  There are different definitions of homological dimension of metric compacta
involving either \v{C}ech homology or exact (Steenrod) homology. In this paper
we investigate the relation between these homological dimensions with respect
to different groups. It is shown that all homological dimensions of a metric
compactum X with respect to any field coincide provided X is homologically
locally connected with respect to the singular homology up to dimension n=dim
X. We also prove that any two-dimensional lc^2 metric compactum X satisfies the
equality dim(X times Y)=dim X+dim Y for any metric compactum Y. This improves
the well known result of Kodama that every two-dimensional ANR is dimensionally
full-valued. Actually, the condition X to be lc^2 can be weaken to the
existence at every point x a neighborhood V of x such that the inclusion
homomorphism H_k(V;S^1)\to H_k(X;S^1)$ is trivial for all k=1,2.
