Journal of the
Korean Mathematical Society JKMS

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 categories of cohomologically $\mathfrak{a}$-torsion $N$-complexes and 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.

Keywords: Koszul $N$-complex, telescope $N$-complex, torsion, completion

MSC numbers: 13D07, 13D30, 13B35

Supported by: This work was financially supported by National Natural Science Foundation of China (Grant No. 11761060,11901463).