Lecture 2: Abelian Categories
Teacher: Benjamin Hennion
NB:
II. Abelian categories
1. Additive categories
- A category
is additive: -
if
is an abelian group-
composition is bilinear:
-
admits a object: an object which is both initial and terminal - the category
admits finite coproducts and products (i.e. all binary co/products exist)
Prop: If
satisfies (1), (2) and (3) and admits finite coproducts, then it is additive and
Notation: If
Proof:
- in any category, if
exists, then -
:We get a morphism
if exists.We have to prove that
satisfies the universal property of the product. , induced by and , is a bijection:Build
:-
take
and -
,
in
, I can takeWe can check that
-
-
Indeed:
look at
:(same for
)
-
-
We have
-
NB:
Let
is additive if:-
, the map is a morphism of abelian groups
Prop: if
is additive, then and : And reciprocally.
Proof:
Use the universal property of
Examples:
field: is additive ring left (or right) -modules form an additive category- Sheaves in an additive category are again an additive category
- Any sort of topological vector spaces (Fréchet/Banach)
2. Abelian categories
Goal of this entire class: talk about exact sequences. In all previous examples it works, except the last one (sort of topological vector spaces (Fréchet/Banach)).
Let
it means that for all
We get:
- An additive category
is called abelian: -
if
- kernels and cokernels always exist
- for all
, the morphism is an isomorphism
Examples:
are abelian : (modules over of finite type) is abelian if is noetherian (if is not noetherian, then kernels are not necessarily of finite type).-
over : additive but not abelian.-
with the
-adic topology:The kernel and cokernel are trivial, the image is
, the coimage
-
: , similar situation
Lemma:
Proof:
is a monomorphism:-
if
is an epimorphism:-
if
NB: there are notions of monomorphisms and epimorphisms in general categories.
In an abelian category, everything behaves like it does in
Actually: any abelian category can be “locally” embedded in a category of the
Exact sequences
-
is exact in if -
A short exact sequence is a sequence
exact in
-
A sequence is
split (scindée in fr) if one of the following equivalent assertion holds:
admits a section ( st ) admits a retract ( st )- the sequence is isomorphic to
Exercise: Prove the assertions are equivalent
Lemma: a split sequence is exact
Examples:
- in
, any short exact sequence splits -
does not split in is -mod exact, -mod split
NB:
3. Exact functors
We say that
the sequence
is exact (resp.
Lemma:
is left exact iff it preserves kernels: for all ,
is right exact iff it preserves cokernels
Proof: Up to changing
Then
is exact iff
Examples:
-
: is always right exactIt’s left exact iff
is flat (definition) -
left exact (if is commutative) -
group, ringis left exact.
III. Complexes
Fix an abelian category.
Definitions
- A complex in
: -
is a sequence
in st ( )
We will denote it by
A morphism
All the rectangles
commute
Denote by
We denote by
if for if for if or for
NB:
is a fully faithful functor.
Prop:
are abelian categories.
Proof:
- Abelian group structure on them: ok
: ok-
coprod, prod =
because they are computed degree-wise: - kernels and cokernels are computed degree-wise
Cohomology
Let
- The cohomology of
is: -
the family
NB: Quotient = cokernel of the map
-
If
,-
then we call cocycles the elements of
-
we call coboundaries the elements of
-
-
A complex
is called acyclic/exact if
Examples:
-
manifoldde Rham complex.
is called the de Rham cohomology of -
Cech cohomology
-
Singular cohomology
-
Dolbeault complex
NB: Cohomology vs homology:
-
difference = natural complexes:
- increasing the degree: cochain complexes ⟶ cohomology
- decreasing the degree: chain complexes ⟶ homology
⟹ equivalent categories, no big difference
-
usually: for historical reasons, some theories are called homology while others are called cohomology: this has to do with the variance of the functor
Group cohomology
Fix a group
Linear map:
:-
is defined as the complex where
(all maps: no structure preserved)
Exercise: Show that
- The group cohomology of
with coefficients in : -
is the cohomology of this complex
It’s denoted by
NB:
And what is
So
Assume, for the sake of simplicity, that
Pick such a
We have an exact sequence of groups:
Theorem:
NB: you can compute group extensions thanks to this theorem (this is part of the reason why cohomology matters)
Sketch of the proof:
Start with an exact sequence
Choose a splitting
Define
Actually:
Check that
where
If you had chosen another section
Other examples of complexes
Shift
- Shift (décalage in fr):
-
:
NB:
Mapping cone
we have
exact sequence in
Complex of maps
Define
What is
-
- so
Functors, homotopies
Problem: in general,
We say that
Two morphisms
We write
A complex
NB: if
Exercises:
-
is an exact sequenceCompare the data of splitting this sequence with the data of a homotopy
. -
Recall that
Lemma: If two morphisms
Proof: Enough to tackle the case
Corollary:
is acyclic: , hence
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