Lectures on derived and triangulated categories
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES
BEHRANG NOOHI
These are the notes of three lectures given in the International Workshop on
Noncommutative Geometry held in I.P.M., Tehran, Iran, September 11-22.
The first lecture is an introduction to the basic notions of abelian category theory,
with a view toward their algebraic geometric incarnations (as categories of modules
over rings or sheaves of modules over schemes).
In the second lecture, we motivate the importance of chain complexes and work
out some of their basic properties. The emphasis here is on the notion of cone of
a chain map, which will consequently lead to the notion of an exact triangle of
chain complexes, a generalization of the cohomology long exact sequence. We then
discuss the homotopy category and the derived category of an abelian category, and
highlight their main properties.
As a way of formalizing the properties of the cone construction, we arrive at
the notion of a triangulated category. This is the topic of the third lecture. Af-
ter presenting the main examples of triangulated categories (i.e., various homo-
topy/derived categories associated to an abelian category), we discuss the prob-
lem of constructing abelian categories from a given triangulated category using
t-structures.
A word on style. In writing these notes, we have tried to follow a lecture style rather
than an article style. This means that, we have tried to be very concise, keeping the
explanations to a minimum, but not less (hopefully). The reader may find here and
there certain remarks written in small fonts; these are meant to be side notes that
can be skipped without affecting the flow of the material. With a few insignificant
exceptions, the topics are arranged in linear order.
A word on references. The references given in these lecture notes are mostly sugges-
tions for further reading and are not meant for giving credit or to suggest originality.
Acknowledgement. I would like to thank Masoud Khalkhali, Matilde Marcolli,
and Mehrdad Shahshahani for organizing the excellent workshop, and for inviting
me to participate and lecture in it. I am indebted to Masoud for his continual
help and support. At various stages of preparing these lectures, I have benefited
from discussions with Snigdhayan Mahanta . I thank Elisenda Feliu for reading
the notes and making useful comments and, especially for creating the beautiful
diagrams that I have used in Lecture 2!
http://arXiv.org/abs/0704.1009v1
2 BEHRANG NOOHI
Contents
Lecture 1: Abelian categories 3
1. Products and coproducts in categories 3
2. Abelian categories 4
3. Categories of sheaves 7
4. Abelian category of quasi-coherent sheaves on a scheme 9
5. Morita equivalence of rings 10
6. Appendix: injective and projective objects in abelian categories 11
Lecture 2: Chain complexes 13
1. Why chain complexes? 13
2. Chain complexes 14
3. Constructions on chain complexes 15
4. Basic properties of cofiber sequences 17
5. Derived categories 20
6. Variations on the theme of derived categories 21
7. Derived functors 23
Lecture 3: Triangulated categories 24
1. Triangulated categories 24
2. Cohomological functors 27
3. Abelian categories inside triangulated categories; t-structures 28
4. Producing new abelian categories 30
5. Appendix I: topological triangulated categories 31
6. Appendix II: different illustrations of TR4 31
References 32
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 3
Lecture 1: Abelian categories
Overview. In this lecture, we introduce additive and abelian categories, and dis-
cuss their most basic properties. We then concentrate on the examples of abelian
categories that we are interested in. The most fundamental example is the cate-
gory of modules over a ring. The next main class of examples consists of various
categories of sheaves of modules over a space; a special type of these examples, and
a very important one, is the category of quasi-coherent sheaves on a scheme. The
idea here is that one can often recover a space from an appropriate category of
sheaves on it. For example, we can recover a scheme from the category of quasi-
coherent sheaves on it. This point of view allows us to think of an abelian category
as “a certain category of sheaves on a certain hypothetical space”. One might also
attempt to extract the “ring of functions” of this hypothetical space. Under some
conditions this is possible, but the ring obtained is not unique. This leads to the
notion of Morita equivalence of rings. The slogan is that the “ring of functions” on
a noncommutative space is well defined only up to Morita equivalence.
1. Products and coproducts in categories
References: [HiSt], [We], [Fr], [Ma], [GeMa].
Let C be a category and {Xi}i∈I a set of objects in C.
The product
i∈I Xi is an object in C, together with a collection of morphisms
i∈I Xi → Xi, satisfying the following universal property:
Given any collection of morphisms fi : Y → Xi,
i∈I Xi
The coproduct
i∈I Xi is an object in C, together with a collection of morphisms
ιi : Xi →
i∈I Xi, satisfying the following universal property:
Given any collection of morphisms gi : Xi → Y ,
i∈I Xi
Y Xigi
Remark 1.1. Products and coproducts may or may not exist, but if they do they
are unique up to canonical isomorphism.
Example 1.2.
4 BEHRANG NOOHI
1. C = Sets:
= disjoint union,
=cartesian product.
2. C = Groups:
= free product,
=cartesian product.
3. C = UnitalCommRings:
finite
= ⊗Z,
=cartesian product.
4. C = Fields:
= does not exist,
=does not exist.
5. C = R-Mod=left R-modules, R a ring:
finite
finite
= ⊕, the
usual direct sum of modules.
Exercise. Show that in R-Mod there is a natural morphism
i∈I Xi →
i∈I Xi,
and give an example where this is not an isomorphism.
2. Abelian categories
References: [HiSt], [We], [Fr], [Ma], [GeMa].
We discuss three types of categories: Ab-categories, additive categories, and
abelian categories. Each type of category has more structure/properties than the
previous one.
AnAb-category is a category C with the following extra structure: each HomC(X,Y )
is endowed with the structure of an abelian group. We require that composition is
linear:
u(f + g) = uf + ug, (f + g)v = fv + gv.
An additive functor between Ab-categories is a functor that induces group ho-
momorphisms on Hom-sets.
An additive category is a special type of Ab-category. More precisely, an additive
category is an Ab-category with the following properties:
◮ There exists a zero object 0 such that
∀X, HomC(0, X) = {0} = HomC(X, 0).
◮ Finite sums exist. (Equivalently, finite products exist; see Proposition 2.1.)
Before defining abelian categories, we discuss some basic facts aboutAb-categories.
Proposition 2.1. Let C be an Ab-category with a zero object, and let {Xi}i∈I be a
finite set of objects in C. Then
Xi exists if an only if
Xi exists. In this case,
Xi and
Xi are naturally isomorphic.
Proof. Exercise. (Or See [HiSt].) �
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 5
Notation. Thanks to Proposition 2.1, it makes sense to use the same symbol ⊕ for
both product and coproduct (of finitely many objects) in an additive category.
Exercise. In Example 1.2, which ones can be made into additive categories?
A morphism f : B → C in an additive category is called a monomorphism if
−→ C, f ◦ g = 0 ⇒ g = 0.
A kernel of a morphism f : B → C in an additive category is a morphism i : A→ B
such that f ◦ i = 0, and that for every g : X → B with f ◦ g = 0
Proposition 2.2. If i is a kernel for some morphism, then i is a monomorphism.
The converse is not always true.
We can also make the dual definitions.
A morphism f : B → C in an additive category is called an epimorphism if
−→ D, h ◦ f = 0 ⇒ h = 0.
A cokernel of a morphism f : B → C in an additive category is a morphism
p : C → D such that p ◦ f = 0, and such that for every h : B → Y with h ◦ f = 0
Proposition 2.3. If p is a cokernel for some morphism, then p is an epimorphism.
The converse is not always true.
Remark 2.4. Kernels and cokernels may or may not exist, but if they do they are
unique up to a canonical isomorphism.
Example 2.5.
1. In A = R-Mod kernels and cokernels always exist.
6 BEHRANG NOOHI
2. Let A be the category of finitely generated free Z-modules. Then kernels
and cokernels always exist. (Exercise. What is the cokernel of Z
−→ Z?)
3. Let A be the category of C-vector spaces of even dimension. Then kernels
and cokernels do not always exist in A. (Give an example.) The same thing
is true if A is the category of infinite dimensional vector spaces.
An abelian category is an additive category A with the following properties:
◮ Kernels and cokernels always exist in A.
◮ Every monomorphism is a kernel and every epimorphism is a cokernel.
Main example. For every ring R, the additive category R-Mod is abelian.
Remark 2.6. Note that an Ab-category is a category with an extra structure. How-
ever, an additive category is just an Ab-category which satisfies some property (but
no additional structure). An abelian category is an additive category which satisfies
some more properties.
Exercise. In Example 2.5 show that (1) is abelian, but (2) and (3) are not. In
(2) the map Z
−→ Z is an epimorphism that is not the cokernel of any morphism,
because it is also a monomorphism!
Proposition 2.7. Let f : B → C be a morphism in an abelian category. Let
i : ker(f) → B be its kernel and p : C → coker(f) its cokernel. Then there is
a natural isomorphism coker(i) ∼−→ ker(p) fitting in the following commutative
diagram:
ker(f)
coker(f)
coker(i)
ker(p)
Corollary 2.8. Every morphism f : B → C in an abelian category has a unique
(up to a unique isomorphism) factorization
fmono
Corollary 2.9. In an abelian category, mono + epi ⇔ iso.
The object I (together with the two morphisms fepi and fmono) in the above corol-
lary is called the image of f and is denoted by im(f). The morphism fmono factors
through every monomorphism into C through which f factors. Dually, fepi factors
through every epimorphism originating from B through which f factors. Either of
these properties characterizes the image.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 7
In an abelian category, a sequence
0 −→ A
−→ C −→ 0
is called exact if f is a monomorphism, g is an epimorphism, and im(f) = ker(g).
An additive functor between abelian categories is called exact if it takes exact
sequences to exact sequences.
We saw that R-modules form an abelian category for every ring R. In fact, every
small abelian category is contained in some R-Mod. (A category is called small if
its objects form a set.)
Theorem 2.10 (Freyd-Mitchell Embedding Theorem [Fr, Mi1]). Let A be a small
abelian category. Then there exists a unital ring R and an exact fully faithful functor
A → R-Mod.
3. Categories of sheaves
References: [Ha1], [KaSch], [Iv], [GeMa].
The main classes of examples of abelian categories are categories of sheaves
over spaces. We give a quick review of sheaves and describe the abelian category
structure on them.
Let C be an arbitrary category (base) and A another category (values).
A presheaf on C with values in A is a functor F : Cop → A. A morphism f : F → G
of presheaves is a natural transformation of functors. The category of presheaves
is denoted by PreSh(C,A).
Typical example. Let X be a topological space, and let C = OpenX be the
category whose objects are open sets of X and whose morphisms are inclusions.
Let A = Ab, the category of abelian groups. A presheaf F of abelian groups on X
consists of:
⊲ A collection of abelian groups
F(U), ∀ U open;
⊲ “Restriction” homomorphisms
F(U) → F(V ), ∀ V ⊆ U.
Restriction homomorphisms should respect triple inclusions W ⊆ V ⊆ U
and be equal to the identity for U ⊆ U .
Example 3.1.
1. (Pre)sheaf of continuous functions on a topological space X. The assign-
U 7→ OcontX (U) = {continuous functions on U}
8 BEHRANG NOOHI
is a presheaf on X . The restriction maps are simply restriction of func-
tions. In fact, this is a presheaf of rings because restriction maps are ring
homomorphisms.
2. Constant presheaf. Let A be an abelian group. The assignment
U 7→ A
is a presheaf of groups. The restriction maps are the identity maps.
Variations. There are many variations on these examples. For instance, in (1) one can take
the (pre)sheaf of Cr functions on a Cr-manifold, or (pre)sheaf of holomorphic functions
on a complex manifold, and so on. These are called structure sheaves. Idea: Structure
sheaves encode all the information about the structure in question (e.g, Cr, analytic,
holomorphic, etc.). So, for instance, a complex manifold X can be recovered from its
underlying topological space Xtop and the sheaf OholoX . That is, we can think of the pair
(Xtop, OholoX ) as a complex manifold.
Exercise. Formulate the notion of a holomorphic map of complex manifolds purely
in terms of the pair (Xtop,OholoX ).
Proposition 3.2. Let C be an arbitrary category, and A an abelian category. Then
PreSh(C,A) is an abelian category.
The kernel and cokernel of a morphism f : F → G of presheaves are given by
ker(f) : U 7→ ker
−→ G(U)
coker(f) : U 7→ coker
−→ G(U)
A presheaf F, say of abelian groups, rings etc., on X is called a sheaf if for every
open U ⊆ X and every open cover {Uα} of U the sequence
F(Uα) −→
F(Uα ∩ Uβ)
(fα) 7→ (fα|Uα∩Uβ − fβ|Uα∩Uβ )
is exact.
Example 3.3. Structure sheaves (e.g., Example 3.1.1) are sheaves! More generally,
every vector bundle E → X gives rise to a sheaf of vector spaces via the assignment
U 7→ E(U), where E(U) stands for the space of sections of E over U . Is a constant
presheaf (Example 3.1.2) a sheaf?
For an abelian category A, we denote the full subcategory of PreSh(X,A) whose
objects are sheaves by Sh(X,A).
Proposition 3.4. The category Sh(X,A) is abelian.
Remark 3.5.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 9
1. Monomorphisms in Sh(X,A) are the same as the ones in PreSh(X,A), but
epimorphisms are different: f : F → G is an epimorphism if for every open
U and every a ∈ G(U),
∃ {Uα}, open cover of U, such that:
∀α, a|Uα is in the image of f(Uα) : F(Uα) → G(Uα).
2. Kernels in Sh(X,A) are defined in the same way as kernels in PreSh(X,A),
but cokernels are defined differently: if f : F → G is a morphism of sheaves,
the cokernel of f is the
sheaf associated to the presheaf U 7→ coker
−→ G(U)
We remark that there is a general procedure for producing a sheaf Fsh out of a
presheaf F. This is called sheafification. The sheaf Fsh is called the sheaf associated
to F and it comes with a natural morphism of presheaves i : F → Fsh which is
universal among morphisms F → G to sheaves G. More details on this can be found
in [Ha1].
4. Abelian category of quasi-coherent sheaves on a scheme
References: [Ha1], [GeMa].
We give a super quick review of schemes. We then look at the category of
quasi-coherent sheaves on a scheme.
The affine scheme (SpecR,OR) associated to a commutative unital ring R is a
topological space SpecR, the space of prime ideals in R, together with a sheaf of
rings OR, the structure sheaf, on it. Recall that, in the topology of SpecR (called
the Zariski topology) a closed set is the set V (I) of all prime ideals containing a
given ideal I. The sheaf OR is uniquely determined by the fact that, for every
f ∈ R, the ring of sections of OR over the open set Uf := SpecR − V (f) is the
localization R(f), that is, OR(Uf ) = R(f). In particular, OR(SpecR) = R. (Note:
the open sets Uf form a basis for the Zariski topology on SpecR.)
A scheme X consists of a pair (XZar,OX) where X is a topological space and OX
is a sheaf of rings on XZar. We require that X can be covered by open sets U such
that each (U,OU ) is isomorphic to an affine scheme. (Here, OU is the restriction of
OX to U .)
Remark 4.1. This definition is modeled on Example 3.1.1; in particular, read the
paragraph after the example.
10 BEHRANG NOOHI
A sheaf of modules over a scheme X is a sheaf F of abelian groups on XZar such
that, for every open U ⊆ XZar, F(U) is endowed with an OX(U)-module structure
(and restriction maps respect the module structure).
A sheaf of modules F over X is called quasi-coherent if for every inclusion of the
form SpecS = V ⊆ U = SpecR of open sets in XZar we have
F(V ) ∼= S ⊗R F(U).
Proposition 4.2. The category OX-Mod of OX-modules on a scheme X is an
abelian category. The full subcategory Quasi-CohX is also an abelian category.
Example 4.3. To an R-module M there is associated a quasi-coherent sheaf M̃ on
X = SpecR which is characterized by the property that M̃(Uf ) = M(f), for every
f ∈ R. In fact, every quasi-coherent sheaf on SpecR is of this form. More precisely,
we have an equivalence of categories ([Ha1], Corollary II.5.5)
Quasi-CohSpecR ∼= R-Mod.
The category Quasi-CohX is a natural abelian category associated with a
scheme X . This allows us to do homological algebra on schemes (e.g., sheaf coho-
mology). The following reconstruction theorem states that, indeed, Quasi-CohX
captures all the information about X .
Proposition 4.4 (Gabriel-Rosenberg Reconstruction Theorem [Ro]). A scheme X
can be reconstructed, up to isomorphism, from the abelian category Quasi-CohX .
More generally, Rosenberg [Ro], building on the work of Gabriel, associates to
an abelian category A a topological space Spec A together with a sheaf of rings
OA on it. In the case where A = Quasi-CohX , the pair (Spec A,OA) is naturally
isomorphic to (XZar,OX).
The above theorem is a starting point in non-commutative algebraic geometry.
It means that one can think of the abelian category Quasi-CohX itself as a space.
5. Morita equivalence of rings
References: [We], [GeMa].
We saw that, by the Gabriel-Rosenberg Reconstruction Theorem, we can regard
the abelian category Quasi-CohX as being “the same” as the scheme X itself. The
Gabriel-Rosenberg Reconstruction Theorem has a more classical precursor.
Theorem 5.1 (Gabriel [Ga, Fr]). Let R be a unital ring (not necessarily com-
mutative). Then R-Mod has a small projective generator (e.g., R itself), and is
closed under arbitrary coproducts. Conversely, let A be an abelian category with
a small projective generator P which is closed under arbitrary coproducts. Let
R = (EndP )op. Then A ∼= R-Mod.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 11
Remark 5.2. This, however, is not exactly a reconstruction theorem: the projective gen-
erator P is never unique, so we obtain various rings S = EndP such that A ∼= S-Mod.
Nevertheless, all such rings are regarded as giving the same “noncommutative scheme”.
So, from the point of view of noncommutative geometry they are the same. By the
Gabriel-Rosenberg Reconstruction Theorem, if such R and S are both commutative, then
they are necessarily isomorphic.
Two rings R and S are called Morita equivalent if R-Mod ∼= S-Mod.
Example 5.3. For any ring R, the ring S = Mn(R) of n × n matrices over R is
Morita equivalent to R. (Proof. In the above theorem take P = R⊕n, or use the
next theorem.)
The following characterization of Morita equivalence is important. The proof is
not hard.
Theorem 5.4. Let R and S be rings. Then the following are equivalent:
i. The categories R-Mod and S-Mod are equivalent.
ii. There is an S−R bimodule M such that the functor M⊗R− : R-Mod → S-
Mod is an equivalence of categories.
iii. There is a finitely generated projective generator P for R-Mod such that
S ∼= EndP .
6. Appendix: injective and projective objects in abelian categories
We will need to deal with injective and projective objects in the next lecture, so
we briefly recall their definition.
An object P in an abelian category is called projective if it has the following
lifting property:
Equivalently, P is projective if the functor HomA(P,−) : A → Ab takes exact se-
quences in A to exact sequences of abelian groups.
An object I in an abelian category is called injective if it has the following exten-
sion property:
12 BEHRANG NOOHI
Equivalently, I is injective if the functor HomA(−, I) : A
op → Ab takes exact se-
quences in A to exact sequences of abelian groups.
We say that A has enough projectives (respectively, enough injectives), if
for every object A there exists an epimorphism P → A where P is projective
(respectively, a monomorphism A→ I where I is injective).
In a category with enough projectives (respectively, enough injectives) one can
always find projective resolutions (respectively, injective resolutions) for objects.
The category R-Mod has enough injectives and enough projectives. The categories
PreSh(C,Ab), PreSh(C, R-Mod), Sh(C,Ab), Sh(C, R-Mod), and Quasi-CohX
have enough injectives, but in general they do not have enough projectives.
Exercise. Show that the abelian category of finite abelian groups has no injective
or projective object other than 0. Show that in the category of vector spaces every
object is both injective and projective.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 13
Lecture 2: Chain complexes
Overview. Various cohomology theories in mathematics are constructed from
chain complexes. However, taking the cohomology of a chain complex kills a lot
of information contained in that chain complex. So, it is desirable to elevate the
cohomological constructions to the chain complex level. In this lecture, we introduce
the necessary machinery for doing so. The main tool here is the mapping cone
construction, which should be thought of as a homotopy cokernel construction. We
discuss in some detail the basic properties of the cone construction; we will see, in
particular, how it allows us to construct long exact sequences on the chain complex
level, generalizing the usual cohomology long exact sequences.
In practice, one is only interested in chain complexes up to quasi-isomorphism.
This leads to the notion of the derived category of chain complexes. The cone
construction is well adapted to the derived category. Derived categories provide
the correct setting for manipulating chain complexes; for instance, they allow us to
construct derived functors on the chain complex level.
1. Why chain complexes?
Chain complexes arise naturally in many areas of mathematics. There are two
main sources for (co)chain complexes: chains on spaces and resolutions. We give
an example for each.
Chains on spaces. Let X be a topological space. There are various ways to
associate a chain complex to X . For example, singular chains, cellular chains (if
X is a CW complex), simplicial chains (if X is triangulated), de Rham complex (if
X is differentiable), and so on. These complexes encode a great deal of topological
information about X in terms of algebra (e.g., homology and cohomology).
Observe the following two facts:
• Such constructions usually give rise to a chain complex of free modules.
• Various chain complexes associated to a given space are chain homotopy
equivalent (hence, give rise to the same homology/cohomology).
Resolutions. Let us explain this with an example. Let R be a ring, and M and N
R-modules. Recall how we compute Tori(M,N). First, we choose a free resolution
· · · → P−2 → P−1 → P 0
︸ ︷︷ ︸
Then, we define
Tori(M,N) = H
−i(P • ⊗N).
Observe the following two facts:
14 BEHRANG NOOHI
• The complex P • is a complex of free modules.
• Any two resolutions P • and Q• of M are chain homotopy equivalent (hence
Tor is well-defined).
Conclusion. In both examples above, we replaced our object with a chain complex
of free modules that was unique up to chain homotopy. We could then extract infor-
mation about our object by doing algebraic manipulations (e.g., taking homology)
on this complex. Therefore, the real object of interest is the (chain homotopy class
of) a chain complex (of, say, free modules). Of course, instead of working with the
complex itself, one could choose to work with its (co)homology, but one loses some
information this way.
Remark 1.1. Complexes of projective modules (e.g, projective resolutions) work
equally well as complexes of free modules. Sometimes, we are in a dual situation
where complexes of injective modules are more appropriate (e.g., in computing Ext
groups).
2. Chain complexes
References: [We], [GeMa], [Ha2], [HiSt], [KaSch], [Iv], and any book on algebraic topology.
We quickly recall a few definitions. We prefer to work with cohomological index-
ing, so we work with cochain complexes.
A cochain complex C in an abelian category A is a sequence of objects in A
· · · −→ Cn−1
−→ Cn
−→ Cn−1 −→ · · · , d2 = 0.
Such a sequence is in general indexed by Z, but in many applications one works
with complexes that are bounded below, bounded above, or bounded on both sides.
A chain map f : B → C between two cochain complexes is a sequence of fn : Bn →
Cn of morphisms such that the following diagram commutes
· · ·
· · ·
· · ·
· · ·
A null homotopy for a chain map f : B → C is a sequence sn : Bn → Cn−1 such
fn = sn+1 ◦ dn + dn−1 ◦ sn, ∀n
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 15
We say two chain maps f, g : B → C are chain homotopic if f − g is null ho-
motopic. We say f : B → C is a chain homotopy equivalence if there exists
g : C → B such that f ◦ g and g ◦ f are chain homotopy equivalent to the corre-
sponding identity maps.
To any chain complex C one can associate the following objects in A:
Zn(C) = ker(Cn
−→ Cn+1)
Bn(C) = im(Cn−1
−→ Cn)
Hn(C) = Zn(C)/Bn(C)
Exercise. Show that a chain map f : B → C induces morphisms on each of the above
objects. In particular, we have induced morphisms Hn(f) : Hn(B) → Hn(C) for
all n. Show that if f and g are chain homotopic, then they induce the same map
on cohomology. In particular, a chain homotopy equivalence induces isomorphisms
on cohomologies.
A chain map f : B → C is called a quasi-isomorphism if it induces isomorphisms
on all cohomologies. (So every chain homotopy equivalence is a quasi-isomorphism.)
Exercise. When is a chain complex C quasi-isomorphic to the zero complex? (Hint:
exact.) When is C chain homotopy equivalent to the zero complex? (Hint: split
exact.)
A short exact sequence of chain complexes is a sequence
0 → A→ B → C → 0
of chain maps which is exact at every n ∈ Z. Such an exact sequence gives rise to
a long exact sequence in cohomology
· · ·
(A) → H
(B) → H
(A) → H
(B) → H
→ · · ·
A short exact sequence as above is called pointwise split (or semi-split) if every
epimorphism Bn → Cn admits a section, or equivalently, every monomorphism
An → Bn is a direct summand.
Exercise. Does pointwise split imply that the maps ∂ in the above long exact
sequence are zero? Is the converse true? (Answer: both implications are false.)
3. Constructions on chain complexes
References: [We], [GeMa], [Ha2], [HiSt], [KaSch], [Iv].
In the following table we list a few constructions that are of great importance in
algebraic topology. The left column presents the topological construction, and the
right column is the cochain counterpart.
The right column is obtained from the left column as follows: imagine that the
spaces are triangulated and translate the topological constructions in terms of the
simplices; this gives the chain complex picture. The cochain complex picture is ob-
tained by the appropriate change in indexing (from homological to cohomological).
16 BEHRANG NOOHI
The reader is strongly encouraged to do this as an exercise. It is extremely
important to pay attention to the orientation of the simplices, and keep track of the
signs accordingly.
Spaces Cochains
f : X → Y continuous map f : B → C cochain map
Cyl(f) = (X × I) ∪f YX Y Mapping cylinderCyl(f)n := Bn ⊕ Bn+1 ⊕ CnBn ⊕ Bn+1 ⊕ Cn d−→ Bn+1 ⊕ Bn+2 ⊕ Cn+1(b′, b, c) 7→ `dB(b′) − b,−dB(b), f(b) + dC(c)´
dB − idB 0
0 −dB 0
0 f dC
Cone(f) = Cyl(f)/XX Y Mapping coneCone(f) = Cyl(f)/B
Cone(f)n := Bn+1 ⊕ Cn
Bn+1 ⊕ Cn
−→ Bn+2 ⊕ Cn+1
(b, c) 7→
− dB(b), f(b) + dC(c)
Cone(X) = Cone(idX) = X × I/X × {0}X ConeCone(B) := Cone(idB)Cone(B)n = Bn+1 ⊕ Bn
Bn+1 ⊕ Bn
−→ Bn+2 ⊕ Bn+1
(b, b′) 7→
− d(b), b + d(b′)
Σ(X) = Cone(X)/X = X × I/X × {0, 1}X Shift (Suspension)B[1] := Cone(B)/BB[1]n := Bn+1
B[1]n
−→ B[1]n+1
b 7→ −d(b)
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 17
Some history. These constructions were originally used by topologists (e.g., Puppe [Pu])
as a unified way of treating various cohomology theories for topological spaces (the so-
called generalized cohomology theories [Ad]), and soon they became a standard tool in
stable homotopy theory. Around the same time, Grothendieck and Verdier developed the
same machinery for cochain complexes, to be used in Grothendieck’s formulation of duality
theory and the Riemann-Roch theorem for schemes [Ha2]. This led to the invention of
derived categories and triangulated categories [Ve]. As advocated by Grothendieck, it has
become more and more apparent over time that working in derived categories is a better
alternative to working with cohomology. Nowadays people consider even richer structures
such as dg-categories [BoKa, Ke3], A∞-categories [Ke4], (stable) model categories ([Ho],
Section 7), etc. The idea is that, the higher one goes in this hierarchy, the more higher-
order cohomological information (e.g., Massey products, Steenrod operations, or other
cohomology operations) is retained. In these lectures we will not go beyond derived
categories.
An important fact. Taking the mapping cone of f : X → Y is somewhat like
taking cokernels: we are essentially killing the image of f , but softly (i.e., we make
it contractible). That is why Cone(f) is sometimes called the homotopy cofiber or
homotopy cokernel of f ; see [May1], §8.4, especially, the first lemma on page 58.
Proposition 3.1. In the following diagram of cochain complexes, if g and h are
chain equivalences, then so is the induced map on the mapping cones
Cone(f)
C′ Cone(f
A similar statement is true for topological spaces.
Remark 3.2. The above proposition is still true if we use quasi-isomorphisms instead
of chain homotopy equivalences. Similarly, the topological version remains valid if
we use weak equivalences (i.e., maps that induce isomorphisms on all homotopy
groups) instead of homotopy equivalences.
Exercise. Give an example (in both the topological and chain complex settings)
to show that the above proposition is not true if we use the strict cofiber Y/f(X)
instead of Cone(f). (Therefore, since we are interested in chain homotopy equiv-
alence classes, or quasi-isomorphism classes, of cochain complexes, Cone(f) is a
better-behaved notion than Y/f(X).)
Exercise. Formulate a universal property for the mapping cone construction.
4. Basic properties of cofiber sequences
References: [We], [GeMa], [KaSch], [Ha2], [Ve].
The sequence
−→ C −→ Cone(f)
18 BEHRANG NOOHI
(or any sequence quasi-isomorphic to such a sequence) is called a cofiber sequence.
The same definition can be made with topological spaces. We list the basic prop-
erties of cofiber sequences.
1. Exact sequence, basic form. An important property of cofiber sequences is
that they give rise to long exact sequences. Here is a baby version which is very
easy to prove and is left as an exercise.
Proposition 4.1. Let B → C → Cone(f) be a cofiber sequence. Then the sequence
Hn(B) → Hn(C) → Hn(Cone(f))
is exact for every n.
Remark 4.2. This is of course part of a long exact sequence
· · ·
→ Hn(B) → Hn(C) → Hn(Cone(f))
→ Hn+1(B) → Hn+1(C) → Hn+1(Cone(f))
→ · · ·
that we will discuss shortly. If we use the fact that a short exact sequence of
cochain complexes gives rise to a long exact sequence of cohomology groups, this
is easily proven by showing that the cofiber sequence B → C → Cone(f) is chain
homotopy equivalent to the short exact sequence B → Cyl(f) → Cone(f). The
equivalence C ∼ Cyl(f) is given by
C −→ Cyl(f)
c 7→ (0, 0, c)
Cyl(f) −→ C
Bn ⊕Bn+1 ⊕ Cn −→ Cn
(b′, b, c) 7→ f(b′) + c.
Remark 4.3. We have a similar statement for a cofiber sequence X → Y → Cone(f) of
topological spaces. The corresponding long exact sequence in (co)homology should then be
interpreted as the long exact sequence of relative (co)homology groups. More specifically,
if f is an inclusion, we have
H∗(Cone(f)) ∼= H∗(Y, X) and H
(Cone(f)) ∼= H
(Y, X),
where the right hand sides are relative (co)homology groups.
2. A short exact sequence is a cofiber sequence. We pointed out in Re-
mark 4.2 that a cofiber sequence is chain equivalent (hence quasi-isomorphic) to a
pointwise split short exact sequence. The converse is also true.
Proposition 4.4. Consider a short exact sequence of cochain complexes
0 → B
→ C → C/B → 0.
Then the natural chain map ϕ : Cone(f) → C/B defined by
Bn+1 ⊕ Cn → Cn/Bn
(b, c) 7→ c̄
is a quasi-isomorphism. If the sequence is pointwise split, with s : Cn/Bn → Cn a
choice of splitting, then ϕ is a chain equivalence. The inverse is given by ψ : C/B →
Cone(f)
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 19
Cn/Bn → Bn+1 ⊕ Cn
sd(x) − ds(x), s(x)
Exercise. Let f : B →֒ C be an inclusion ofR-modules, viewed as cochain complexes
concentrated in degree 0. Show that ϕ : Cone(f) → C/B defined above is a chain
equivalence if and only if f is split.
3. Iterate of a cofiber sequence. Consider a cofiber sequence
−→ Cone(f).
Observe that g : C → Cone(f) is a pointwise split inclusion. This implies that
the mapping cone of g is naturally chain equivalent to Cone(f)/C = B[1]. More
precisely, we have the following commutative diagram:
Cone(f)
Cone(f)
Cone(g).
ch. eqϕ
Note that we have natural chain equivalences
Cone(∂) ∼ Cone(h) ∼ coker(h) = C[1].
Indeed, a careful chasing through the above chain equivalences (for which we have
given explicit formulas) shows that the sequence
Cone(f)
−→ B[1]
−f [1]
−→ C[1]
is a cofiber sequence. We can now iterate this process forever and produce a long
sequence
· · ·
∂[−1]
−→ Cone(f)
−→ B[1]
−f [1]
−→ C[1]
−g[1]
−→ Cone(f)[1]
−∂[1]
−→ · · · (⋆)
in which every three consecutive terms form a cofiber sequence.
Remark 4.5. A similar long exact sequence can be constructed for a map f : X → Y of
topological spaces, and it is called a Puppe sequence (or cofiber sequence [May1] §8.4).
A Puppe sequence, however, only extends to the right. The reason for this is that the
suspension functor on the category of topological spaces is not invertible (as opposed to
the shift functor for cochain complexes). The Puppe sequence can be used, among other
things, to give a natural construction of the long exact (co)homology sequence of a pair.
4. Long exact sequence of a cofiber sequence. Applying Proposition 4.1 to
the above long exact sequence of chain complexes, and observing that Hi(B[n]) =
Hi+n(B), we obtain a long exact sequence of cohomology groups
· · ·
→ Hn(B) → Hn(C) → Hn(Cone(f))
→ Hn+1(B) → Hn+1(C) → Hn+1(Cone(f))
→ · · · .
20 BEHRANG NOOHI
Moral. By exploiting the notion of cone of a map in a systematic way, we can
elevate many basic cohomological constructions to the level of chain complexes.
For this to work conveniently, one needs to invert chain equivalences so that one
can treat such morphisms as isomorphisms. Indeed, for various reasons, inverting
only chain equivalences is usually not enough. For example, recall that one of our
motivations for working with chain complexes was that we wanted to be able to
replace an objectM ∈ A with a better behaved resolution (projective or injective) of
it. Since the resolution map is only a quasi-isomorphism and not a chain equivalence
in general, it is only after inverting quasi-isomorphisms that we can regard an
M ∈ A and its resolution as the “same”.
5. Derived categories
References: [We], [GeMa], [Ha2], [KaSch], [Iv], [Ve], [Ke1], [Ke2].
Let A be an abelian category.
The homotopy category K(A) is the category obtained by inverting all chain
equivalences in the category Ch(A) of chain complexes. More precisely, there is a
natural functor Ch(A) → K(A) which has the following universal property:
If a functor F : Ch(A) → C sends chain
equivalences to isomorphisms, then
Ch(A)
If in the above definition we replace ‘chain equivalence’ by ‘quasi-isomorphism’, we
arrive at the definition of the derived category D(A). We list the basic properties
of K(A) and D(A).
1. Explicit construction of K(A). We can, alternatively, define the homotopy
category K(A) to be the category whose objects are the ones of Ch(A) and whose
morphisms are defined by
HomK(A)(B,C) := HomCh(A)(B,C)/N
where N stands for the group of null homotopic maps. (Exercise. Show that this
category satisfies the required universal property.) This, in particular, implies that
K(A) is an additive category.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 21
2. Cohomology functors. The cohomology functors Hn : Ch(A) → A factor
through K(A) and D(A):
Ch(A)
3. Explicit construction of D(A). The objects of D(A) can again be taken to
be the ones of Ch(A). Description of morphisms in D(A) is, however, slightly more
involved than the case of K(A). It requires the notion of calculus of fractions in a
category, which we will not get into for the sake of brevity.
Just to give an idea how this works, first we observe that D(A) can, equiva-
lently, be defined by inverting quasi-isomorphisms in K(A). The class of quasi-
isomorphisms in K(A) satisfies the axioms of the calculus of fractions ([GeMa],
Definition III.2.6), and this allows one to give an explicit description of morphisms
in D(A) (see [GeMa], Section III.2, [Ha2], Section I.3, or [We], Section 10.3). A
consequence of this is that, for every morphism f : B → C in D(A), one can find
morphisms s, t, g and h in K(A) such that s and t are quasi-isomorphisms and the
following diagrams commute in D(A):
Also, for any two morphisms f, f ′ : B → C, now in K(A), which become equal in
D(A), there are s and t as above such that f ◦ s = f ′ ◦ s and t ◦ f = t ◦ f ′.
4. Cofiber sequences. There is a notion of cofiber sequence in both K(A) and
D(A). Every morphism f : B → C fits in a sequence ⋆ as in page 19 in which every
three consecutive terms form a cofiber sequence.
6. Variations on the theme of derived categories
References: [We], [GeMa], [Ha2], [KaSch], [Iv], [Ke1], [Ke2].
To an abelian category A we can associate various types of homotopy or derived
categories by imposing certain boundedness conditions on the chain complexes in
question.
The bounded below derived category D+(A) is the category obtained from in-
verting the quasi-isomorphisms in the category of bounded below chain complexes
22 BEHRANG NOOHI
Ch+(A). Equivalently, D+(A) is obtained from inverting the quasi-isomorphisms
in the homotopy category K+(A) of Ch+(A).
In the same way one can define bounded derived categories D−(A) and Db(A),
where b stands for bounded on two sides.
Remark 6.1. One can use the calculus of fractions in K+(A) (respectively, K−(A),
Kb(A)) to give a description of morphisms in D+(A) (respectively, D−(A), Db(A)).
It follows that each of these bounded derived categories can be identified with a full
subcategory of D(A). In particular, Db(A) = D+(A) ∩ D−(A). We will say more
on this in the next lecture.
There is an alternative way of computing morphisms in bounded derived cate-
gories using injective or projective resolutions. It is summarized in the following
theorems.
Theorem 6.2 ([GeMa], § III.5.21, [We], Theorem 10.4.8). Let I+(A) ⊂ K+(A) be
the full subcategory consisting of complexes whose terms are all injective. Then the
composition
I+(A) K+(A) D+(A)
is fully faithful. It is an equivalence if A has enough injectives. The same thing is
true if we replace + by − and injective by projective:
P−(A) K−(A) D−(A) .
The (first part of the) above theorem says that, if I and J are bounded below
complexes of injective objects, then we have
HomD(A)(J, I) = HomK(A)(J, I).
(This is actually true if J is an arbitrary bounded below complex; see [We], Corollary
10.4.7). In particular, if I and J are quasi-isomorphic, then they are chain equiva-
lent. This is good news, because we have seen that computing Hom is much easier
in K(A). Now if A has enough injectives, it can be shown that for every bounded
C, there exists a bounded below complex of injectives I, and a quasi-isomorphism
C ∼−→ I. This is called an injective resolution for C. So, by virtue of the above
fact, injective resolutions can be used to compute morphisms in derived categories:
HomD(A)(B,C) = HomK(A)(B, I).
Exercise. Set Homi
D(A)(B,C) := Hom(B,C[i]). Let B and C be objects in A,
viewed as complexes concentrated in degree 0. Assume A has enough injectives.
Show that
D(A)(B,C)
∼= Ext
i(B,C).
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 23
Use the second part of the theorem to give a way of computing Ext groups using a
projective resolution for C, if they exist. Compute Homi
K(A)(B,C) and compare it
with Homi
D(A)(B,C).
Exercise. Give an example of a non-zero morphism in D(A), with A your favorite
abelian category, which induces zero maps on all cohomologies.
7. Derived functors
References: [We], [GeMa], [Ha2], [KaSch], [Iv], [Ke1], [Ke2].
To keep the lecture short, we will skip the very important topic of derived func-
tors and confine ourselves to an example: the derived tensor
⊗. The reader is
encouraged to consult the given references for the general discussion of derived
functors, especially the all important derived hom RHom.
Let A = R-Mod. The idea is that we want to have a notion of tensor product
for chain complexes which is well defined on D(A).
The usual tensor product of chain complexes (Definition [We], §2.7.1) does not pass
to derived categories. This is because tensor product is not exact. More precisely,
if A → A′ is a quasi-isomorphism, then A ⊗ B → A′ ⊗ B may no longer be. (For
a counterexample, let A be a short exact sequence, A′ = 0, and B a complex
concentrated in degree 0.)
The usual tensor product of chain complexes DOES pass to homotopy categories.
This is because a null homotopy s for a chain map A → A′ gives rise to a null
homotopy s⊗B for A⊗B → A′ ⊗B (exercise).
In particular, taking all complexes to be (bounded-above) complexes of projective
modules, we get a well-defined tensor product − ⊗ − : P−(A) × P−(A) → P−(A).
Since R-Mod has enough projectives, we obtain, via the equivalence P−(A) ∼=
D−(A) of Theorem 6.2, the desired tensor product on the bounded above derived
category:
⊗− : D−(A) × D−(A) → D−(A).
More explicitly, A
⊗B is defined to be P ⊗Q, where P → A and Q→ B are certain
chosen projective resolutions. (In fact, it is enough to resolve only one of A or B.)
Exercise. Show that if M,N ∈ A are viewed as complexes concentrated in degree
0, then
H−i(M
⊗N) ∼= Tori(M,N).
24 BEHRANG NOOHI
Lecture 3: Triangulated categories
Overview. The main properties of the cone construction for chain complexes
can be formalized into a set of axioms. This leads to the notion of a triangulated
category, the main topic of this lecture. A triangulated category is an additive
category in which there is an abstract notion of mapping cone. The cohomology
functors on chain complexes can also be studied at an abstract level in a triangulated
category.
The main example of a triangulated category is the derived category D(A) of
chain complexes in an abelian category A. Using a t-structure on a triangulated
category, we can produce an abelian category, called the heart of the t-structure.
For example, there is a standard t-structure on D(A) whose heart is A. The t-
structure is not unique, and by varying it we can produce new abelian categories.
We show how using a torsion theory on an abelian category A we can produce a new
t-structure on D(A). The heart of this t-structure is then a new abelian category B.
For example, this method has been used in noncommutative geometry to “deform”
the abelian category A of coherent sheaves on a torus X . The new abelian category
B can then be thought of as the category of coherent sheaves on a “noncommutative
deformation” of X , a noncommutative torus.
1. Triangulated categories
References: [We], [GeMa], [Ha2], [Ne], [BeBeDe], [KaSch], [Iv], [Ve], [Ke1], [Ke2].
We formalize the main properties of the mapping cone construction and define
triangulated categories.
Let T be an additive category equipped with an auto-equivalence X 7→ X [1] called
shift (or translation).
By a triangle in T we mean a sequence X → Y → Z → X [1]. We sometimes write
this as
A triangulation on T is a collection of triangles, called exact (or distinguished)
triangles, satisfying the following axioms:
TR1. a) X
−→ X −→ 0 −→ X [1] is an exact triangle.
b) Any morphism f : X → Y is part of an exact triangle.
c) Any triangle isomorphic to an exact triangle is exact.
TR2. X
→ X [1] exact ⇔ Y
→ X [1]
−f [1]
→ Y [1] exact.
TR3. In the diagram
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 25
X Y Z X [1]
X ′[1]
if the horizontal rows are exact, and the left square commutes, then the
dotted arrow can be filled (not necessarily uniquely) to make the diagram
commute.
TR4 (short imprecise version). Given f : X → Y and g : Y → Z, we have the
following commutative diagram in which every pair of same color arrows is
part of an exact triangle
U V W
Idea: “(Z/X)/(Y/X) = Z/Y )”. See Appendix II for a more precise state-
ment.
A triangulated category is an additive category T equipped with an auto-
equivalence X 7→ X [1] and a triangulation.
A triangle functor (or a exact functor)1 is a functor between triangulated cate-
gories which commutes with the translation functors (up to natural transformation)
and sends exact triangles to exact triangles.
Remark 1.1. A triangulated category can be thought of as a category in which
there are well-behaved notions of homotopy kernel and homotopy cokernel2 (hence
also various other types of homotopy limits and colimits).
Some immediate consequences. Let T be a triangulated category, and let
X → Y → Z → X [1] be an exact triangle in T. Then, the following are true:
0. The opposite category Top is naturally triangulated.
1. Any length three portion of the sequence
· · · −→ Z[−1]
−h[−1]
−→ X[1]
−f [1]
−→ Y [1]
−g[1]
−→ Z[1]
−h[1]
−→ · · ·
1This is not a very good terminology, because there is also a notion of t-exact functor with
respect to a t-structure.
2Indeed not quite well-behaved (read, functorial), essentially due to the fact that the morphism
whose existence is required in TR3 may not be unique. This turns out to be problematic. A way
to remedy this is to work with DG categories; see [BoKa].
26 BEHRANG NOOHI
is an exact triangle.
2. In the above exact sequence, any two consecutive morphisms compose to
zero. Proof. Enough to check g ◦ f = 0. Apply TR1.a and TR3 to
X [1]
X [1]
3. Let T be any object. Then,
Hom(T,X) → Hom(T, Y ) → Hom(T, Z)
is exact. Therefore, the long sequence of (1) gives rise to a long exact
sequence of abelian groups. The same thing is true for
Hom(X,T ) → Hom(Y, T ) → Hom(Z, T ).
Remark 1.2. In fact, TR3 and half of TR2 follow from the rest of the axioms. For this,
see [May2]. Also see [Ne] for another formulation of TR4; especially, Remark 1.3.15 and
Proposition 1.4.6.
The main example. The proof of the following theorem can be found in [GeMa]
or [KaSch]. (We have sketched some of the ideas in Lecture 2.)
Theorem 1.3. Let A be an abelian category. Then K(A) and D(A) are triangulated
categories.
In both cases, the distinguished triangles are the ones obtained from cofiber
sequences. Equivalently, the distinguished triangles are the ones obtained from
the pointwise split short exact sequences. (To see the equivalence, note that one
implication follows from Proposition 4.4 of Lecture 2. The other follows from the
fact that the cofiber sequence B → C → Cone(f) is chain homotopy equivalent to
the pointwise split short exact sequence B → Cyl(f) → Cone(f); see Remark 4.2
of Lecture 2.) In the case of D(A) we get the same triangulation if we take all short
exact sequences. This also follows from Proposition 4.4 of Lecture 2.
The following proposition allows us to produce more triangulated categories from
the above basic ones.
Proposition 1.4. Let A be an abelian category, and let C be a full additive sub-
category of Ch(A). Let K ⊂ K(A) be the corresponding quotient category and D
the localization of K with respect to quasi-isomorphisms. Assume C is closed under
translation, quasi-isomorphisms, and forming mapping cones. Then K and D are
triangulated categories and we have fully faithful triangle functors
K → K(A)
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 27
D → D(A).
Proof. The proof in the case of K is straightforward. The case of D → D(A)
follows easily from the existence of calculus of fractions on K(A). (Our discussion
of calculus of fractions in Lecture 2 is enough for this.) �
Exercise. Show that D−(A) can, equivalently, be defined by inverting the quasi-
isomorphisms in the category C of chain complexes with bounded above cohomology.
The similar statement is true in the case of D+(A) and Db(A). (Hint: construct a
right inverse to the inclusion ι : Ch
(A) →֒ C by choosing an appropriate truncation
of each complex in C; see page 29 to learn how to truncate. Show that this becomes
an actual inverse to ι when we pass to derived categories.)
Corollary 1.5. We have the following fully faithful triangle functors:
−(A),K+(A),Kb(A) →֒ K(A),
−(A),D+(A),Db(A) →֒ D(A).
Proof. This is an immediate corollary of the previous proposition. (We also need
to use the previous exercise.) �
Important examples to keep in mind:
1. A = R-Mod.
2. A = sheaves of R-modules over a topological space.
3. A = presheaves of R-modules over a topological space.
4. A = sheaves of OX -modules on a scheme X .
5. A = quasi-coherent sheaves on a scheme X .
2. Cohomological functors
References: [We], [GeMa], [BeBeDe] [Ha2], [KaSch], [Ve].
Let T be a triangulated category and A an abelian category. An additive functor
H : T → A is called a cohomological functor if for every exact triangle X →
Y → Z → X [1],
H(X) → H(Y ) → H(Z) is exact in A.
If we set Hn(X) := H(X [n]), we obtain the following long exact sequence:
· · · −→ Hn−1(Z) → Hn(X) → Hn(Y ) → Hn(Z) → Hn+1(X) → Hn+1(Y ) → Hn+1(Z) → · · ·
Example 2.1.
28 BEHRANG NOOHI
1. Let T be any of K∗(A) or D∗(A), ∗ = ∅,−,+, b. Then the functorH : T → A,
X 7→ H0(X) is a cohomological functor.
2. For any T , Hom(T,−) : T → A is a cohomological functor. Similarly,
Hom(−, T ) : T → Aop is a cohomological functor.
Exercise. Show that (1) is a special case of (2).
3. Abelian categories inside triangulated categories; t-structures
References: [GeMa], [KaSch], [BeBeDe].
The triangulated categories associated to an abelian category A contain A as a
full subcategory. More precisely,
Proposition 3.1. Let T be any of K∗(A) or D∗(A), ∗ = ∅,−,+, b. Then the
functor A → T defined by
A 7→ · · · → 0 → A→ 0 → · · ·
(A is sitting in degree zero) is fully faithful.
Observe that this is not the only abelian subcategory of T. For example, we have
Z many copies of A in T (simply take shifts of the above embedding). We may also
have abelian categories in T that are not isomorphic to A.
As we saw above, the triangulated structure of T is not sufficient to recover
the abelian subcategory A. What extra structure do we need on T in order to
reconstruct A?
A t-structure on a triangulated category T is a pair (T≤0,T≥0) of saturated (i.e.,
closed under isomorphism) full subcategories such that:
t1. If X ∈ T≤0, Y ∈ T≥1, then Hom(X,Y ) = 0.
t2. T≤0 ⊆ T≤1, and T≥1 ⊆ T≥0.
t3. For every X ∈ T, there is an exact triangle
A→ X → B → A[1]
such that A ∈ T≤0 and B ∈ T≥1.
Notation: T≤n = T≤0[−n] and T≥n = T≥0[−n].
Main example. Let T = D(A). Set
≤0 = {X ∈ D(A) | Hi(X) = 0, ∀i > 0},
≥0 = {X ∈ D(A) | Hi(X) = 0, ∀i < 0}.
The proof that this is a t-structure is not hard. The axioms (t1) and (t2) are
straightforward. To verify (t3), we define truncation functors
τ≤0 : T → T
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 29
τ≥1 : T → T
τ≤0(X) := · · · → X
−2 → X−1 → ker d→ 0 → 0 → · · ·
τ≥1(X) := X/τ≤0(X) = · · · → 0 → 0 → X0/ kerd→ X
1 → X2 → · · · .
It is easy to see that
τ≤0X → X → τ≥1X → τ≤0X [1]
is exact (Lecture 2, Proposition 4.4). So, we can take A = τ≤0X and B = τ≥1X .
Remark 3.2. This t-structure induces t-structures on each of D−,+,b(A).
Exercise. Show that this does not give a t-structure on K(A). (Hint: show that
(t1) fails by taking X = Y to be an exact complex that is not contractible, e.g., a
non-split short exact sequence.)
The truncation functors discussed above can indeed be defined for any t-structure.
Proposition 3.3 ([BeBeDe], §1.3). Let T be a triangulated category equipped with
a t-structure (T≤0,T≥0).
i. The inclusion T≤n →֒ T admits a right adjoint τ≤n : T → T
≤n, and the
inclusion T≥n →֒ T admits a left adjoint τ≥n : T → T
ii. For every X ∈ T and every n ∈ Z, there is a unique d that makes the
triangle
τ≤nX −→ X −→ τ≥n+1X
−→ τ≤nX [1]
exact.
Remark 3.4. An aisle in a triangulated category is a full saturated subcategory U that
is closed under extensions, U[1] ⊆ U, and such that U → T admits a right adjoint. For
example, U = T≤0 is an aisle. Conversely, any aisle U is equal to T≤0 for a unique
t-structure (T≤0, T≥0); see [KeVo].
Proposition 3.5 ([BeBeDe], §1.3). Let a ≤ b. Then τ≥a and τ≤b commute, in the
sense that
τ≤bX X τ≥aX
τ≥aτ≤bX
τ≤bτ≥aX
is commutative. Furthermore, ϕ is necessarily an isomorphism.
We set τ[a,b] := τ≥aτ≤bX . We call T
≤0∩T≥0 the heart (or core) of the t-structure.
Theorem 3.6 ([BeBeDe], §1.3). The heart C := T≤0 ∩ T≥0 is an abelian category.
The functor H0 := τ[0,0] : T → C is a cohomological functor.
Example 3.7. The heart of the standard t-structure on D(A) is A, and the functor
τ[0,0] is the usual H
30 BEHRANG NOOHI
Remark 3.8. In the above theorem, T may not be equivalent to any of D∅,−,+,b(C).
The moral of the story is that, by varying t-structures inside a triangulated
category (e.g., D(A)), we can produce new abelian categories.
4. Producing new abelian categories
Reference: [HaReSm].
One can use t-structures to produce new abelian categories out of given ones.
This technique appeared for the first time in [BeBeDe] where they alter the t-
structures on various derived categories of sheaves on a space to produce new abelian
categories of perverse sheaves (see loc. cit. §2 and also Theorem 1.4.10).
Another way to produce t-structures is via torsion theories. For an application
of this in noncommutative geometry see [Po].
A torsion theory on an abelian category A is a pair (T,F) of full additive subcat-
egories of A such that:
• For every T ∈ T and F ∈ F, we have Hom(T, F ) = 0.
• For every A ∈ A, there is a (necessarily unique) exact sequence
0 → T → A→ F → 0, T ∈ T, F ∈ F.
Example 4.1. Let A = Ab be abelian categories. Take T to be the torsion abelian
groups and F the torsion-free abelian groups.
Exercise. Prove that for a torsion theory (T,F) we have T⊥ = F and ⊥F = T, that
F = {F ∈ A | ∀T ∈ T, Hom(T, F ) = 0},
T = {T ∈ A | ∀F ∈ F, Hom(T, F ) = 0}.
Out of a given torsion theory (T,F) on an abelian category A we can construct
a new abelian category B in which the roles of T and F are interchanged.
We construct B as follows. Consider the following t-structure on D = D(A):
≤0 = {B ∈ D≤0 | H0(B) ∈ T},
≥0 = {B ∈ D≥−1 | H−1(B) ∈ F}.
This is easily seen to be a t-structure, so, by Theorem 3.6, B := D≤0 ∩ D≥0 is an
abelian category. More precisely, B is equivalent to the full subcategory of D(A)
consisting of complexes d : A−1 → A0 such that ker d ∈ F and cokerd ∈ T.
Remark 4.2. Let T′ and F′ be essential images of F[1] and T in B. Then (T′,F′) is
a torsion theory for B. (However, if we repeat the same process as above for this
torsion theory, we may not get back A.)
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 31
5. Appendix I: topological triangulated categories
References: ([Ho], Section 7), [Ma], ([We], Section 10.9).
In our examples, we considered only triangulated categories that come from
algebra (e.g, derived categories). There are certain triangulated categories that
arise from topology (e.g, stable homotopy categories) that we have not discussed in
these notes but which are worthy of attention. In line with our topological-vs-chain-
complex comparison picture of Lecture 2, we say a few words about the homotopy
category of spectra.
Spectra. Recall from Lecture 2 that the cone construction (which is the main input
in the construction of derived categories) was motivated by a topological construc-
tion on spaces. One may wonder then whether one could repeat the construction
of the derived categories in the topological setting. The problem that arises here is
that the suspension functor X 7→ SX is not an auto-equivalence of the category of
topological spaces (while its chain complex counterpart, the shift functor C 7→ C[1],
is). To remedy this, one can formally “invert” the suspension functor. This gives
rise to the category of Spanier-Whitehead spectra ([Ma], Chapter 1) from which we
can construct a triangulated category by inverting weak equivalences, in the same
way we obtained the derived category by inverting quasi-isomorphisms.
What made topologists unhappy about the category of Spanier-Whitehead spec-
tra is that it is not closed under arbitrary colimits. 3 Nowadays there are bet-
ter categories of spectra that do not have this deficiency. They go under various
names such as spectra, Ω-spectra, symmetric spectra, etc. All these categories come
equipped with a notion of weak equivalence, and inverting the weak equivalences in
any of these categories gives rise to the same triangulated category, the homotopy
category of spectra. Understanding the homotopy category of spectra is an area
of active research that is called stable homotopy theory. To see how triangulated
categories emerge in this context consult the given references.
6. Appendix II: different illustrations of TR4
One way of illustrating TR4 is via the following “braid” diagram:
Z W U [1]
. . . Y
V Y [1] . . .
W [−1] U X [1] Z[1]
The axiom TR4 can be stated as saying that, given the three solid strands
of braids, the dotted strand can be filled so that the diagram commutes. It is
understood that each of the four long sequences is exact (i.e., obtained from a
distinguished triangle). Note that we are requiring the existence of only three
3Actually neither are the bounded derived categories!
32 BEHRANG NOOHI
consecutive dotted arrows; the rest of the sequence is uniquely determined by these
three.
Another depiction of the axiom TR4 is via the following octahedron:
The axiom requires that the dotted arrows could be filled. It is understood that
the four mixed-color triangles are commutative, and the four unicolor triangles are
exact.
References
[Ad] J. F. Adams, Stable homotopy and generalised homology, Chicago Lectures in Mathematics.
University of Chicago Press, Chicago, IL, 1995.
[BeBeDe] A. A. Beilinson, J. Bernstein, P. Deligne, Faisceaux pervers, Analysis and topology on
singular spaces, I (Luminy, 1981), 5–171, Astérisque, 100, Soc. Math. France, Paris, 1982
[BoKa] A. I. Bondal, M. M. Kapranov, Framed triangulated categories, (Russian) Mat. Sb. 181
(1990), no. 5, 669–683; translation in Math. USSR-Sb. 70 (1991), no. 1, 93–107.
[Fr] P. Freyd, Abelian categories, Harper, 1966.
[Ga] P. Gabriel, Des catégories abéliennes, Bull. Soc. Math. France 90 (1962), 323–448.
[GeMa] S. I. Gelfand, Y. I. Manin, Methods of homological algebra, Springer-Verlag, Berlin, 1996.
[Gr] A. Grothendieck, Sur quelques points d’algébre homolgiques, Tôhoku Math. J. 9 (1957),
119–221.
[HaReSm] D. Happel, I. Reiten, S. O. Smalø, Tilting in abelian categories and quasitilted algebras,
Mem. Amer. Math. Soc. 575.
[Ha1] R. Hartshorne, Algebraic geometry, Graduate Texts in Mathematics 52, Springer-Verlag,
New York–Heidelberg, 1977.
[Ha2] R. Hartshorne, Residues and duality, Lecture Notes in Math. 20, Springer-Verlag, 1966.
[HiSt] P. Hilton, U. Stammbach, A course in homological algebra, Graduate Texts in Mathematics
4, Springer-Verlag, New York, 1997.
[Ho] M. Hovey, Model categories, Mathematical Surveys and Monographs 63, American Mathe-
matical Society, Providence, RI, 1999.
[Iv] B. Iversen, Cohomology of sheaves, Universitext, Springer-Verlag, Berlin, 1986.
[Bernhard Keller’s Webpage] http://www.math.jussieu.fr/%7Ekeller/publ/index.html
[Ke1] B. Keller, Introduction to abelian and derived categories, in Representations of reductive
groups, edited by R. W. Carter and M. Geck, Cambridge University Press 1998, 41-62 (avail-
able on Keller’s webpage).
[Ke2] B. Keller, Derived categories and their uses, in Handbook of algebra, edited by
M. Hazewinkel, Elsevier 1996 (available on Keller’s webpage).
[Ke3] B. Keller, On differential graded categories, preprint (available on Keller’s webpage).
[Ke4] B. Keller, Introduction to A∞-algebras and modules, Homology, Homotopy, and Applica-
tions 3 (2001), 1–35 (available on Keller’s webpage).
[KeVo] B. Keller, D. Vossieck, Aisles in derived categories, Bull. Soc. Math. Belg. Ser. A 40 (1988),
no. 2, 239-253.
[KaSch] M. Kashiwara, P. Schapira, Sheaves on Manifolds, Grundlehren der Mathematichen Wis-
senschaften 292, Springer-Verlag, 1990.
LECTURES ON DERIVED AND TRIANGULATED CATEGORIES 33
[Li] J. Lipman, Notes on derived categories and derived functors, available at
http://www.math.purdue.edu/∼lipman.
[Ma] S. Mac Lane, Categories for the working mathematician, Second edition. Graduate Texts in
Mathematics 5, Springer-Verlag, New York, 1998.
[Ma] H. R. Margolis, Spectra and the Steenrod algebra North-Holland Mathematical Library 29,
North-Holland Publishing Co., Amsterdam, 1983.
[May1] J. P. May A concise course in algebraic topology, Chicago Lectures in Mathematics, Uni-
versity of Chicago Press, Chicago, IL, 1999.
[May2] J. P. May, The axioms for triangulated categories, preprint.
[Mi1] B. Mitchel, The full embedding theorem, Am. J. Math. 86 (1964), 619–637.
[Mi2] B. Mitchel, Rings with several objects, Adv. in Math. 8 (1972), 1–161.
[Ne] A. Neeman, Triangulated categories, Annals of Mathematics Studies, 148. Princeton Univer-
sity Press, Princeton, NJ, 2001.
[Po] A. Polishchuk, Noncommutative two-tori with real multiplication as noncommutative projec-
tive varieties, J. Geom. Phys. 50, no. 1-4, 162-187.
[Pu] D. Puppe, On the formal structure of stable homotopy theory, in Colloq. Alg. Topology,
Aarhus University (1962), 65-71.
[Ro] A. Rosenberg, The spectrum of abelian categories and reconstructions of schemes, in Rings,
Hopf Algebras, and Brauer groups, 257–274, Lectures Notes in Pure and Appl. Math. 197,
Marcel Dekker, New York, 1998.
[Ve] J.-L. Verdier, Des catégories dérivées des catégories abéliennes, Astérisque 239, 1996.
[We] C. A. Weibel, Introduction to homological algebra, Cambridge Studies in Advanced Mathe-
matics 38, Cambridge University Press, 1994.
E-mail address: behrang@alum.mit.edu
Mathematics Department, Florida State University, 208 Love Building, Tallahassee,
FL 32306-4510, U.S.A.
	Lecture 1: Abelian categories
	1. Products and coproducts in categories
	2. Abelian categories
	3. Categories of sheaves
	4. Abelian category of quasi-coherent sheaves on a scheme
	5. Morita equivalence of rings
	6. Appendix: injective and projective objects in abelian categories
	Lecture 2: Chain complexes
	1. Why chain complexes?
	2. Chain complexes
	3. Constructions on chain complexes
	4. Basic properties of cofiber sequences
	5. Derived categories
	6. Variations on the theme of derived categories
	7. Derived functors
	Lecture 3: Triangulated categories
	1. Triangulated categories
	2. Cohomological functors
	3. Abelian categories inside triangulated categories; t-structures
	4. Producing new abelian categories
	5. Appendix I: topological triangulated categories
	6. Appendix II: different illustrations of TR4
	References
