The importance of being isolated

TL;DR

This paper investigates conditions under which the derived category of a commutative ring is generated by residue fields, highlighting cases where the local-to-global principle holds or fails, with implications for polynomial rings.

Contribution

It provides new criteria for when the derived category is generated by residue fields and characterizes the local-to-global principle based on the spectrum's topology.

Findings

Residue fields do not generate the derived category of polynomial rings in infinitely many variables.

A sufficient condition and an obstruction for generation by residue fields are established.

The local-to-global principle depends solely on the topology of the spectrum.

Abstract

We give both a sufficient condition for and an obstruction to the derived category of a commutative ring being generated by its residue fields. As an illustration, we exhibit a ring for which Foxby's small support classifies localizing subcategories despite the failure of the local-to-global principle. We also conclude that the residue fields do not generate for a polynomial ring in infinitely many variables. Finally, we give a necessary and sufficient criterion for the derived category to satisfy the local-to-global principle; it turns out to depend solely on the topology of the spectrum.