  We introduce the notions of Koszul $N$-complex, $\check{\mathrm{C}}$ech
$N$-complex and telescope $N$-complex, explicit derived torsion and derived
completion functors in the derived category $\mathbf{D}_N(R)$ of $N$-complexes
using the $\check{\mathrm{C}}$ech $N$-complex and the telescope $N$-complex.
Moreover, we give an equivalence between the category of cohomologically
$\mathfrak{a}$-torsion $N$-complexes and the category of cohomologically
$\mathfrak{a}$-adic complete $N$-complexes, and prove that over a commutative
noetherian ring, via Koszul cohomology, via RHom cohomology (resp. $\otimes$
cohomology) and via local cohomology (resp. derived completion), all yield the
same invariant.
