  In this paper, we give an algebra-combinatorics formula of the M\"obius
transform for an abstract simplicial complex $K$ on $[m]=\{1, ..., m\}$ in
terms of the Betti numbers of the Stanley-Reisner face ring of $K$.
Furthermore, we employ a way of compressing $K$ to estimate the lower bound of
the sum of those Betti numbers by using this formula. As an application,
associating with the moment-angle complex $\mathcal{Z}_K$ (resp. real
moment-angle complex ${\Bbb R}\mathcal{Z}_K$) of $K$, we show that the
Halperin-Carlsson conjecture holds for $\mathcal{Z}_K$ (resp. ${\Bbb
R}\mathcal{Z}_K$) under the restriction of the natural $T^m$-action on
$\mathcal{Z}_K$ (resp. $({\Bbb Z}_2)^m$-action on ${\Bbb R}\mathcal{Z}_K$).
