Homological properties of the homology algebra of the Koszul complex of a local ring: Examples and questions

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 Avramov spectral sequence. When the embedding dimension of R is four and when R can be presented with quadratic relations we have found 101 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 Table A, Table B, Table C and Table D at the end of the paper. We also study some higher embedding dimensions. Among the methods used are the programme BERGMAN by Jörgen Backelin et al., the Macaulay2-package DGAlgebras by Frank Moore, combined with results by Govorov, Clas Löfwall, Victor Ufnarovski and others.