Complete intersections and derived categories

TL;DR

This paper defines when a triangulated category can be considered a complete intersection and proves that for derived categories of local rings, this condition aligns exactly with the classical notion of complete intersections.

Contribution

It introduces a new categorical definition of complete intersections and establishes its equivalence with the classical algebraic concept for derived categories of local rings.

Findings

The categorical condition matches classical complete intersections for local rings.

Provides a new perspective linking triangulated categories to classical algebraic geometry.

Uses existing work of Avramov and Gulliksen to establish the equivalence.

Abstract

We propose a definition of when a triangulated category should be considered a complete intersection. We show (using work of Avramov and Gulliksen) that for the derived category of a complete local Noetherian commutative ring R, the condition on the derived category D(R) holds precisely when R is a complete intersection in the classical sense.