HOMEWORK 2
Exercise 1. Finish the proof of the 5–lemma, i.e. prove that in the following commutative
diagram of Abelian groups
A //
a

B //
b

C //
c

D //
d

E
e

A // B // C // D // E
where the rows are exact sequences and a, b, , d, e are isomorphisms, the morphism c
is an epimorphism.
Exercise 2. Given the short exact sequence of Abelian groups
0 // A
i // B
j
// C // 0
the following conditions are equivalent:
(1) There exists p: C → B such that j ◦ p = idC.
(2) There exists q: B → A such that q ◦ i = idA.
(3) There are p, q as above such that i ◦ q + p ◦ j = idB
We have shown that (1) ⇒ (2) and (3). Prove that (2) ⇒ (1) and (3)
Exercise 3. For the short exact sequence od chain complexes
0 → A∗ → B∗ → C∗ → 0,
there is a long exact sequence of homology groups
· · · → Hn+1(C∗) → Hn(A∗) → Hn(B∗) → Hn(C∗) → Hn−1(A∗) → . . .
Prove the exactness in Hn(C∗).
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