The finite type of modules of bounded projective dimension and Serre's conditions

IF 0.8 3区数学 Q2 MATHEMATICS

Bulletin of the London Mathematical Society Pub Date : 2024-06-01 DOI:10.1112/blms.13099

Michal Hrbek, Giovanna Le Gros

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引用次数: 0

Abstract

Let $R$ be a commutative Noetherian ring. For a natural number $k$ , we prove that the class of modules of projective dimension bounded by $k$ is of finite type if and only if $R$ satisfies Serre's condition $(S_{k})$ . In particular, this answers positively a question of Bazzoni and Herbera in the specific setting of a Gorenstein ring. Applying similar techniques, we also show that the $k$ -dimensional version of the Govorov–Lazard theorem holds if and only if $R$ satisfies the ‘almost’ Serre condition $(C_{k + 1})$ .