= Standard projective resolution of a quiver representation
Set $A=kQ$, $P_i=Ae_i$, $X_i=e_iX$. There is an exact sequence
$$
0\to\bigoplus_{\rho:i\to j}P_j\otimes_kX_i\xrightarrow{d}\bigoplus_iP_i\otimes_kX_i\to X\to0,
$$
where $d(p\otimes x)=p\rho\otimes x-p\otimes f_\rho(x)$ in the source and target summands, and the augmentation is $p\otimes x\mapsto px$. Each $P_i$ is a summand of $A$, so the two terms are <projective modules>. This also works for infinite-dimensional modules.
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