  Given a factor map $p : (X,T) \to (Y,S)$ of Cantor minimal systems, we study
the relations between the dimension groups of the two systems. First, we
interpret the torsion subgroup of the quotient of the dimension groups
$K_0(X)/K_0(Y)$ in terms of intermediate extensions which are extensions of
$(Y,S)$ by a compact abelian group. Then we show that, by contrast, the
existence of an intermediate non-abelian finite group extension can produce a
situation where the dimension group of $(Y,S)$ embeds into a proper subgroup of
the dimension group of $(X,T)$, yet the quotient of the dimension groups is
nonetheless torsion free.
  Next we define higher order cohomology groups $H^n(X \mid Y)$ associated to
an extension, and study them in various cases (proximal extensions, extensions
by, not necessarily abelian, finite groups, etc.). Our main result here is that
all the cohomology groups $H^n(X \mid Y)$ are torsion groups. As a consequence
we can now identify $H^0(X \mid Y)$ as the torsion group of $ K_0(X)/K_0(Y)$.
