  We study a Leech homology of a locally bounded free partially commutative
monoid $M(E,I)$. Given a contravariant natural system of abelian groups $F$ on
$M(E,I)$ we build a precubical set $T(E,I)$ with a homological system of
abelian groups $\bar{F}$ and prove that the Leech homology groups
$H_n(M(E,I),F)$ are isomorphic to the cubical homology groups
$H_n(T(E,I),\bar{F})$, $n\geq 0$. As a consequence we have confirmed a
conjecture that if the free partially commutative monoid does not contain $>n$
mutually commuting generators, then its the homological dimension $\leq n$. We
have built the complexes of finite length for a computation of the Leech
homology of such monoids and the Hochschild homology of their monoid rings. The
results are applied to the homology of asynchronous transition systems. We give
the positive answer to a question that the homological dimension of the
asynchronous system does not greater than the maximal number of its mutually
independent events. We have built the complex for computing the integral
homology groups of an asynchronous transition system by the Smith normal form
of integer matrices.

Category Theory;Algebraic Topology