Finiteness Properties of Local Cohomology Modules and Generalized Regular Sequences

Abstract

Let R be a commutative Noetherian ring, 𝔞 an ideal of R, and M an R-module. The purpose of this paper is to show that if M is finitely generated and dim M/𝔞M > 1, then the R-module ∪{N|N is a submodule of [Formula: see text] and dim N ≤ 1} is 𝔞-cominimax and for some x ∈ R is Rx + 𝔞-cofinite, where t ≔ gdepth (𝔞, M). For any nonnegative integer l, it is also shown that if R is semi-local and M is weakly Laskerian, then for any submodule N of [Formula: see text] with dim N ≤ 1 the associated primes of [Formula: see text] are finite, whenever [Formula: see text] for all i < l. Finally, we show that if (R, 𝔪) is local, M is finitely generated, [Formula: see text] for all i < l, and [Formula: see text] then there exists a generalized regular sequence x1, …, xl ∈ 𝔞 on M such that [Formula: see text].

Technology and language learning

“…Both of them were developed by Jurkovič (2019) in which a few minor adjustments to the Indonesian context were made. The last instrument was a self-assessment language proficiency grid according to the CEFRL classified into three broad levels including Basic User, Independent User, and Proficient User which can be broken down into six levels (A1-C2) (COUNCIL OF EUROPE, 2001). This self-assessment, which was on a 1-6 scale Belo Horizonte, v. 13, n. 3, p. 103-120, set.-dez.…”