Let $Q$ be a finite acyclic quiver, and $X$ some representation of $Q$. For $i \in Q_0$ define the $kQ$-modules $P_i = kQe_i$, and $X(i) = e_i X$. The representation $X$ has a canonical projective resolution, called the Ringel resolution, given by 
$$
0 
\longrightarrow 
\bigoplus_{a \in Q_1} P_{h(a)} \otimes X(t(a))
\xrightarrow{\quad d \quad}
\bigoplus_{i \in Q_0} P_i \otimes X(i)
\xrightarrow{\quad f \quad}
X
\longrightarrow 
0
$$
where in each direct summand the maps are given by 
$$
d(p \otimes x) = p \otimes (a \cdot x) - pa \otimes x
\qquad\quad
f(p \otimes x) = p \cdot x
$$

Does anyone have some intuition for this resolution that they could share? Can the modules and the maps in this resolution be nicely interpreted in terms of the paths in $Q$? And can this interpretation be extended to get us projective resolutions for quivers with relations? Or for non-acylic quivers?

For more details on this resolution, see [these lecture notes by Harm Derksen][HD].

[HD]: http://www.math.lsa.umich.edu/~hderksen/math711.w01/quivers/lecture4.pdf