Cohomological dimension of certain algebraic varieties

Abstract:

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.