a new lower bound for cohomological dimension

Let (r,m) be a noetherian local ring, m a finitely generated r-module, and a an ideal of r. we define the a-minimum dimension d(a,m) of m by d(a,m)=min{dimr/p+a:p∈asshr(m)}. in this paper, we show that cd(a,m)≥dimm−d(a,m) and we give some sufficient conditions and characterization for the equality to hold true.