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

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.

10 pages, comments welcome