Summary: Rings and Algebras Problem set #12. Dec. 8, 2011.
1. Show that the exact sequence 0 # A # B # C # 0 is homotopic with the 0­complex if and
only if it is split exact. (Observe, that at the same time all the homology modules are 0.)
2. Show the naturality of the connecting homomorphism in the long exact sequence of homologies.
3. (3 × 3­lemma) Consider the following commutative diagram with exact rows:
0 0 0
# # #
0 # A # # B # # C # # 0
# # #
0 # A # B # C # 0
# # #
0 # A ## # B ## # C ## # 0
# # #
0 0 0
Prove that if the middle column is exact then the first column is exact if and only if the last
column is exact.
4. Suppose we have the following commutative diagram with exact rows:
· · · # A n
i n
# B n

Source: Ágoston, István - Institute of Mathematics, Eötvös Loránd University

Collections: Mathematics