TITLE: Projective objects in the category of chain complexes 
QUESTION [16 upvotes]: Excercise 2.2.1 in Weibel ("An Introduction to Homological Algebra") states that an object $P$ in the category of chain complexes over an abelian category is projective if and only it is a 
split exact complex of projectives. 
I was able to solve the only-if-part but I have touble with the if-part and would be glad if 
someone can give me some help. This is no homework! 
What have I a tried so far ? Given an epimorphism $\pi: X \to Y$ and a morphism $f: P \to Y$, it has to be shown that there is a morphism $g: P \to X$ s.t. $\pi \circ g=f$. 
Weibel hints to consider the special case  $0 \to P_1 \cong P_0 \to 0$. It's easy to construct $g$ in this case: $\pi$ epi means that each $\pi_i:X_i \to Y_i$ is epi. By projectivity of $P_1$ there is a hom. $g_1: P_1 \to X_1$ s.t. $\pi_1 \circ g_1 = f_1$. If $d^P$ resp. $d^X$ denotes the differential in $P$ resp. $X$, set $$g_0 := d^X_1 \circ g_1 \circ (d^P_1)^{-1}: P_0 \to X_0,\qquad g_i = 0: P_i \to X_i\; (i\neq 0,1)$$
Then $g=(g_i): P \to X$ is a morphisms s.t. $\pi \circ g=f$.
But I have no idea how to generalize this procedure to the general case where $d^P$ can not be expected to be an isomorphism.

REPLY [6 votes]: The question as asked has been answered, but to understand where bounded below enters the picture, it is 
helpful to think model categorically (as in Dwyer and Spalinsky
or, more recently, chapter 18 of More Concise Algebraic Topology,
by Kate Ponto and myself).  With the usual model structure (there
are others in the latter reference), a chain complex is acyclic
and cofibrant if and only if it is a projective object.   If it
is cofibrant (not necessarily acyclic) then it is degreewise
projective.  If it is degreewise projective and bounded below,
then it is cofibrant.  However, it can be acyclic and degreewise projective
and yet not cofibrant if it is not bounded below. There is a nice
example in the paper [K] that TJ refers to:  work over the ring
$Z/4$ and take all $P_n$ to be free on one generator, with all 
differentials given by multiplication by $2$:  $P$  is acyclic and
degreewise free, but it is not cofibrant and not a projective object.
Split exactness rules out such examples and is automatic when $P$ 
is exact, degreewise projective, and bounded below.
Incidentally, the role of $R$-split exactness becomes really interesting
model theoretically when $R$ is commutative and not a field and one considers 
model structures on DG modules over a DG $R$-algebra.  There are (at least) six 
different interesting projective type model structures, and the usual one is arguably not the most useful one (this is a shameless
advertisement for a paper in the writing stage by Tobi Barthel, Emily Riehl, 
and myself).