Unbounded derived categories and the finitistic dimension conjecture

Jeremy Rickard

doi:10.1016/j.aim.2019.106735

Unbounded derived categories and the finitistic dimension conjecture

Jeremy Rickard

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Abstract

We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but prove that they do for any Noetherian commutative ring. For non-commutative finite dimensional algebras the question is open, and we prove that if injectives generate for such an algebra, then the finitistic dimension conjecture holds for that algebra.

Original language	English
Article number	106735
Number of pages	21
Journal	Advances in Mathematics
Volume	354
Early online date	23 Jul 2019
DOIs	https://doi.org/10.1016/j.aim.2019.106735
Publication status	Published - 1 Oct 2019

Keywords

Derived categories
Finite dimensional algebras
Finitistic dimension conjecture

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10.1016/j.aim.2019.106735

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This is the accepted author manuscript (AAM). The final published version (version of record) is available online via Elsevier at https://doi.org/10.1016/j.aim.2019.106735 . Please refer to any applicable terms of use of the publisher.
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abstract = " We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but prove that they do for any Noetherian commutative ring. For non-commutative finite dimensional algebras the question is open, and we prove that if injectives generate for such an algebra, then the finitistic dimension conjecture holds for that algebra. ",

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AB - We consider the question of whether the injective modules generate the unbounded derived category of a ring as a triangulated category with arbitrary coproducts. We give an example of a non-Noetherian commutative ring where they don't, but prove that they do for any Noetherian commutative ring. For non-commutative finite dimensional algebras the question is open, and we prove that if injectives generate for such an algebra, then the finitistic dimension conjecture holds for that algebra.

KW - Derived categories

KW - Finite dimensional algebras

KW - Finitistic dimension conjecture

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Unbounded derived categories and the finitistic dimension conjecture

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Professor Jeremy C Rickard

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Unbounded derived categories and the finitistic dimension conjecture

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Professor Jeremy C Rickard

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