  In this paper, we consider homology groups induced by the exterior algebra
generated by a simplicial compliment of a simplicial complex $K$. These
homology groups are isomorphic to the Tor-groups $\mathrm{Tor}_{i,
J}^{\mathbf{k}[m]}(\mathbf{k}(K),\mathbf{k})$ of the face ring $\mathbf{k}(K)$,
which is very useful and much studied in toric topology. By using
$\check{C}ech$ homology theory and Alexander duality theorem, we prove that
these homology groups have dualities with the simplicial cohomology groups of
the full subcomplexes of $K$. Then we give a new proof of Hochster's theorem.
