ′(n) → C is

defined such that H∗γ(n) is onto, we make C ′(n+1) by killing all cycles which map to boundariesin C:

C ′(n)� _

��

γ(n) // C

C ′(n+1)

==

Now put C ′ := lim−→C ′(n) and γ := lim−→ γ(n).Next we check that γ is a quasi-isomorphism. Since directed colimit is an exact functor,

we have H∗C′ = lim−→H∗C

′(n), and H∗γ = lim−→H∗γ(n). Consider the diagram

KerH∗γ(n)� _

��

0

''

KerH∗γ(n+1)� _

��

0

%%

KerH∗γ� _

��· · · // H∗C

′(n)

//

H∗γ(n)��

H∗C′(n+1)

//

H∗γ(n+1)��

· · ·lim−→ // H∗C

′

H∗γ��

H∗C H∗C H∗C

Here, each column of vertical arrows extends to an exact sequence, so by exactness of directedcolimits, we have

KerH∗γ = lim−→KerH∗γ(n) = 0

22 IGOR KRIZ

since the connecting maps in the diagram are zero, and

CokerH∗γ = lim−→CokerH∗γ(n) = 0,

since every CokerH∗γ(n) = 0 �

Note. The cell approximation C 7→ C ′ is a functor, and γ : (−)′ ⇒ Id is a natural transfor-mation.

2.3. Derived category of R-chain complexes. R-chain homotopy is a congruence rela-tion on Ch(R), i.e. an equivalence relation preserved by compositions. Therefore, we mayform the quotient category hCh(R) of Ch(R) by R-chain homotopy. In other words, we have

• Obj hCh(R) := Obj Ch(R);• hCh(R)(C,D) := [C,D].

We define DCh(R) as follows:

• Obj DCh(R) := Obj hCh(R) = Obj Ch(R);• DCh(R)(C,D) := [C ′, D′] = hCh(R)(C ′, D′), where (−)′ denotes the cell approxima-

tion functor.

We can construct a co-localization functor (see Section 3.4 for a precise definition)

Φ : hCh(R) −→ DCh(R)

by the functoriality of cell approximation and the naturality of γ. In fact, given f : C → Din hCh(R), Φ(f) is the unique morphism in hCh(R) such that the square

C ′

∼��

Φ(f)// D′

∼��

Cf// D

commutes, where uniqueness follows from the cellularity of C ′.

Remark. This is an “upgrade” of the uniqueness of Tor and Ext to all of R-chain complexes.

Theorem 2.3.1. The functor Φ : hCh(R) −→ DCh(R) sends quasi-isomorphisms to iso-morphisms. Moreover, for every functor F : hCh(R) −→ C which sends quasi-isomorphismsto isomorphisms, there exists a unique functor F ′ : DCh(R) → C completing the followingdiagram:

hCh(R)

Φ��

F // C

DCh(R)

F ′

77

Sketch of proof. For the first statement, note that if f : C → D is a quasi-isomorphism, thenΦ(f) is a quasi-isomorphism between cell R-chain complexes; Lemma 3.4.1 implies that it isan R-chain homotopy equivalence.

For the second statement, it is equivalent to that DCh(R) is a derived category of hCh(R).This follows from Theorem 3.4.2 since Φ is a co-localization, as a consequence of Theorems2.2.3 and 2.2.5. ♦


Remark. We are not really interested in strict commutativity of functors, but in commutati-vity up to natural isomorphism, and in categories up to equivalence. In this sense, “DCh(R)is just the category of cell R-chain complexes and R-chain homotopy classes of maps.” Thisis true up to equivalence of categories.

Exercise 12. For R-modules M and N , show that ExtnR(M,N) = DCh(R)(M,N [n]).

24 IGOR KRIZ

3. Derived categories and derived functors

3.1. Derived categories.

Definition. If C is a category and E ⊆ Mor C is a class of morphisms (called equivalences),the a derived category DC = DE C = E −1C with respect to E , if one exists, is a categorywith a functor Φ : C → DC such that:

(1) Φ sends morphisms in E to isomorphisms, and(2) for any functor F : C → D which sends morphisms in E to isomorphisms, there is a

unique functor F ′ : DC → D such that F = F ′ ◦ Φ.

C

Φ��

F // D

DCF ′

77

Note. Just like any concepts defined by universal properties, a derived category, if one exists,is unique.

Example. By Theorem 2.3.1, the category DCh(R) is the derived category of hCh(R) withrespect to quasi-isomorphisms.

Proposition 3.1.1. DCh(R) is also the derived category of Ch(R) with respect to quasi-isomorphisms.

Proof. Suppose F : DCh(R)→ C sends quasi-isomorphisms to isomorphisms. Then we havediagrams of functors

Ch(R)

��

F // C

hCh(R)

��

F

77

DCh(R)

F′

>>

To show that F factors uniquely through DCh(R), it suffices to show that it factors throughthe quotient category hCh(R). This is equivalent to that, if f ' g : C → D are R-chainhomotopic R-chain maps, then F (f) = F (g). Using the setting and notations in Section2.1.1, we see that F (i0) and F (i1) are isomorphisms with a common left inverse F (Id⊕ Id),hence must coincide. Therefore,

F (f) = F (h) ◦ F (i0) = F (h) ◦ F (i1) = F (g). �

Lemma 3.1.2. If a derived category DE C of C with respect to E exists, then Φ is a bijectionon objects.

Proof. Define a new category D′C as follows:

• Obj D′C = Obj C ;• D′C (X, Y ) = DC (ΦX,ΦY ).

We claim that D′C also satisfies the universal property of the derived category, hence wehave an isomorphism of categories DC ∼= D′C . In particular, Φ is bijective on objects.

To this end, we will construct:


(1) a functor Φ′ : C → D′C that sends morphisms in E to isomorphisms, and

(2) a unique factorization into Φ as CΦ′−→ D′C

Γ−→ DC .

For (1), we define Φ to be identity on objects and Φ on morphisms. This forces Γ, in orderfor Φ = Γ ◦ Φ′, to be Φ on objects and identities on morphisms. Yet, this indeed defines afunctor, and the proof is complete. �

3.1.1. Derived categories in the 2-categorical sense. In this section, we relax the notion of de-rived categories above, which we shall refer to as “strict” derived categories. A 2-categoricalderived category is defined similarly to a strict derived category, with the condition (2)replaced by

(2’) (a) for any functor F : C → D which sends morphisms in E to isomorphisms, there

is a functor F ′ : DC → D , along with a natural isomorphism ι : F∼=

=⇒ F ′ ◦ Φ:

C

Φ

��

F //

∼=ι

�%

D

DC

F ′

>>

(b) for any functor F ′′ : DC → D and natural isomorphism κ : F∼=

=⇒ F ′′ ◦ Φ, there

is a natural isomorphism η : F ′∼=

=⇒ F ′′ such that ι′ = ι ◦ (ηΦ):

C

Φ

��

F //

∼=κ

�"

∼=ι

�!

D

∼=η

�!

DC

F ′>>

F ′′

PP

Clearly, this notion is invariant under equivalence of categories: it is in the 2-categorywhere 2-morphisms are isomorphisms of functors, while equivalences of categories are equi-valences in this 2-category.

Proposition 3.1.3. A strict derived category is a 2-categorical derived category.

Proof. We only need to discuss the uniqueness, and it follows immediately with η = κ. �

3.2. Kan extensions. A functor F : C → D is left adjoint to G if there is a naturalisomorphism of functors

D(F−,−) ∼= C(−, G−) : Cop ×D −→ Set.

In this definition, for every x ∈ ObjC, the identity morphism Fx → Fx corresponds tothe unit of adjunction ηx : x → GFx, whereas for every y ∈ ObjD, the identity morphism

26 IGOR KRIZ

Gy → Gy corresponds to the counit of adjunction εy : FGy → y. Moreover, they definenatural transformations

η : Id =⇒ GF, and ε : FG =⇒ Id .

Equivalently, the adjunction between F and G may be defined as the existence of suchnatural transformations which satisfy the identities

FGFεF

""

GFGGε

""F

Fη<<

Id// F G

ηG<<

Id// G

Moreover, both of them are equivalent to a universal property: for every object x ∈ ObjC,there exists a morphism x

ι−→ Gy in C universal in the following sense:

xι //

∀ϕ

Gy

Gϕ′

��

y

∃!ϕ′��

Gz z

Dually and equivalently, for every object y ∈ ObjD, there exists a universal morphismsFx→ y in D:

w

∃!ψ′

��

Fw

Fψ′

��

∀ψ

x Fx

θ// y

Kan extensions give rise to a very important class of examples of adjoint functors. For theeasiest example, consider an inclusions i : H ↪→ G of groups. Consider the category G-Set,where objects are (left) G-set, and morphisms are G-maps. There is a “forgetful functor”

i∗ : G-Set −→ H-Set,

and it has both a left and a right adjoint.Write µ : H×G→ G for the multiplication on G restricted to G×H, and α : H×X → X

the H-action on X. The left adjoint (left Kan extension) i] is

G×H − : H-Set −→ G-Set,

which is defined for any H-set X by the coequalizer

G×H X := coeq

[G×H ×X

G×α−−−−−−⇒µ×X

G×X],

which inherits the G-set structure from G × X. To see the adjunction, define the unit ofadjunction for an H-set X by

η : X −→ G×H Xx 7−→ (1, x)

It is an H-map since

η(hx) = (1, hx) = (h, x) = h(1, x) = h(η(x)).


Then for any H-set X and any G-set Y , we have a bijection

G-Set(G×H X, Y )∼=←→ H-Set(X, i∗Y )

f 7−→ f ◦ η[(g, x) 7→ gϕ(x)] ←− [ ϕ

On the other hand, the right adjoint (right Kan extension) i∗ is

HomH(G,−) := H-Set(G,−) : G-Set −→ H-Set,

where G is regarded as a left H-set by multiplication. Note that from the diagram

H ×Gµ

��

H×f // H ×Xα

��G

f// X

of H-equivariance, there is an equalizer formula

HomH(G,X) := eq

[XG

α◦(H×−)−−−−−−−−−−⇒−◦µ

XH×G],

dual to G ×H X. In particular, HomH(G,X) inherits the G-set structure from XG. To seethe adjunction, define the counit of adjunction for an H-set X by

ε : HomH(G,X) −→ Xf 7−→ f(1)

It is an H-map since

h(ε(α)) = h(α(1)) = α(h) = α(1 · h) = (hα)(1) = ε(hα).

Exercise 13. Prove that i∗ is right adjoint to i∗.

Remark. There is a parallel example with group morphism i : H → G replaced by ringmorphism i : R → S, and G-sets replaced by R-modules. Then G ×H − and HomH(G,−)are replaced by S ⊗R − and HomR(S,−), and called extension and coextension of scalars,respectively.

Recall that a group G can be viewed as a category on one object, then a G-set correspondsto a functor G → Set, and a G-map is a natural transformation between such functors.Therefore, to generalize this example, we could replace the group G by a (small) categoryC , the inclusion i : H ↪→ G by a functor i : C → D , and the category G-Set by the functorcategory C Set, which we still denote by C -Set. For simplicity, we assume that i is identityon objects. (This assumption is unnecessary, but it will be the case of our interest.) Then iinduces a “forgetful functor”

i∗ : D-Set −→ C -Set.

The left Kan extension along i is the functor

i] : C -Set −→ D-Set

which is left adjoint to i∗. It is defined by

(i]F )(x) := coeq

∐g∈D(i(y),x)

∐h∈C (z,y)

F (z) ⇒∐

g∈D(i(y),x)

F (y)

,

28 IGOR KRIZ

where the first arrow sends F (z)(g,h) to F (h)(F (z))(g) and the second arrow sends it toF (z)(g◦i(h)). Spelling out the details,

(i]F )(x) ∼={(g, t)|g ∈ D(i(y), x), t ∈ F (y)}

{(gi(h), s) ∼ (g, F (h)s)|h ∈ C (z, y), s ∈ F (z)}.

In particular,(i]F )(γ)(g, t) = (γg, t)

for γ ∈ Mor D .

Exercise 14. Prove that i] defines a functor.

Dually, the right Kan extension along i is the functor

i∗ : C -Set −→ D-Set

which is right adjoint to i∗. It is defined by

(i∗F )(x) := eq

∏g∈D(x,i(y))

F (y) ⇒∏

g∈D(x,i(z))

∏h∈C (y,z)

F (z)

,where for each F (z)(g,h), the first arrow projects the product onto the factor F (h)(F (y))(g),while the second projects to the factor F (z)(g◦i(h)). Spelling out the details,

(i∗F )(x) ={{ϕy : D(x, i(y))→ F (y)}y∈Obj C

∣∣∣∀h ∈ C (y, z), F (h) ◦ ϕy = ϕz ◦D(x, i(h))}

In particular, for γ : x ∈ x′ ∈ MorD,

(i∗F )(γ)({ϕy : D(x, i(y))→ F (y)}y

)= {ψy : D(x′, i(y))→ F (y)}y ,

whereψy(g) = ϕy(gγ).

Exercise 15. Describe the counit of adjunction.

3.3. Derived functors. Recall that, if C is a category and E ⊆ Mor C is a class of mor-phisms (called equivalences), then the derived category DC , if one exists, comes with afunctor Φ : C → DC which sends E to isomorphisms, and whenever F : C → D sends E toisomorphisms, there is a unique functor F ′ : DC → D such that F = F ′ ◦ Φ.

Now let’s drop the assumption that F sends E to isomorphisms, and we wish to “extend”F to DC .

Definition. The left Kan extension of F to DC , if one exists, is denoted by RF and calledthe total right derived functor of F . Dually, the right Kan extension of F to DC , if oneexists, is denoted by LF and called the total left derived functor of F .

In particular, there is a natural transformation

ε : LF ◦ Φ =⇒ F,

which is universal in the sense that, for every functor G : DC → D with a natural transfor-mation κ : G ◦ Φ⇒ F , there exists a unique natural transformation λ : G⇒ LF such that


κ = ε ◦ λΦ:G

λ

��

G ◦ Φ

λΦ

��

κ

�'F

LF LF ◦ Φ

ε

7?

The case for RF is dual.

Note. In an arbitrary (not necessarily small) abelian category A , the total derived functorExt = RHom may not exist, as the Ext-groups could be proper classes instead of sets.

3.4. Co-localization and localization.

Definition. In a category C , an object z is co-local with respect to a class of morphisms Eif every f : x→ y in E induces a bijection

C (z, f) : C (z, x)∼=−→ C (z, y).

A co-localization of C with respect to E , if one exists, is a functor (−)′ : C → C with targetsin B, and a natural transformation γ : (−)′ ⇒ Id whose components γx : x′ → x are in E .

Remark. If we assume E satisfies the “two-of-three property”, that is, if f and g are compo-sable, and two of the three maps f , g and gf are in E , then so is the third, then it suffices toassume the existence of γx : x′ → x ∈ E for each x ∈ Obj C , and functoriality and naturalityhold automatically.

Lemma 3.4.1. Let C be a category, and x, y ∈ Obj C co-local with respect to a class ofmorphisms E . If f : x→ y is in E , then f is an isomorphism.

Proof. We prove the co-local case, as it is directly relevant in this section; the local case isproved in exactly the same way. Using the co-locality of z, we have

C (y, f) : C (y, x) −−� C (y, y)? 7−→ Id

so there exist g ∈ C (y, x) with fg = Id. On the other hand, the co-locality of α implies that

C (x, f) : C (x, x) ↪−→ C (x, y)gf, Id 7−→ f

so gf = Id. Hence f is an isomorphism. �

Theorem 3.4.2. If a co-localization of C with respect ot E exists, then the derived categoryDE C = DC exists, and is given by

• Obj DC = Obj C ;• DC (x, y) = C (x′, y′),

Moreover, the functor Φ : C → DC is identity on objects and f 7→ f ′ on hom-sets.

Proof. We prove that for every functor F : C → D which sends morphisms in E to isomor-phisms, there exists a unique functor F ′ : DC → D such that F = F ′ ◦ Φ.

Given any morphism f : x→ y in C , we are forced to define

F ′(f ′) = F ′(Φ(f)) = F (f).

30 IGOR KRIZ

In particular, since

(x′)′ //

γx′

��

x′

γx

��x′ γx

// x

can be completed by both γx′ and (γx)′, the uniqueness due to co-localization forces them to

coincide, and thusF ′(γx′) = F ′((γx)

′) = F (γx).

Now for any morphism x → y in DC represented by g : x′ → y′, we have commutativesquares

(x′)′g′ //

γx′

��

(y′)′

γy′

��⇒

F ′(x′)′F ′(g′)

//

F ′(γx′ ) ∼=��

F ′(y′)′

F ′(γy′ )∼=��

x′ g// y′ F ′x′

F ′(g)// F ′y′

so we are forced to define

F ′(g) = F ′(γy′) ◦ F ′(g′) ◦ F ′(γx′)−1

= F (γy) ◦ F (g) ◦ F (γx)−1.

In particular, setting g = Idx, we see that F ′ agrees with F on objects.Conversely, it is clear that such definition makes F ′ a functor. It remains to check F ′(f ′) =

F (f) for consistency, which follows by applying F to the commutative square

x′f ′ //

γx

��

y′

γy

��x

f// y

�

Theorem 3.4.3. If a co-localization of C with respect ot E exists, then the left derivedfunctor LF of any functor F : C → D exists, and is given by

LF (x) := F (x′), LF (f) := F (f ′).

Proof. Define a natural transformation ε : LF ◦ Φ⇒ F by

εx = F (γx) : LF ◦ Φ(x) = F (x′) −→ F (x)

for any x ∈ Obj C , and we show that it is universal. Suppose G : DC → D is a functor,and κ : G ◦ Φ ⇒ F . In particular, for any x ∈ Obj C , the morphism γx : x′ → x induces acommutative diagram

G(x′)

GΦ(γx) ∼=��

κx′ // F (x′)

F (γx)

��

(LF ◦ Φ)(x)

(G ◦ Φ)(x) G(x)

λx

;;

κx// F (x)

by naturality of κ. We set

λx = κx′ ◦G(γx)−1 : G(x) = GΦ(x) −→ F (x′) = LF (x).


Now the square with dashed sides in the diagram

G(x′)κx′ //

GΦ(γx)

∼=

##GΦ(f ′)

��

F (x′)F (γx)

##

F (f ′)

��

G(x)κx //

GΦ(f)

��

λx;;

F (x)

F (f)

��

G(y′)κy′ //

GΦ(γy)

∼=

##

F (y′)F (γy)

##G(y) κy

//λy

;;

F (y)

commutes, so(LF ◦ Φ)(f) ◦ λx = λy ◦ (GΦ)(f).

For a general morphism g : x→ y ∈ DC corresponding to a morphism g : x′ → y′, then

g = (γy′) ◦ g′ ◦ (γx′)−1

= (γ′y) ◦ g′ ◦ (γ′x)−1

= Φ(γy) ◦ Φ(g) ◦ Φ(γx)−1,

in DC , so the naturality condition of λ follows from the special case of morphisms of the formΦ(f), taking f = γy, g and γx. Therefore, this defines a natural transformation λ : G⇒ LF .Moreover, it is clear that ε ◦ (λΦ) = κ.

To see that such λ is unique, consider the ladder

G(x′′)

GΦ(γx′ ) ∼=��

κx′′ // F (x′′)

F (γx′ )∼=��

G(x′)

GΦ(γx) ∼=��

κx′ //

λ′x′

::

F (x′)

F (γx)

��G(x)

λ′x

::

κx// F (x)

Here γx′ : x′′ → x′ ∈ E with x′, x′′ ∈ B, so γx′ , and thus F (γx′), are isomorphism. Thisforces

λ′x′ = F (γx′)−1 ◦ κx′ = κx′′ ◦ κx′′ = λx′ .

Finally, the commutative parallelogram with dashed arrows forces

λ′x = F (γx′) ◦ λx′ ◦GΦ(γx) = λx. �

We can also consider a symmetric situation as a device to compute right derived functors.

Definition. An object z is local with respect to E if every f : x→ y in E induces a bijection

C (f, z) : C (y, z)∼=−→ C (x, z).

A localization of C with respect to E , if one exists, is a functor (−)′ : C → C with sourcesin B, and a natural transformation γ : Id⇒ (−)′ with components γx : x→ x′ in E .

32 IGOR KRIZ

Remark. Again, if E satisfies the “two-of-three property”, then it suffices to assume theexistence of γx : x→ x′ ∈ E for each x ∈ Obj C .

The following results are completely dual to the co-local case.

Lemma 3.4.4. Let C be a category, and x, y ∈ Obj C local with respect to a class ofmorphisms E . If f : x→ y is in E , then f is an isomorphism. �

Theorem 3.4.5. If a localization of C with respect ot E exists, then the derived categoryDE C = DC exists, and is given by

• Obj DC = Obj C ;• DC (x, y) = C (x′, y′). �

Theorem 3.4.6. If a localization of C with respect ot E exists, then the right derived functorRF of any functor F : C → D exists, and is given by

RF (x) := F (x′), RF (f) := F (f ′). �

3.4.1. Example: Tor and Ext as left derived functors. Take C to be hCh(R), E the collectionof all quasi-isomorphisms, and B the collection of cell R-chain complexes. Then we cancompute in this way a left derived functor of any functor on Ch(R) which takes R-chainhomotopic class maps to the same morphism.

Example (Tor). Consider

F = (−)⊗R N : hCh(R) −→ hCh(R)

where N is some R-module, then

LF (X) = X ′ ⊗R N.In particular, take X = M , the R-chain complex with an R-module M concentrated indegree 0. One can take X ′ to be, up to isomorphism in hCh(R), a free resolution of M , sothe total left derived functor

LF : DCh(R) −→ hCh(R) −→ DCh(R)

evaluated on M isLF (M) = X ′ ⊗R N.

Its ith homology is thenHi(LF (M)) = TorRi (M,N),

as previously defined in 1.4.

For this reason, for a functor F : hCh(R)→ hCh(R), one sometimes writes

LiF (X) = Hi(LF (X)).

Remark. In fact, one could consider any F : hCh(R) → C , where C is any category withsome notions of “homotopy groups”; in Ch(R), the correct notions are homology groups, asExercise 10 suggests.

Example (Ext). For an R-module N , consider

F = HomR(−, N) : hCh(R) −→ hCh(R)op,


where the target category is the category of R-cochain complexes and R-cochain homotopyclasses of R-cochain maps, which is isomorphic to hCh(R) by re-indexing by the Note in 1.2.Then we can compute the total left derived functor

LF : DCh(R) −→ hCh(R)op −→ DCh(R)op

byLF (X) = HomR(X ′, N).

If X = M , then we can take X ′ to be, up to isomorphism in hCh(R), a free resolution of M .Taking the ith cohomology, or equivalently the (−i)th homology, we have

L−iF (M) = H−i(LF (M)) = ExtiR(M,N),

as previously defined in 1.4.

3.5. Co-localization in abelian categories. Recall that an object P in an abelian cate-gory A is projective if any diagram

P

��B // // C // 0

a lift (the dashed arrow) exists. Equivalently, P is projective if and only if A(P,−) is anexact functor.

Definition. Suppose the abelian category A has enough projectives. A (projective) cellchain complex in A is a chain complex C that is the directed union of a sequence of chaincomplexes in A

0 = C(−1) ⊆ C(0) ⊆ C(1) ⊆ · · · ,where C(n+1) is the mapping cone of a chain map

P(n) =⊕i

Pn,i[i] −→ C(n),

where each Pn,i is a projective object in A , and P(n) is regarded as a chain complex in Awith zero differentials.

Exercise 16. Prove that these cell R-chain complexes are co-local in hCh(R) with respectto quasi-isomorphisms.

Theorem 3.5.1. If an abelian category A has colimits and enough projectives, then hCh(A )has co-localization with respect to quasi-isomorphisms, with co-local objects being the pro-jective cell A -chain complexes. �

In particular, we have TorAn and ExtnA , which are (small) abelian groups.

Note. Abelian sheaves (see !! for a precise definition) on a general space X, however, do nothave enough projectives.

There is a dual notion of injectives, which we discuss in Section 3.6.

34 IGOR KRIZ

3.6. Localization in abelian categories. An object I in an abelian category A is injectiveif any diagram

0 // A �� //

��

B

��I

a lift (the dashed arrow) exists. Equivalently, I is injective if and only if A (−, I) is an exactfunctor. In particular, this implies that an arbitrary product of injective objects is injective.

Theorem 3.6.1 ([1, Tag 01D7]). An abelian group A is injective if and only if it is divisible,i.e., for any a ∈ A and n ∈ N, there exists b ∈ A such that nb = a.

Example. Q and Q/Z are divisible, hence injective.

Corollary 3.6.2. The category Ab has enough injectives.

Proof. Fix any abelian group A, and consider any a ∈ A. If a is a torsion element of ordern, there is an abelian group morphism

0 // Z/n � � 17→a //

17→1/n��

A

ιa~~

Q/Z

by the injectivity of Q/Z, and if a has order ∞, there is an abelian group morphism

0 // Z/n � � 17→a //

17→1��

A

ιa~~

Qby the injectivity of Q. Therefore, we have constructed an abelian group morphism

A

∏a∈A

ιa

−−−→∏a∈A|a|<∞

(Q/Z)×∏a∈A|a|=∞

Q,

with kernel ⋂a∈A

Ker ιa = 0.

This gives an injection of A into an injective abelian group. �

Exercise 17. The forgetful functor R-Mod→ Ab given by the restriction of scalars functorinduced from the canonical ring map Z → R has a right adjoint, the right Kan extension,given by

A 7−→ HomZ(R,A).

Prove that this is an injection, and if I is an injective abelian group, then HomZ(R, I) is aninjection R-module.

Corollary 3.6.3. The category R-Mod has enough injectives.


Proof. Any R-module M embeds into an injective abelian group I, say via an abelian groupmorphism f : M → I. Define an R-module map

F : Mf−→ I −→ HomZ(R, I)

m 7−→ f(m) 7−→ [r 7→ f(rm)]

Moreover, this is injective since f and HomZ(R, Id) (from the Exercise above) are, andHomZ(R, I) is an injective R-module also from the Exercise above. �

Note. The category of abelian sheaves on a space X also has enough injectives. We postponethe proof until !!.

Definition. The mapping co-cone Pf of a chain map f ∈ Mor Ch(A ) is Cf [−1], themapping cone of f shifted by −1.

Definition. Suppose an abelian category A has limits and enough injectives. A co-cellA -chain complex is an A -chain complex X that is the directed inverse limit of a sequenceof A -chain complexes:

X = lim←−n

X(n) −→ · · · −→ X(1) −→ X(0) −→ X(−1) = 0,

where X(n+1) is the mapping co-cone (it turns out to be the mapping cone shifted by −1) ofa chain map

X(n) −→ Q(n),

where each Q(n) is injective on each degree and has zero differentials.

Proposition 3.6.4. If A has limits and enough injectives, then co-cell A -chain complexesare local with respect to quasi-isomorphisms.

Note. A bounded above co-cell A -chain complex is a bounded above complex of injectiveobjects.

Lemma 3.6.5. If Q is an A -chain complex which is graded injective and has zero differen-tials, then for every exact A -chain complex X we have [X,Q] = 0.

Proof. By assumption, we have

Q ∼=∏n∈Z

Qn[n],

where each Qn is regarded as an A -chain complex concentrated in degree 0. Therefore,

[X,Q] =∏n

[X,Qn[n]] =∏n

[X[−n], Q] =∏n

H0HomA (X[−n], Qn).

Since Qn is injective, the functor A (−, Qn) is exact. Recall that direct product is exact inAb, so we conclude that HomA (−, Qn) is also exact. Exactness of X implies that X[−n] ∼ 0,so HomA (X[−n], Qn) = 0. It follows that [X,Q] = 0. �

Proof of Proposition 3.6.4. Let J be a co-cell A-chain complex. By the use of mappingco-cone, it suffices to prove that [X, J ] = 0 for any exact complex X in A , just as theproof of the Whitehead Theorem for Ch(R) (Theorem 2.2.3). For each n, we have fromJ(n) → Q(n) → J(n+1)[1] a cofiber sequence (Corollary 2.1.3)

· · · −→ [X[1], Q(n)] −→ [X, J(n+1)] −→ [X, J(n)] −→ [X,Q(n)] −→ · · · .

36 IGOR KRIZ

The Lemma above implies that [X, J(n+1)] ∼= [X, J(n)], and it furthermore follows by inductionthat [X, J(n)] ∼= 0 for all n.

Recall that[X, J ] ∼= H0HomA (X, J) ∼= H0 lim←−

n

HomA

(X, J(n)

)since HomA (X,−) preserves all limits.

Claim. There is a short exact sequence

0 −→ lim←−n

H1HomA

(X, J(n)

)−→ [X, J ] −→ lim←−

n

[X, J(n)] −→ 0.

By the argument above, the left and right terms are already zero before taking the limit(??); it follows that middle term is also zero.

Finally, we note that the Claim follows from the more general Milnor sequence (Proposition3.6.8) discussed in Section 3.6.1. �

Theorem 3.6.6. If an abelian category A has limits and enough injectives, then hCh(A )has localization with respect to quasi-isomorphisms, with local objects being the co-cell A -chain complexes.

Proof. In light of Proposition 3.6.4, it remains to show that for every A -chain complex Xthere exists a co-cell A -chain complex J and a quasi-isomorphism γ : X → J .

We define J inductively on co-cell skeleta. For the base case n = 0, consider the diagram

Z∗X� � //

��

X

��0 // H∗X

� � //� _

��

X/B∗X

zzJ(0)

where J(0) is graded injective and has zero differentials, and the dashed arrow is extendeddegree-wise by injectivity of J(0). The composite γ(0) : X → X/B∗X → J(0) is an A -chainmap since the boundaries are forced to map to zero. Moreover, the induced map on homologyis injective.

Now suppose the map γ(n) : X → J(n) is defined such that it is injective on homology.Consider the diagram

Z∗J(n)� � //

��

J(n)

��

0 //H∗J(n)

H∗X� � //

� _

��

J(n)/B∗J(n)

ImH∗X

yyQ(n)

with Q(n) graded injective and has zero differentials. Again this gives an A -chain mapJ(n) → Q(n), and we define J(n+1) to be the mapping co-cone of it.


We need to show that X → J(n) factors through J(n+1) in Ch(A ). By construction, weknow that the composite

f : X −→ J(n) −→ Q(n)

is zero on homology, and we want to show that f is null-homotopic. Since Q has zerodifferentials, this is equivalent to

f = dQh+ hdX = hdX .

Note that f being an A -chain map means fdX = dQf = 0, thus f restricts to zero onKer dX . We may define h to be zero on Im dX , and extend to X by the graded injectivity ofQ(n):

0 // B∗X //

0��

X

h}}Q(n)

Therefore, the map γ(n)X → J(n) factors through J(n+1) as

Xγ(n+1)−−−→ J(n+1)

α(n)−−−→ J(n).

By induction, we get a limit A -chain complex J = lim←− J(n), which is co-cell by construction,and a limit A -chain map

γ = lim←−n

γ(n) : X −→ J.

It remains to prove that H∗γ is an isomorphism. First observe that lim←−H∗J(n) = H∗X(??). Note that the connecting maps α(n) of the inverse system of co-cell skeleta {J(n)} aresurjective, so the Milnor sequence (Proposition 3.6.8) reads

0 // lim←−n

1H∗+1J(n)// H∗ lim←−

n

J(n)// lim←−n

H∗J(n)// 0

H∗JH∗γ // H∗X

It remains to show that lim←−1 vanishes for the inverse system {H∗J(n), H∗α(n)}. By con-

struction, we have a commutative diagram

· · · // H∗J(n+1)

H∗α(n) // H∗J(n)//

��

H∗Q(n)// · · ·

H∗J(n)

ImH∗γ(n)

, �

::

where the row is the (co)fiber long exact sequence, so it follows that

ImH∗α(n) = ImH∗γ(n).

38 IGOR KRIZ

In particular, for every i ≥ n, the diagram

H∗X

H∗γ(i)�� H∗γ(n+1) ))

H∗γ(n)

++H∗J(i)

// · · · // H∗J(n+1)H∗α(n)

// H∗J(n)

and we haveImH∗γ(n) ⊆ Im[H∗J(i) → H∗J(n)] ⊆ ImH∗α(n),

so equality holds everywhere, and the Mittag–Leffler condition (see the digression below inSection 3.6.1) is satisfied. Therefore lim←−

1H∗+1J(n) = 0 by Theorem 3.6.9. This completesthe proof of that γ is a quasi-isomorphism. �

3.6.1. Digression: lim1. Suppose {A(k), fk}k∈N is an inverse system of abelian groups. Then

lim←−k

A(k)∼= Ker

[∏k∈N

A(k)f−Id−−−→

∏k∈N

A(k)

].

We define

lim←−k

1A(k)∼= Coker

[∏k∈N

A(k)f−Id−−−→

∏k∈N

A(k)

].

Lemma 3.6.7. If all the connecting maps fk of an inverse system {A(k)} are onto, then

lim←−1A(k) = 0.

Proof. We show that f − Id is onto. Let a ∈∏A(k), and we inductively choose bk+1 ∈ A(k+1)

to be a lift of ak + bk ∈ A(k). Then

(f − Id)(b) =∑k

[fk+1(bk+1)− bk] =∑k

ak = a. �

Exercise 18. Prove for a short exact sequence of inverse systems

0 −→ A(k) −→ B(k) −→ C(k) −→ 0

of abelian groups, there is a long exact sequence

0 −→ lim←−k

A(k) −→ lim←−k

B(k) −→ lim←−k

C(k) −→ lim←−k

1A(k) −→ lim←−k

1B(k) −→ lim←−k

1C(k) −→ 0

Suppose we have a tower

· · · −→ X(k)ϕk−→ X(k−1) −→ · · · −→ X(1)

ϕ1−→ X(0)

of Z-chain complex, and we are interested in the homology of lim←−X(k).

Proposition 3.6.8 (Milnor sequence). If the connecting maps ϕk of the tower of Z-chaincomplexes {X(k), ϕk} are surjective, then there are short exact sequences

0 −→ lim←−k

1Hn+1X(k) −→ Hn

(lim←−k

X(k)

)−→ lim←−HnX(k) −→ 0.


Proof. First of all, from surjectivity of ϕk we obtain a short exact sequence

0 −−−→ lim←−k

X(k) −−−→∏k∈N

X(k)ϕ−Id−−−→

∏k∈N

X(k) −−−→ 0.

Since (arbitrary) product is exact in Ch(Ab), the induced long exact sequence reads

· · · Hn+1ϕ−Id−−−−−−→∏k

Hn+1X(k) −−−→ Hn

(lim←−k

X(k)

)−−−→

∏k

HnX(k)Hnϕ−Id−−−−→ · · · .

Splitting this sequence around Hn

(lim←−X(k)

)gives the desired short exact sequences. �

Definition. An inverse system {A(k), fk}k∈N of abelian groups satisfies the Mittag–Lefflercondition if for all k there exists j ≥ k such that for all i ≥ j ≥ k, we have

Im[A(i) → A(k)] ∼= Im[A(j) → A(k)].

Note that the Mittag–Leffler condition is satisfied in particular if all morphisms fk aresurjective.

Theorem 3.6.9. If the Mittag–Leffler condition holds, then lim←−1A(k) = 0.

Proof. We will show that the map f − Id is onto. Let B(k) ⊆ A(k) be the eventual image

lim←−i

Im[A(i) → A(k)] = Im[A(j) → A(k)].

Then we have a short exact sequence of inverse systems

0 −→ {B(k)} −→ {A(k)} −→ {A(k)/B(k)} −→ 0.

Note that the inverse system {B(k)} has surjective connecting maps, so lim←−1B(k) = 0. From

the long exact sequence in Exercise 18, it suffices to show that lim←−1A(k)/B(k) = 0.

Observe that the inverse system {C(k)} = {A(k)/B(k)} satisfies a stronger Mittag–Lefflercondition: for all k there exists j ≥ k such that for all i ≥ j ≥ k, we have Im[C(i) → C(k)] = 0.Now let c ∈

∏C(k), and set

c′k = −ck − ck+1 − · · · − cj−1,

where ci is the image of ci ∈ C(i) in C(k). Then

fk+1(c′k+1) = −ck+1 − ck+2 − · · · − cj−1

since cj = 0 by the stronger Mittag–Leffler condition, so

(f − Id)(c′) =∑k

[fk+1(c′k+1)− c′k] =∑k

ck = c. �

Exercise 19. Calculate lim←−1

ipiZ of the inverse system

{piZ} = · · · −→ Z p−→ Z p−→ Z.

40 IGOR KRIZ

3.7. Total Tor and Ext functors. For M ∈ Obj A , the functor HomA (M,−) : Ch(A )→Ab preserves A -chain homotopies, so we obtain a functor

F = HomA (M,−) : hCh(A ) −→ hCh(Ab) −→ DCh(Ab).

ForN ∈ Obj A , considered as an A -chain complex concentrated in degree 0, we can computeRHomA (M,N) via an injective resolution of N . Specifically, we construct inductively shortexact sequences

0 −→ Ni −→ Qi −→ Ni+1 −→ 0

in A with Qi injective, starting from N0 = N . Then we splice Qi into an A -chain complexQ with Q0 at degree 0, and we obtain a co-cell A -chain complex Q, and the chain mapN → Q is a quasi-isomorphism. Now we can form

RHomA (M,N) := HomA (M,Q),

Rn HomA (M,N) := H−n HomA (M,Q).

Proposition 3.7.1. For any commutative ring R, we have

Rn HomR(M,−)(N) ∼= ExtnR(M,N) := L−n HomR(−, N)(M). �

Exercise 20. Prove that if A is an abelian category with arbitrary products and coproducts,and enough injectives and projectives, then

[LHomA (−, Y )](X) ∼= [RHomA (X,−)](Y ).

In particular, the Proposition above follows.

Recall that we also have an internal tensor product of chain complexes, defined in Section??.

Remark. Such structure is not present in general abelian category. There are concepts toadd this structure, for instance, tensor category.

One can prove that, for calculating TorR, one can resolve in either coordinate. In fact, thesame strategy to prove Exercise 20, that is, resolving both coordinates, also works. This isequivalent to the following Lemma:

Lemma 3.7.2. If X is a cell R-chain complex, then X ⊗R− preserves quasi-isomorphisms.

Proof. Again, it suffices to show that for any exact R-chain complex Y , the R-chain complexX ⊗R Y is also exact.

If P is a graded projective R-chain complex with zero differentials, then P is graded R-flat, and hence H∗(P ⊗R Y ) = 0. Consider the cell skeleta X(n) of X; we show by inductionthat H∗(X(n) ⊗R Y ) = 0 for all n. The base case n = −1 is trivial. Suppose the claim isestablished for X(k) with k ≤ n. Since X(n+1) is the mapping cone of some map

P(n)fn−→ X(n)

with P(n) graded projective with zero differentials. Moreover, there is a short exact sequence

0 −→ X(n) −→ X(n+1) −→ P(n)[1] −→ 0,

which is furthermore split-exact since P(n)[1] is graded projective and has zero differentials.Therefore, applying −⊗R Y gives a short exact sequence

0 −→ X(n) ⊗R Y −→ X(n+1) ⊗R Y −→ P(n)[1]⊗R Y −→ 0.


Since H∗(P(n)[1]⊗R Y ) = 0, applying the 5-lemma to the induced long exact sequence showsthat

H∗(X(n+1) ⊗R Y

)= H∗

(X(n) ⊗R Y

)= 0

by inductive hypothesis.Finally, recall that directed colimit commutes with tensor product and is exact in R-Mod,

so

H∗(X ⊗R Y ) = H∗((

lim−→X(n)

)⊗R Y

)= H∗

(lim−→X(n) ⊗R Y

)= lim−→H∗

(X(n) ⊗R Y

)= 0. �

Corollary 3.7.3. For any commutative ring R, we have

[Ln(M ⊗R −)](N) ∼= TorRn (M,N) := [Ln(−⊗R N)](M). �

Exercise 21. Prove that for any short exact sequence

0 −→M1 −→M2 −→M3 −→ 0

of R-modules and any R-module N , there are long exact sequences

0 −→ TorRn (M1, N) −→ TorRn (M2, N) −→ TorRn (M3, N) −→ 0,

0 −→ ExtnR(N,M1) −→ ExtnR(N,M2) −→ ExtnR(N,M3) −→ 0,

0 −→ ExtnR(M3, N) −→ ExtnR(M2, N) −→ ExtnR(M1, N) −→ 0.

42 IGOR KRIZ

Appendix A. Solutions to Exercises

Exercise 1. Recall that ZS ∼= Z⊕S . Observe that −⊗ A is a left adjoint, hence preserves colimits,and in particular coproducts. �Exercise 2.

(1) Z/m admits the free resolution

C = · · · −→ 0 −→ Z ·m−−→ Z,so

C ⊗ Z/n = · · · −→ 0 −→ Z/n ·m−−→ Z/n;

it is clear that H1(C ⊗ Z/n) = Ker[Z/n ·m−→ Z/n] = Z/ gcd(m,n).(2) It suffices to show that B is Z-flat modules. If B is finitely generated, then the structure

theorem for finitely generated abelian groups implies that B is free, hence in particular Z-flat. For general B, observe that it is a filtered colimit of its finitely generated subgroups,and recall that tensoring is a left adjoint and thus preserves colimits, and that filteredcolimit is an exact functor. �

Exercise 3.

(1) Take the same free resolution C as above, then

Hom(C,Z/n) = · · · ←− 0←− Z/n ·m←−− Z/n;

by the isomorphism theorems, H1(Hom(C,Z/n)) = Coker[Z/n ·m−→ Z/n] = Z/ gcd(m,n).(2) Z is a free resolution of itself, hence there is no higher Ext’s. �

Exercise 5. Clearly

C := · · · −→ R−·x−−→ R

−·x−−→ R

is a free R-resolution of K, and we have

C ⊗R K = · · · −→ K0−−→ K

0−−→ K

andHomR(C,K) = · · · ←− K 0←−− K 0←−− K.

Therefore, we haveTorRn (K,K) = K = ExtnR(K,K)

for any n. �

Exercise 6. The canonical free Z[Z/2]-resolution for the trivial Z[Z/2]-module Z has the form

C = · · · −→ Z[Z/2]1+α−−→ Z[Z/2]

1−α−−→ Z[Z/2]1+α−−→ Z[Z/2]

1−α−−→ Z[Z/2].

SinceC ⊗Z[Z/2] Z = · · · −→ Z 0−−→ Z 2−−→ Z 0−−→ Z 2−−→ Z

andHomZ[Z/2](C, Z) = · · · ←− Z 0←−− Z 2←−− Z 0←−− Z 2←−− Z,

we have

Hn(Z/2; Z) =

{Z/2 n ≥ 0 even,

0 else,Hn(Z/2; Z) =

{Z/k n ≥ 0 odd,

0 else,�


Exercise 9. First note that

Id−pnqn =

1 0 00 0 00 fn 1

.

Define a collection of maps hn : (Ci)n → (Ci)n+1 by

hn =

0 0 00 0 0

(−1)n 0 0

,

then

dn+1hn + hn−1dn =

(−1)2n 0 00 0 0

(−1)n(dC)n + (−1)n−1(dC)n fn (−1)2n

=

1 0 00 0 00 fn 1

= Id−pnqn,

therefore Id ' pq. �

Exercise 11. Suppose C is cell. Since C is exact, the unique maps 0 → C → 0 are quasi-isomorphisms. Then the Whitehead theorem implies that they are R-chain homotopy equivalences.In particular, we conclude that TorRn (C,K) ∼= TorRn (0,K) = 0 for all n. However, since

C ⊗R K = · · · −→ K0−−→ K

0−−→ K −→ · · · ,

we have TorRn (C,K) ∼= K for all n. This is a contradiction. �

Exercise 12. Let M ′ be the canonical cell approximation of M , which is necessarily free, we have

ExtnR(M,N) = Hn HomR(M ′, N)

∼= [R[−n],HomR(M ′, N)]

∼= [R[−n]⊗RM ′, N ]

= [M ′[−n], N ] ∼= [M ′, N [n]].

Here, the second isomorphism is the tensor-hom adjunction, which descends to hCh(R) since bothtensor product and hom preserve R-chain homotopies. The last isomorphism is due to [n] being anautomorphism on Ch(R) and hence hCh(R).

Now let N ′ be the canonical cell approximation of N , then N ′[n] is a the canonical cell approxi-mation of N [n]. The Whitehead theorem implies that

ExtnR(M,N) ∼= [M ′, N [n]] ∼= [M ′, N ′[n]] ∼= DCh(R)(M,N [n]). �

Exercise 13. For any H-set X and any G-set Y , we have a bijection

G-Set(Y,HomH(G,X))∼=←→ H-Set(Y,X)

f 7−→ ε ◦ f[y 7→ [g 7→ ϕ(gy)]] ←− [ ϕ

�

Exercise 14. For f : x→ x′ ∈ Mor D , map each coproduct over g ∈ D(i(y), x) to a coproduct overfg ∈ D(i(y), x′); this induces a canonical map

(i]F )(f) : (i]F )(x) −→ (i]F )(x′).

Functoriality is clear from the universal property of coequalizer. �

44 IGOR KRIZ

Exercise 15. The counit of adjunction has components

(i∗i∗F )(y) = (i∗F )(i(y)) −→ F (y){ϕy : D(i(y), i(z))}z 7−→ ϕy(idi(y))

�

Exercise 4. Suppose f, g : C → D are R-chain maps, and h : f ' g is an R-chain homotopy. Thenf ⊗R N ' g ⊗R N via {hn ⊗R N}n, and HomR(f,N) ' HomR(g,N) via {HomR(hn, N)}n. �

Exercise 10. We first show [R,X] ∼= H0X; the general case follows since

HiX ∼= H0(X[−i]) ∼= [R,X[−i]] ∼= [R[i], X].

Consider the diagram

· · · // 0 //

��

R //

f

��

h

}}

0 //

��||

· · ·

· · · // X1// X0

// X−1// · · ·

Then the commutativity forces f to land in Z0; this shows HomR(R,X) = Z0X. On the otherhand, any null homotopy h of a map from R to X has the form f = d1h ∈ B0X. Therefore,

[R,X] ∼= Z0X/B0X = H0X. �

Exercise 16. �

Exercise 17. It is clear that this map is given by a 7−→ [r 7→ ra]; it is injective since a 7−→ 0 meansit takes 1 ∈ R to a = 0.

Since I is injective, for a monomorphism f : M → N in R-Mod the map HomZ(f, I). From thetensor-hom adjunction∗, we have an isomorphism of functors

HomZ(−, I) ∼= HomR(−,HomZ(R, I)).

Applying this functor to f , we see that HomR(f,HomZ(R, I)) is an epimorphism, so HomZ(R, I)is an injective R-module. �

Exercise 18. Applying the snake lemma to the diagram

0 //∏k A(k)

//

f−id��

∏k B(k)

//

g−id��

∏k C(k)

//

h−id��

0

0 //∏k A(k)

//∏k B(k)

//∏k C(k)

// 0

gives the desired long exact sequence. �

∗ Recall the bimodule tensor-hom adjunction: for (not necessarily commutative) rings R, S, T and U , and(R,S)-bimodule M , (S, T )-bimodule N , and (U, T )-bimodule P , we have an isomorphism of (U,R)-modules

HomMod-T (M ⊗S N,P ) ∼= HomMod-S (M,HomMod-T (N,P )) .

The desired isomorphism is obtained by specializing to the case (R,S, T, U) = (R,R,Z,Z) and (M,N,P ) =(−, R, I).


Exercise 19. Form a short exact sequence of inverse systems

...

��

...

��

...

��0 // Z

p

��

pi+1

// Z // // Z/pi+1Z

/p��

// 0

0 // Z

p

��

pi // Z // // Z/piZ

/p��

// 0

0 // Zpi−1

// Z // // Z/pi−1Z // 0

...

��...

��...

��

Note that the last two inverse systems both satisfy the Mittag–Leffler condition since the connectingmaps are surjective, so lim←−

1 vanish. The associated long exact sequence then reads

0 −→ lim←−p

piZ −→ lim←−p

Z −→ lim←−p

Z/piZ −→ lim←−p

1piZ −→ 0.

Clearly the middle map is Z→ Zp, which is nonzero, so it follows that

lim←−p

piZ = 0, lim←−p

1piZ = Zp/Z. �

Exercise 20. By the assumption of enough projectives and injectives, we may choose a cell approx-imation γ : X ′ → X and a co-cell approximation δ : Y → Y ′. Then we have isomorphisms

RHomA (X,Y ) := HomA (X,Y ′)[γ,Y ′]−−−→∼= HomA (X ′, Y ′)

[X′,δ]−−−→∼= HomA (X ′, Y ) =: LHomA (X,Y ).

�

Exercise 21. Let P•∼−→ N be a projective resolution and N

∼−→ I• an injective resolution of N .Then the functors − ⊗R Pn, HomR(Pn,−) and HomR(In,−) are exact, so we obtain short exactsequences of chain complexes

0 −→M1 ⊗R P• −→M2 ⊗R P• −→M3 ⊗R P• −→ 0,

0 −→ HomR(P•,M1) −→ HomR(P•,M2) −→ HomR(P•,M3) −→ 0,

0 −→ HomR(M3, I•) −→ HomR(M2, I•) −→ HomR(M3, I•) −→ 0.

The induced long exact sequences are the desired ones, by definitions of Tor and Ext as well asProposition 3.7.1. �

Exercise 7. Again, we proceed by induction. For the base case n = 0 we set h0 = 0. For theinductive step, it suffices to define h on the universal element (σ0, τ0) = (Id∆k , Id∆`) ∈ Sk(∆k) ⊗S`(∆

`). Again, the only requirement is

d×n+1hn(σ0 ⊗ τ0) = (ϕ′n − ϕn − hn−1d⊗n )(σ0 ⊗ τ0),

but the right-hand side is a cycle, hence a boundary by acyclicity of C(∆k)⊗Z C(∆`), since

[d×n (ϕ′n − ϕn)− d×n hn−1d⊗n ](σ0 ⊗ τ0) = [(ϕ′n−1 − ϕn−1)d⊗n − d×n hn−1d

⊗n ](σ0 ⊗ τ0)

= [(d×n hn−1 + hn−2d⊗n−1)d⊗n − d×n hn−1d

⊗n ](σ0 ⊗ τ0)

= hn−2d⊗n−1)d⊗n (σ0 ⊗ τ0) = 0. �

46 IGOR KRIZ

Exercise 8. Recall that

Hn(RPk;Z/2) =

{Z/2 n ≤ k0 else

so we haveq2 Z/2 Z/2 Z/2 Z/21 Z/2 Z/2 Z/2 Z/20 Z/2 Z/2 Z/2 Z/2

Hp(RP3;Z/2)⊗Hq(RP2;Z/2) 0 1 2 3 p

Therefore

Hn(RP3 × RP2;Z/2) =

Z/2 n = 0, 5

(Z/2)⊕2 n = 1, 4

(Z/2)⊕3 n = 3, 4

0 else

�

References

[1] The Stacks Project Authors, exitStacks Project, http://stacks.math.columbia.edu, 2017.

http://stacks.math.columbia.edu

Documents

A COURSE IN MODERN ALGEBRAIC TOPOLOGY...A COURSE IN MODERN ALGEBRAIC TOPOLOGY IGOR KRIZ Abstract. This is a course in algebraic topology for anyone who has seen the fundamental group