Cohomology of torsion and completion of N-complexes

TL;DR

This paper develops new tools for understanding the cohomology of N-complexes, introducing explicit functors and establishing equivalences between torsion and completion categories in the derived setting.

Contribution

It introduces explicit derived torsion and completion functors for N-complexes and proves an equivalence between cohomologically torsion and complete N-complexes over noetherian rings.

Findings

Defined Koszul, Čech, and telescope N-complexes.

Established equivalence between torsion and complete N-complex categories.

Unified various cohomology theories via N-complexes.

Abstract

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.