The finite type of modules of bounded projective dimension and Serre’s conditions. (English) Zbl 07902269
Summary: 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 \((C_{k+1})\).
© 2024 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
© 2024 The Author(s). The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.
MSC:
| 13C05 | Structure, classification theorems for modules and ideals in commutative rings |
| 13D07 | Homological functors on modules of commutative rings (Tor, Ext, etc.) |
| 13C60 | Module categories and commutative rings |
| 16E65 | Homological conditions on associative rings (generalizations of regular, Gorenstein, Cohen-Macaulay rings, etc.) |
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