Cohomological dimension of certain algebraic varieties

Let $\mathfrak a$ be an ideal of a commutative Noetherian ring $R$. For finitely generated $R$-modules $M$ and $N$ with $\operatorname{Supp} N\subseteq\operatorname{Supp} M$, it is shown that $\mathrm{cd}(\mathfrak {a},N)\leq \mathrm{cd}(\mathfrak {a},M)$. Let $N$ be a finitely generated module over a local ring $(R,\mathfrak m)$ such that $\operatorname{Min}_{\hat{R}}\hat{N}=\operatorname{Assh}_{\hat{R}}\hat{N}$. Using the above result and the notion of connectedness dimension, it is proved that $\mathrm{cd}(\mathfrak {a},N)\geq\dim N-c(N/\mathfrak {a} N)-1.$ Here $c(N)$ denotes the connectedness dimension of the topological space $\operatorname{Supp} N$. Finally, as a consequence of this inequality, two previously known generalizations of Faltings' connectedness theorem are improved.