Cohomology of torsion and completion of N-complexes

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.