Homological Algebra - Problem Set 1 (due Tuesday, April 21)
Problem 1. Consider a commutative diagram of R-modules of the form
/
X0
/
0
f
/
X
/
Y0
g
/
Y
/
Z0
0
h
Z
with exact rows. Show that there is an exact sequence
ker(f ) → ker(g) → ker(h) → coker(f ) → coker(g) → coker(h).
Problem 2. Consider a commutative diagram of R-modules of the form
A0
/
B0
a
A
/
b
B
/
/ D0
C0
c
/C
/
/
E0
d
e
/E
D
with exact rows. Show that if b and d are monic and a is epic, then c is monic. Show that
if b and d are epic and e is monic, then c is epic. Deduce that if b and d are isomorphisms,
a is epic and e is monic, then c is an isomorphism.
Problem 3. Let A be an additive category and let X,Y be objects of A.
(1) Denote by e the neutral element of the abelian group HomA (X, Y ). Denote by e0 the
composite of the unique maps X → 0 and 0 → Y , where 0 denotes a zero object of A.
Show that e0 = e. In what follows we will refer to e as 0.
(2) We denote by X ⊕ Y the sum of X and Y , equipped with inclusions i1 : X → X ⊕ Y ,
i2 : Y → X ⊕ Y and projections p1 : X ⊕ Y → X, p2 : X ⊕ Y → Y which characterize
X ⊕ Y as both a product and a coproduct, respectively. We define the diagonal map
δX : X → X ⊕ X
to be the map uniquely determined by the properties p1 ◦ δX = idX and p2 ◦ δX = idX .
Dually, we define the codiagonal map
σY : Y ⊕ Y → Y
determined by σY ◦ i1 = idY and σY ◦ i2 = idY . Given morphisms f : X → Y ,
g : X 0 → Y 0 , we define the map
f ⊕ g : X ⊕ X0 → Y ⊕ Y 0
determined by p1 ◦ (f ⊕ g) ◦ i1 = f , p2 ◦ (f ⊕ g) ◦ i2 = g, and pj ◦ (f ⊕ g) ◦ ik = 0
for j 6= k. Show that, for f, g : X → Y , the sum f + g, computed with respect to the
group structure on HomA (X, Y ), equals the composition
δ
f ⊕g
σ
X
Y
X −→
X ⊕ X −→ Y ⊕ Y −→
Y.
(3) Show that the abelian group structure on the Hom-sets of A is uniquely determined by
the ordinary category underlying A. Conclude that an additive category is a category
which satisfies certain properties as opposed to a category with additional structure.
Formulate those properties, giving an alternative definition to the one in class. Extend
this statement to abelian categories.
Problem 4. Let A be an abelian category.
(1) Let I be a small category. Show that the category AI of (covariant) functors from I
to A is an abelian category.
(2) Let X be a topological space and let C be a category. We define a presheaf F on X
with values in C to consist of the following data:
• for each open subset U of X, an object F (U ) of C,
• for each inclusion U ⊂ V of open subsets of X, a morphism ρVU : F (V ) → F (U )
called restriction from V to U ,
satisfying the following conditions:
W
• for inclusions U ⊂ V ⊂ W of open subsets of X, we have ρVU ◦ ρW
V = ρU ,
• for every open subset U of X, we have ρUU = idF (U ) .
We define a morphism between presheaves F and G to be given by the following data:
• for each open subset U of X, a morphism fU : F (U ) → G(U ) in C,
such that, for every inclusion U ⊂ V of open subsets of X, the diagram
F (V )
fV
/ G(V )
ρV
U
F (U )
fU
/
ρV
U
G(U )
commutes. We denote the resulting category of presheaves on X with values in C by
PSh(X, C) Reformulate the definition of a presheaf and use (2) to deduce that the
category of presheaves on X with values in an abelian category A is abelian.
2