The generating hypothesis in the derived category of R-modules

Details The generating hypothesis in the derived category of R-modules The generating hypothesis in the derived category of R-modules

2005-11-21

arxiv.org/abs/math/0511534v1

Mathematics

Algebraic Topology

16 pages, submitted to JPAA

Scientific paper

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.

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