Algorithmic homological algebra
Exercise sheet 4
Prof. Dr. Mohamed Barakat, Sebastian Posur
Exercise 1. (Exactness)
Let M −→φ N −→ψ L be a differential sequence in an Abelian category. Prove that the following statements are equivalent:
• The sequence is exact at N.
• The image mono of φ is a kernel mono of ψ.
• The image epi of ψ is a cokernel epi of φ.
Exercise 2. (Chain complexes)
LetAbe an additive orAbelian category. Show that the categories Ch•A, Ch≥0A, Ch+A, Ch≤0A, Ch−A, ChbAare again additive orAbelian, respectively. Furthermore, show that if M• is a complex, then H•(M•) is a subfactor ofM• with trivial differentials.
This exercise sheet will be discussed on 8.12.2016.
