Let R be an associative ring with a unit, and consider the standard projective model structure of non-negatively graded (left) R-module, $Ch\_R$. A map $f:M\to N$ in $Ch\_R$ is a *weak equivalence* if it induces isomorphisms $H\_kM\to H\_kN$ on all homologies, and a *cofibration* if each $f\_k:M\_k\to N\_k$ is a monomorphism, with a projective R-module as its cokernel.


Let A be a projective R-*module*. The chain complex $(\*)$ $0\to A\to A\to0$ is a projective object in the world of chain complexes, $Ch\_R$. So is any direct sum of such chain complexes. It is a known result (see [DS95, 7.10] for instance) that any acyclic object in $Ch\_R$ which is level-wise projective is isomorphic to a direct sum of chains complexes as in $(\*)$, with $A$'s projective.


We have here two notions, of (i) a chain complex which is level-wise project, and of (ii) a projective object in $Ch\_R$. My gut's feeling is that (i) *doesn't* imply (ii). In the other way around I'm not sure. I would be glad to have an answer in both directions. :-)


[DS95] = Dwyer & Spalinski, *Homotopy theories and model categories*.



On the difference between a projective chain complex and a level-wise projective chain complex
**Edit:** I put my answer into a broader perspective. 


In the following [We] refers to Weibel's "An Introduction to Homological Algebra". 


Recall that a chain complex $C$ is *split exact* if it is acyclic and $Z\_n$ (the cycles) is a direct summand of $C\_n$ for each $n$ [We, Def. 1.4.1, Ex. 1.4.2]. Moreover, let $Ch$ be the category of (unbounded) chain complexes over an abelian category with enough projectives. Then, by [We, Ex. 2.2.1] and my answer in 


$\qquad$[https://mathoverflow.net/questions/103056/when-is-an-acylic-chain-complex-contractible](https://mathoverflow.net/questions/103056/when-is-an-acylic-chain-complex-contractible/103071#103071) 


we have: 



> 
> For a chain complex $P$ the following are equivalent: 
> 
> 
> 1. $P$ is a projective object in $Ch$
> 2. $P$ is a split exact complex of projectives
> 3. $P$ is a contractible complex of projectives
> 
> 
> 


Also from the link we obtain the following **examples** where $Ch\_R$ denotes the category of unbounded chain complexes of modules over the ring $R$: 


* If $R$ is hereditary (e.g. PID's, Dedekind domains), then the projective objects of $Ch\_R$ are exactly the acyclic complexes of projective $R$-modules.
* If $R$ is any ring with unit and the acyclic complex $P$ of projective $R$-modules is bounded below, then $P$ is a projective object in $Ch\_R$.


However, not all acyclic complexes of projective or free modules are projective objects in $Ch\_R$. A counter-example (due to Dold) is given in [We, Example 1.4.2]: 


* Over $R=\mathbb{Z}/4$ the following complex is exact 
$$\cdots \to \mathbb{Z}/4 \xrightarrow{2} \mathbb{Z}/4 \xrightarrow{2} \mathbb{Z}/4 \to \cdots $$
But it's no projective object in $Ch\_R$ since $Z\_n = \mathbb{Z}/2$ can't be a direct summand of $C\_n = \mathbb{Z}/4$.




---


**Added:** Let $Ch\_b \subseteq Ch$ be the subcategory of chain complexes that are bounded below. In contrast to $Ch$ the following holds in $Ch\_b$: 



> 
> The projective objects in $Ch\_b$ are exactly the acyclic chain complexes (bounded below) of projectives. 
> 
> 
> 


Proof: Let $P$ be an acyclic chain complex of projectives that is bounded below. We have already seen in example 2 above (compare also [We, Ex. 1.4.1 2.]) that $P$ is projective in $Ch$. Since $P \in Ch\_b$ it's a projective object in $Ch\_b$ 


Now let $P \in Ch\_b$ be a projective object. The same proof as in $Ch$ shows that each $P\_i$ is projective (consider objects of the abelian category as chain complexes concentrated in a single degree). Also the same proof as in $Ch$ can be used to see that $P$ is acyclic: The mapping cone of $id: P[1] \to P[1]$ yields a short exact sequence 
$$0 \to P[1] \to \operatorname{cone}(id\_{P[1]}) \to P \to 0.$$
But $P, P[1] \in Ch\_b$ and by definition $\operatorname{cone}(id)\_i = P\_i\oplus P\_{i+1}$ whence it is also bunded below. So the short exact sequence is in $Ch\_b$ and splits since $P$ is a projective object. Hence $id\_{P[1]}$ is nullhomotopic. So in particular, $P[1]$ and thus $P$ is acyclic. q.e.d. 


