quasi isomorphism between chain complexes of projectives and homotopy equivalence
1. Proposition
Let 
be an abelian category and 
be the category of chain complexes.
Suppose
are non negatively graded chain complex of projectives with a chain map
Then 
is part of a chain homotopy equivalence
2. Proof
2.1. a)
By assumption, there exists an inverse 
on the homology 
Consider
Then since 
is projective, there exists a morphism
Then it holds, that
is the zero morphism
hence by extension of a chain map we get chain maps
Furthermore, define
as evident zero morphism and since
also
??? at least rmod ? ???
a map
Thus by projectivity, we get an
and hence
thus by extension of a chain homotopy it follows, that 
is a chain homotopy equivalence.
2.2. b)
again we get a morphism
and again using analogously
3. meta
- for modules mostly happy
- in general for abelian categories correct ??