A new lower bound for cohomological dimension

Document Type : Research Paper


1 Faculty of Mathematical Sciences Lorestan University Khorram Abad Iran

2 Faculty of Mathematical Sciences, Lorestan University, Khorram Abad, Iran.


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.


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