This may be obvious. My question is short: 


$R$ is the polynomial algebra $\mathbb{k}[X\_{1},\dots , X\_{n}]$. Is the $R$-module $\mathbb{k}$ perfect in the sense that $\mathbb{k}$ is a compact object in the derived category $\mathbf{D}(R)$? You can consider $\mathbb{k}$ to be the field of rationals or complex numbers if it helps. 



perfect modules over polynomial algebra
Yes. The perfect objects in $\mathbf{D}(R)$ are the objects isomorphic to bounded complexes of finitely generated projective modules, and $\mathbb{k}$ is isomorphic in $\mathbf{D}(R)$ to its minimal projective resolution, which is such a complex.


More generally, in the derived category of a noetherian ring with finite global dimension, any finitely generated module is perfect.


