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arXiv · 2405.00152

Homological dimensions, the Gorenstein property, and special cases of some conjectures

Souvik Dey·Rafael Holanda·Cleto B. Miranda-Neto
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Abstract

Our purpose in this work is multifold. First, we provide general criteria for the finiteness of the projective and injective dimensions of a finite module $M$ over a (commutative) Noetherian ring $R$. Second, in the other direction, we investigate the impact of the finiteness of certain homological dimensions of $M$ if $R$ is local, mainly when $R$ is Cohen-Macaulay and with a partial focus on duals. Along the way, we produce various freeness criteria for modules. Finally, we give applications, including characterizations of when $R$ is Gorenstein (and other ring-theoretic properties as well, sometimes in the prime characteristic setting), particularly by means of its anticanonical module, and in addition we address special cases of some long-standing conjectures; for instance, we confirm the 1985 conjecture of Vasconcelos on normal modules in case the module of differentials is almost Cohen-Macaulay.

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Souvik Dey, Rafael Holanda, Cleto B. Miranda-Neto. 2024-04-30. Homological dimensions, the Gorenstein property, and special cases of some conjectures. https://arxiv.org/abs/2405.00152

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