A partial converse ghost lemma for the derived category of a commutative noetherian ring

TL;DR

This paper establishes a criterion involving ghost indices to determine containment among thick subcategories in the derived category of a commutative noetherian ring, extending to a converse coghost lemma for graded DG algebras.

Contribution

It introduces a new condition using ghost indices to detect subcategory containment, and proves a converse coghost lemma for derived categories of graded DG algebras.

Findings

N is in the thick subcategory generated by M iff ghost index of N_p w.r.t. M_p is finite for all primes p.

Provides a converse coghost lemma for the derived category of a non-negatively graded DG algebra.

Establishes a criterion linking local properties to global subcategory containment.

Abstract

In this article a condition is given to detect the containment among thick subcategories of the bounded derived category of a commutative noetherian ring. More precisely, for a commutative noetherian ring $R$ and complexes of $R$-modules with finitely generated homology $M$ and $N$, we show $N$ is in the thick subcategory generated by $M$ if and only if the ghost index of $N_\mathfrak{p}$ with respect to $M_\mathfrak{p}$ is finite for each prime $\mathfrak{p}$ of $R$. To do so, we establish a "converse coghost lemma" for the bounded derived category of a non-negatively graded DG algebra with noetherian homology.