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 R and complexes of R R -modules with finitely generated homology M M and N N , we show N N is in the thick subcategory generated by M M if and only if the ghost index of N p N_\mathfrak {p} with respect to M p M_\mathfrak {p} is finite for each prime p \mathfrak {p} of R 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.