A generalized Serre’s condition

Abstract

Throughout, let R be a commutative Noetherian ring. A ring R satisfies Serre's condition (S ℓ ) if for all p ∈ Spec R, depth Rp ≥ min{ℓ, dim Rp}. Serre's condition has been a topic of expanding interest. In this paper, we examine a generalization of Serre's condition (S j ℓ ). We say a ring satisfies (S j ℓ ) when depth Rp ≥ min{ℓ, dim Rp − j} for all p ∈ Spec R. We prove generalizations of results for rings satisfying Serre's condition.

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

“…An analogous condition to Serre's condition (𝑆 𝑘 ) to characterise almost Cohen-Macaulay rings was introduced by Ionescu [18], see also a more general version of Holmes [15]. Recall that a local ring 𝑅 is almost Cohen-Macaulay if 𝖽𝗂𝗆(𝑅) − 1 ⩽ 𝖽𝖾𝗉𝗍𝗁(𝑅) ⩽ 𝖽𝗂𝗆(𝑅), and for a non-local ring the condition is defined locally.…”