Yazd University Algebraic Structures and Their Applications 2382-9761 7 1 2020 02 01 A new lower bound for cohomological dimension 21 28 1621 10.22034/as.2020.1621 EN Alireza Nazari Faculty of Mathematical Sciences Lorestan University Khorram Abad Iran Asghar Farokhi Faculty of Mathematical Sciences, Lorestan University, Khorram Abad, Iran. Journal Article 2019 07 01 Let $(R,\mathfrak{m})$ be a Noetherian local ring, $M$ a finitely generated $R$-module, and $\mathfrak{a}$ an ideal of $R$. We define the $\mathfrak{a}$-minimum dimension $d(\mathfrak{a},M)$ of $M$ by $$d(\mathfrak{a},M)=Min\{\dim \frac{R}{\mathfrak{p}+\mathfrak{a}}:\mathfrak{p}\in Assh_{R}(M)\}.$$ In this paper, we show that $cd(\mathfrak{a},M)\geq \dim M-d(\mathfrak{a},M)$ and we give some sufficient conditions and characterization for the equality to hold true. https://as.yazd.ac.ir/article_1621_42a77c099b9affb629ee988bbd224dbb.pdf