Let $q$ be a non-negative integer. We prove that perfect field $K$ has cohomological dimension at most $q+1$ if, and only for any finite extension $L$ of homogeneous space $Z$ under smooth linear connected algebraic group over $L$, the $q$-th Milnor $K$-theory is spanned by images norms coming from extensions which rational point. also variant this result imperfect fields.

Let Γ \D be an arithmetic quotient of a symmetric space of non-compact type. A spine D0 is a Γ -equivariant deformation retraction of D with dimension equal to the virtual cohomological dimension of Γ . We explicitly construct a spine for the case of Γ = SU(2,1;Z[i]). The spine is then used to compute the cohomology of Γ \D with various local coefficients. © 2007 Elsevier Inc. All rights reserv...