On the existence of a compact generator on the derived category of a noetherian formal scheme

TL;DR

This paper proves that the derived category of sheaves with quasi-coherent torsion homologies on a noetherian formal scheme is generated by a single compact object, and shows the category of compact objects is skeletally small.

Contribution

It establishes the existence of a compact generator for the derived category of a noetherian formal scheme, advancing understanding of its structure.

Findings

The derived category D_qct(X) has a single compact generator.

The category of compact objects in D_qct(X) is skeletally small.

Provides foundational results for the structure of derived categories on formal schemes.

Abstract

In this paper, we prove that for a noetherian formal scheme X, its derived category of sheaves of modules with quasi-coherent torsion homologies D_qct(X) is generated by a single compact object. In an appendix we prove that the category of compact objects in D_qct(X) is skeletally small.