extension of a chain homotopy
1. Proposition
Let 
be an abelian category and 
be the category of chain complexes.
Suppose 
is a chain complex and 
and a chain complex of projectives and 
chain maps
with 
for 
satisfying the chain homotopy equalities
TFAE:
- the map
for the homology 
induced by
is the zero morphism
- there exists a morphism
satisfying the chain homotpy equalities
2. Proof
2.1. welldefined
it remains to show, that
Applying 
gives
thus it follows, that
showing that 
is welldefined.
2.2. 1) 
2)
gives
results in
2.3. 2) 1)
By assumption,
hence it follows, that following diagram commutes
or that the induced map
