Algebra III exercises, fall semester 2021-22
Giulio Lo Monaco
1. Describe coproduct in the category Ab of abelian groups and group homomorphisms.
What is the relation between these and coproducts in Grp?
[2 points]
2. A generating set for an algebra A is a set X ⊆ A such that ⟨X⟩. A basis
is a generating set X such that for each a ∈ X, then ⟨X \ {a}⟩ ̸= A.
Prove that an algebra need not have a basis. Prove that if A is finitely
generated, then it has a finite basis.
[2 points]
3. Let consider an algebra structure on N with one unary operation −⊙
defined by 0⊙
= 0 and for n > 0, n⊙
= n − 1. Prove that N has no basis.
[1 point]
4. Consider the set {a, b, c, d} with no operations. How many congruences
does it have? Now endow it with a unary operation defined by a → b, b →
a, c → d, d → c. How many congruences are there now?
[1 point]
5. An algebra is simple if its only congruences are the diagonal ∆A ⊆ A × A
and A × A itself.
Consider an algebra A with one ternary operation t defined by
t(x, y, z) =
z if x = y
x if x ̸= y.
Prove that A is simple.
[1 point]
6. Given a monad T on a category C, we know that there are two functors
FT
: C ⇆ Alg(T) : UT
. Prove that these form an adjunction FT
⊣ UT
.
[2 points]
7. A split fork is a diagram a b c
f0
f1
e
for which there exist arrows
a b ct s
in such a way that ef0 = ef1, es = 1c, f0t = 1b and
f1t = se.
Prove that a split fork is always an absolute coequalizer.
[1 point]
8. Prove that the free group monad is finitely accessible.
[2 points]
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9. Describe the algebras of the monad (−)N
.
[1 point]
10. Prove that, if |X| = λ, then the monad (−)X
is λ+
-accessible, but it is
not µ-accessible for any µ ≤ λ.
[2 points]
11. There is a free point monad on sets given by X → X •. The multiplication
X • • → X • is defined by sending X to X and both additional
points to the unique additional points in the codomain. The unit is the
inclusion X → X •.
Establish whether this monad is λ-accessible for some λ, and describe its
algebras.
[2 points]
12. Prove that a sequence of R-modules A → B → C → 0 is exact if and only
if for every R-module N the induced sequence
0 → Hom(C, N) → Hom(B, N) → Hom(A, N)
is exact.
[2 points]
13. Prove that a sequence A → B → C → 0 of R-modules is exact if and only
if the square
A B
0 C
u
v is a pushout.
[2 points]
14. Find an example of a non-flat module.
[2 points]
15. Find two modules M and N with finitely generated submodules M0 and
N0 respectively, such that there is a non-zero tensor mi ⊗ni ∈ M0 ⊗N0
which is zero in M ⊗ N.
[1 point]
16. Remember that a poset I is directed if every finite subposet I0 admits
an upper bound in I. A diagram in a category is called directed if it is
indexed by a directed poset. Give an explicit description of the colimit of
a directed diagram.
(Hint: if all the maps in a given directed diagram are inclusions, then the
colimit is simply the union of all the involved modules.)
[2 points]
17. Show that the operation of tensoring commutes with directed colimits, i.e.
if A is a module and (Bi)i∈I is a directed diagram, then the natural map
colimi∈I(A ⊗ Bi) → A ⊗ colimi∈I Bi
is an isomorphism.
[2 points]
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18. A ring is called a PID (principal ideal domain) if every ideal is of the form
I = (a), i.e. generated by a single element. Show that, if R is a PID, then
a quotient of an injective R-module is injective.
[1 point]
19. Show that every module admits both a projective and an injective resolu-
tion.
[2 points]
20. Show that the homology and the cohomology functors are additive, i.e. for
two maps f, g : C → D between chain complexes, we have Hn(f + g) =
Hn(f) + Hn(g) and similarly for cohomology.
[2 points]
21. Show that, for a left (resp. right) exact functor F there is a natural
isomorphism R0F ∼= F (resp. L0F ∼= F).
[1 point]
22. If B is a Z-module and p is a natural number, compute the modules
Extn
(Z/p, B) for all n. (Hint: Z is a PID.)
Also, compute all modules Torn
(Z/p, B).
[3 points]
23. Prove at least one between (a) and (b) and at least one between (c) and
(d) among the following statements:
(a) A is projective ⇔ ∀B Ext1
(A, B) = 0 ⇔ ∀B ∀n > 0 Extn
(A, B) = 0;
(b) B is injective ⇔ ∀A Ext1
(A, B) = 0 ⇔ ∀A ∀n > 0 Extn
(A, B) = 0;
(c) A is flat ⇔ ∀B Tor1
(A, B) = 0 ⇔ ∀B ∀n > 0 Torn
(A, B) = 0;
(d) B is flat ⇔ ∀A Tor1
(A, B) = 0 ⇔ ∀A ∀n > 0 Torn
(A, B) = 0.
[4 points]
24. Show that if U is an exact functor and F is right exact, then we have
natural isomorphisms ULnF ∼= LnUF.
[2 points]
25. Show that Torn
(A, B) commutes with finite sums in both variables.
[2 points]
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