The derived category of a locally complete intersection ring

TL;DR

This paper characterizes locally complete intersection rings through properties of complexes with finitely generated homology, linking algebraic structure to derived category behavior.

Contribution

It provides a new characterization of locally complete intersection rings using proxy small and virtually small complexes in the derived category.

Findings

A local ring is a complete intersection iff all complexes with finitely generated homology are proxy small.

A noetherian ring is locally a complete intersection iff all such complexes are virtually small.

Answers a question posed by Dwyer, Greenlees, and Iyengar.

Abstract

In this paper, we answer a question of Dwyer, Greenlees, and Iyengar by proving a local ring $R$ is a complete intersection if and only if every complex of $R$-modules with finitely generated homology is proxy small. Moreover, we establish that a commutative noetherian ring $R$ is locally a complete intersection if and only if every complex of $R$-modules with finitely generated homology is virtually small.