extension of a chain map
1. Proposition
Let 
be an abelian category and 
be the category of chain complexes.
Suppose 
is a chain complex of projectives and 
another chain complex
Assume given 
and a chain map 
.
→
TFAE:
- this
extends to a chain map 
- the map
induced by
is the zero morphism
2. Proof
somewhat sloppy written, todo with universal properties
2.1. 1) 
2)
Suppose there exists such a map 
.
Then it follows, that
hence it follows, that
And thus composed with
results in the zero morphism
2.2. 2) 1)
Suppose the morphism is the zero morphism. Then it follows, that
and thus by projectivity, there exists a map
making the diagram commute, hence also making
commute.