  Let $R$ be a local commutative noetherian ring and $HKR$ the homology ring of
the corresponding Koszul complex. We study the homological properties of $HKR$
in particular the so-called Avramov spectral sequence. When the embedding
dimension of $R$ is four and when $R$ can be presented with quadratic relations
we have found 102 cases where this spectral sequence degenerates and only three
cases where it does not degenerate. We also determine completely the Hilbert
series of the bigraded Tor of these $HKR$ in tables A-D of section 5. We also
study some higher embedding dimensions. Among the methods used are the
programme {\tt BERGMAN} by J\"orgen Backelin et al, the {\tt Macaulay2}-package
{\tt DGAlgebras} by Frank Moore, combined with results by Govorov, Clas
L\"ofwall, Victor Ufnarovski and others.
