Borsuk-Sieklucki theorem in cohomological dimension theory

The Borsuk-Sieklucki theorem says that for every uncountable family {X_α}_α∈A of n-dimensional closed subsets of an n-dimensional ANR-compactum, there exist α ≠ β such that dim(X_α ∩ X_β) = n. In this paper we show a cohomological version of that theorem: THEOREM. Suppose a compactum X is clc_ℤⁿ⁺¹, where n ≥ 1, and G is an Abelian group. Let {X_α}_α∈J be an uncountable family of closed subsets of X. If dim_GX = dim_GX_α = n for all α ∈ J, then dim_G(X_α ∩ X_β) = n for some α ≠ β. For G being a countable principal ideal domain the above result was proved by Choi and Kozlowski [C-K]. Independently, Dydak and Koyama [D-K] proved it for G being an arbitrary principal ideal domain and posed the question of validity of the Theorem for quasicyclic groups (see Problem 1 in [D-K]). As applications of the Theorem we investigate equality of cohomological dimension and strong cohomological dimension, and give a characterization of cohomological dimension in terms of a special base.