  We describe the (chiral) BRST-cohomology of matter with central charge
$1<c_M<25$ coupled to a ``Liouville" theory, realized as a free field with a
background charge $Q_L$ such that $c_M+c_L=26$. We consider two cases: a)
matter is realized by one free field with an imaginary background charge, b)
matter is realized by $D$ free fields: $c_M =D$. In case a) the cohomology
states can be labelled by integers $r,s$ of a rotated $c_M =1$ theory, but
hermiticity imposes $r=s$. Thus there is still a discrete set of momenta
$p_M(r,r),\ p_L(r,r)$ such that there are non- trivial (relative) cohomology
states at level $r^2$ with ghost-numbers 0 or 1 (for $r>0$) and ghost-numbers 0
or $-1$ (for $r<0$). The (chiral) ground ring is isomorphic to a subring of the
$c_M =1$ theory which is $(xy)^n,\ n=0,1,2,\ldots$, and there are {\it no}
non-trivial currents acting on the ground ring. In case b) there is no
non-trivial relative cohomology for non-zero ghost numbers and, for zero ghost
number, the cohomology groups are isomorphic to a $(D-1)$-dimensional on-shell
``transverse" Fock space. The only exceptions are at level 1 for vanishing
matter momentum and $p_L=Q_L(1+r)$ with $r=\pm 1$, where one has one more
ghost-number zero and a ghost-number $r$ cohomology state. All these results
follow quite easily from the existing literature.
