Lectures on Derived Categories

236 Pages·2010·1.226 MB·English

by  Dragan Milicic

Lectures on Derived Categories
Dragan Miliˇci´c

Contents
Chapter 1. Localization of Categories 1
1. Localization of categories 1
2. Localization of additive categories 26
Chapter 2. Triangulated Categories 47
1. Triangulated categories 47
Chapter 3. Derived Categories 91
1. Category of complexes 91
2. Homotopic category of complexes 102
3. Derived categories 117
4. Generating classes 135
Chapter 4. Truncations 145
1. t-structures 145
2. Extensions 178
3. Grothendieck Groups 182
Chapter 5. Derived Functors 191
1. Derived functors 191
2. Resolutions 222
3. Derived functors revisited 224
iii

CHAPTER 1
Localization of Categories
1. Localization of categories
1.1. Localization of categories. Let A be a category and S an arbitrary
class of morphisms in A. In this section we establish the following result.
1.1.1. Theorem. There exist a category A[S−1] and a functor Q : A −→
A[S−1] such that
(i) Q(s) is an isomorphism for every s in S;
(ii) for any category B and functor F : A −→ B such that F(s) is an iso-
morphism for any s in S, there exists a unique functor G:A[S−1]−→B
such that F =G◦Q, i.e., we have the following commutative diagram of
functors:
F
A //B
Q(cid:15)(cid:15) xxxxxGxxxx<<
A[S−1]
The category A[S−1] is unique up to isomorphism.
The category A[S−1] is called the localization of A with respect to S.
We first prove the uniqueness. Assume that we have two pairs (C,Q) and
(C′,Q′) satisfying the conditions of the theorem. Then, the universal property
would imply the existence of the functors G: C −→C′ and H : C′ −→C such that
Q′ =G◦QandQ=H◦Q′,i.e.,wewouldhavethefollowingcommutativediagram
of functors:
Q
A //C
EE
G
Q′
H
(cid:15)(cid:15) (cid:6)(cid:6)
C′
This implies that Q′ = (G◦H)◦Q′ and Q = (H ◦G)◦Q. In particular, we have
the following commutative diagram of functors
Q
A //C
idC
Q
H◦G
(cid:15)(cid:15) (cid:6)(cid:6)
xx
C
1
2 1. LOCALIZATION OF CATEGORIES
where id is the identity functor on C. By the uniqueness of the factorization we
C
must have H ◦G = id . Analogously, we get G◦H = id . Therefore, H and G
C C′
are isomorphisms of categories.
It remains to establish the existence of A[S−1]. We put
ObA[S−1]=ObA.
It remains to define morphisms in A[S−1].
We fix two objects M and N in A. Let I =(0,1,...,n), J ={(i,i+1)|0≤
n n
i≤n−1}. A path of length n is
(i) a map L of I into the objects of A such that L =M and L =N;
n 0 n
(ii) a map Φ of J into the morphisms of A such that either Φ(i,i+1)=f :
n i
L −→L or Φ(i,i+1)=s :L −→L with s in S.
i i+1 i i+1 i i
Diagrammatically, a path can be represented by an oriented graph as
f f s f s s
0 - 1 - (cid:27) i−1 i - (cid:27) n−2 (cid:27) n−1
• • ··· • ··· • •
M L L L N
1 i n−1
An elementary transformation of a path is:
(i) Switch of
f f
i−1- i -
··· • • • ···
L L L
i−1 i i+1
and
f ◦f
i i−-1
··· • •··· .
L L
i−1 i+1
(ii) Switch of
s s
- (cid:27)
··· • • • ···
L P L
and
id
L-
··· • •··· .
L L
(iii) Switch of
s s
(cid:27) -
··· • • • ···
L P L
and
id
L-
··· • •··· .
L L
1. LOCALIZATION OF CATEGORIES 3
(iv) Switch of
id s
L- (cid:27)
··· • • • ···
L L P
and
s
(cid:27)
··· • •··· .
L P
Two paths between M and N are equivalent if one can be obtained from the
other by a finite sequence of elementary transformations. This is clearly an equiv-
alence relation on the set of all paths between M and N.
We define morphisms between M and N in A[S−1] as equivalence classes of
paths betweenM andN. The compositionof paths is defined as concatenation. It
clearly induces a composition on equivalence classes. The identity morphism of an
object M is given by the equivalence class the path
id
M-
• • .
M M
It is easy to check that A[S−1] is a category. We define the functor Q from A into
A[S−1] to be the identity on objects, and to map the morphism f :M −→N into
the path
f
-
• • .
M N
Clearly, Q(s) is represented by
s
-
• •
M N
and its inverse is
s
(cid:27)
• • .
M N
Hence, Q(s), s ∈ S, are isomorphisms. We define G to be equal to F on objects.
For a path of length n between M and N, we put
G(P)=G(Φ(n−1,n))◦···◦G(Φ(2,1))◦G(Φ(1,0))
where
F(f ), if Φ(i,i+1)=f :L −→L
i i i i+1
G(Φ(i,i+1))= .
(F(si)−1, if Φ(i,i+1)=si :Li+1 −→Li
If a path P′ is obtained from another path P by an elementary transformation, it
is clear that G(P′)=G(P). Therefore, G is constant on the equivalence classes of
4 1. LOCALIZATION OF CATEGORIES
paths. Hence, it induces a map from Hom (M,N) into Hom (G(M),G(N)).
A[S−1] B
ItiseasytocheckthatG,definedinthis way,is afunctor fromA[S−1]intoB such
that G◦Q = F. Moreover, by the construction, G is uniquely determined by F.
Therefore, the pair (A[S−1],Q) satisfies the conditions of the theorem.
1.2. Localization of the opposite category. Let Aopp be the category
opposite to the category A. Let S be a class of morphisms in A. We can also
view them as morphisms in Aopp. The functor Q : A −→ A[S−1] can be viewed
as a functor from Aopp into the opposite category A[S−1]opp of A[S−1] which we
denote with the same symbol. For any morphism s ∈ S, the morphism Q(s)
is an isomorphism in A[S−1]opp. Hence, the functor Q : Aopp −→ A[S−1]opp
factors through the localization Aopp[S−1] of Aopp, i.e., we have a unique functor
α:Aopp[S−1]−→A[S−1]opp such that the diagram of functors
Q
Aopp //A[S−1]opp
Qopp (cid:15)(cid:15) qqqqqqαqqqq88
Aopp[S−1]
commutes. The functor α is identity on the objects. Moreover, if ϕ : M −→ N is
a morphism represented by a path
f f s f s s
0 - 1 - (cid:27) i−1 i - (cid:27) n−2 (cid:27) n−1
• • ··· • ··· • •
M L L L N
1 i n−1
in Aopp[S−1], the morphism α(ϕ) is the morphism in A[S−1]opp corresponding to
the morphism in A[S−1] represented by the path
s s f s f f
(cid:27) n−1 (cid:27) n−2 i - (cid:27) i−1 1 - 0 -
• • ··· • ··· • •
N L L L M
n−1 i 1
obtainedby invertingthe orderofsegments inthe originalpath. This immediately
leads to the following result.
1.2.1. Theorem. The functor α:Aopp[S−1]−→A[S−1]opp is an isomorphism
of categories.
1.3. Localizing classes of morphisms. LetAbeacategory. IfS isanarbi-
trary class of morphisms, it is very hardto say anything about A[S−1]. Therefore,
we concentrate on special types of classes of morphisms. For such classes, one can
give a more manageable description of morphisms.
AclassofmorphismsSinAisalocalizingclassifithasthefollowingproperties:
(LC1) For any object M in A, the identity morphism id on M is in S.
M
(LC2) If s,t are morphisms in S, their composition s◦t is in S.
1. LOCALIZATION OF CATEGORIES 5
(LC3a) For any pair f in MorA and s in S, there exist g in MorA and t in S
such that the diagram
g
K //L
t s
(cid:15)(cid:15) (cid:15)(cid:15)
M //N
f
is commutative.
(LC3b) For any pair f in MorA and s in S, there exist g in MorA and t in S
such that the diagram
g
K oo L
OO OO
t s
M oo N
f
is commutative.
(LC4) Letf,g :M −→N be twomorphisms. Thenthereexists sinS suchthat
s◦f =s◦g if and only if there exists t in S such that f ◦t=g◦t.
Clearly,ifS isalocalizingclassinA, itisalsoalocalizingclassinthe opposite
category Aopp.
1.3.1. Example. LetS beafamilyofisomorphismsinAwhichsatisfies(LC1)
and (LC2). Then, S is a localizing class in A. To check (LC3a), we put K = M,
t = id and g = s−1 ◦f. The check of (LC3b) is analogous. It is obvious that
M
(LC4) holds.
Let A be a category and S a localizing class in A. Let A[S−1] be the local-
ization of A with respect to S. Then any morphism in A[S−1] is represented as a
composition of several morphisms Q(s)−1, s∈S, and Q(f).
By (LC2), Q(s◦t)−1 = Q(t)−1◦Q(s)−1, hence any morphism in A[S−1] has
the form
Q(f )◦Q(s )−1◦Q(f )◦Q(s )−1◦···◦Q(f )◦Q(s )−1
1 1 2 2 n n
with s ,s ,...,s ∈ S. On the other hand, by (LC3a), for any morphism f and
1 2 n
s ∈ S, there exist g and t ∈ S such that f ◦t = s◦g. Therefore, Q(f)◦Q(t) =
Q(s)◦Q(g), and Q(s)−1◦Q(f) = Q(g)◦Q(t)−1. By induction in n, this implies
thatthatanymorphisminA[S−1]canberepresentedasQ(f)◦Q(s)−1 withs∈S.
Analogously, it can also be represented by Q(s)−1◦Q(f) with s ∈ S. Therefore,
any morphism can be viewed as a left or right “fraction”.
We are going to describe now a more manageable description of morphisms in
A[S−1] which is suitable for computations.
Let A be a category and S a localizing class of morphisms in A. A (left) roof
between M and N is a diagram
L
@
~~}}s}}}∼}}} @@@@f@@(cid:31)(cid:31)
M N
where s is in S. The symbol ∼ denotes that that arrow is in S.
6 1. LOCALIZATION OF CATEGORIES
Analogously, we define a (right) roof between M and N as a diagram
L
}}}}}g}}}>> __@∼@@@@t@@
M N
where t is in S.
Clearly,goingfromAtotheoppositecategoryAopp switchesleftroofsbetween
M and N and right roofs between N and M. Therefore, it is enough to study
properties of left roofs.
If
L K
@ A
~~}}s}}}∼}}} @@@@f@@(cid:31)(cid:31) ~~|||t||∼||| AAAAgAA
M N M N
are two left roofs, we say that they are equivalent if there exist an object H in A
and morphisms p:H −→L and q :H −→K such that the diagram
L
~~|||s||∼||| OOpAAAAAfAAA
M H N
``BBBtBB∼BBB q }}}}g}}>>
(cid:15)(cid:15) }
K
commutes and s◦p=t◦q ∈S.
1.3.2. Remark. If
L K
@ A
~~}}s}}}∼}}} @@@@f@@(cid:31)(cid:31) ~~|||t||∼||| AAAAgAA
M N M N
are two equivalent left roofs, Q(p◦s)= Q(p)◦Q(s) is an isomorphism in A[S−1].
Since Q(s) is also an isomoprphism, Q(p) is an isomorphism too. Analogously, we
see that Q(q) is also an isomorphism. Hence,
Q(f)◦Q(s)−1 =Q(f)◦Q(p)◦Q(p)−1◦Q(s)−1 =Q(f ◦p)◦Q(s◦p)−1
=Q(g◦q)◦Q(t◦q)−1 =Q(g)◦Q(q)◦Q(q)−1◦Q(t)−1 =Q(g)◦Q(t)−1.
This motivates the above definition.
1.3.3. Lemma. The above relation on left roofs is an equivalence relation.