A new lower bound for cohomological dimension

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.