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arXiv · 2112.12632

Generalized local duality, canonical modules, and prescribed bound on projective dimension

Thiago H. Freitas·Victor H. Jorge-Pérez·Cleto B. Miranda-Neto·Peter Schenzel
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Abstract

We present various approaches to J. Herzog's theory of generalized local cohomology and explore its main aspects, e.g., (non-)vanishing results as well as a general local duality theorem which extends, to a much broader class of rings, previous results by Herzog-Zamani and Suzuki. As an application, we establish a prescribed upper bound for the projective dimension of a module satisfying suitable cohomological conditions, and we derive some freeness criteria and questions of Auslander-Reiten type. Along the way, we prove a new characterization of Cohen-Macaulay modules which truly relies on generalized local cohomology, and in addition we introduce and study a generalization of the notion of canonical module.

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Thiago H. Freitas, Victor H. Jorge-Pérez, Cleto B. Miranda-Neto, Peter Schenzel. 2021-12-23. Generalized local duality, canonical modules, and prescribed bound on projective dimension. https://arxiv.org/abs/2112.12632

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