  In 1989 Happel conjectured that for a finite-dimensional algebra $A$ over an
algebraically closed field $k$, $\gl A< \infty$ if and only if $\hch A <
\infty$. Recently Buchweitz-Green-Madsen-Solberg gave a counterexample to
Happel's conjecture. They found a family of pathological algebra $A_q$ for
which $\gl A_q = \infty$ but $\hch A_q=2$. These algebras are pathological in
many aspects, however their Hochschild homology behaviors are not pathological
any more, indeed one has $\hh A_q = \infty=\gl A_q$. This suggests to pose a
seemingly more reasonable conjecture by replacing Hochschild cohomology
dimension in Happel's conjecture with Hochschild homology dimension: $\gl A <
\infty$ if and only if $\hh A < \infty$ if and only if $\hh A = 0$. The
conjecture holds for commutative algebras and monomial algebras. In case $A$ is
a truncated quiver algebras these conditions are equivalent to the quiver of
$A$ has no oriented cycles. Moreover, an algorithm for computing the Hochschild
homology of any monomial algebra is provided. Thus the cyclic homology of any
monomial algebra can be read off in case the underlying field is characteristic
0.
