[HTML][HTML] The generating hypothesis in the derived category of R-modules

KH Lockridge - Journal of Pure and Applied Algebra, 2007 - Elsevier
KH Lockridge
Journal of Pure and Applied Algebra, 2007Elsevier
In this paper, we prove a version of Freyd's generating hypothesis for triangulated
categories: if D is a cocomplete triangulated category and S∈ D is an object 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 …
In this paper, we prove a version of Freyd’s generating hypothesis for triangulated categories: if D is a cocomplete triangulated category and S∈D is an object 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.
Elsevier
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