The generating hypothesis in the derived category of R-modules

TL;DR

This paper establishes a criterion for the generating hypothesis in triangulated categories, linking it to von Neumann regularity of endomorphism rings, with implications for derived categories of rings.

Contribution

It proves a version of Freyd's generating hypothesis in triangulated categories and characterizes when it holds in derived categories of commutative rings.

Findings

Generating hypothesis holds iff the endomorphism ring is von Neumann regular.

In derived categories of commutative rings, the hypothesis is true iff the ring is von Neumann regular.

Characterization of Noetherian stable homotopy categories where the hypothesis holds.

Abstract

In this paper, we prove a version of Freyd's generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S is an object in D whose endomorphism ring is graded commutative and concentrated in degree zero, then S generates (in the sense of Freyd) the thick subcategory determined by S if and only if the endomorphism ring of S is von Neumann regular. As a corollary, we obtain that the generating hypothesis is true in the derived category of a commutative ring R if and only if R is von Neumann regular. We also investigate alternative formulations of the generating hypothesis in the derived category. Finally, we give a characterization of the Noetherian stable homotopy categories in which the generating hypothesis is true.