## Homological properties of the homology algebra of the Koszul complex of a local ring: examples and questions.(English)Zbl 1347.13007

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 Tables A-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 DG Algebras by Frank Moore combined with results by V. E. Govorov [Math. Notes 12, 552–556 (1973; Zbl 0253.16003)], C. Löfwall [Mat. Inst., Stockh. Univ. 5 (1976; Zbl 0429.13008)], V. A. Ufnarovskij [Itogi Nauki Tekh., Ser. Sovrem. Probl. Mat., Fundam. Napravleniya 57, 5–177 (1990; Zbl 0706.16001)] and others.

### MSC:

 13D03 (Co)homology of commutative rings and algebras (e.g., Hochschild, André-Quillen, cyclic, dihedral, etc.) 13P20 Computational homological algebra 68W30 Symbolic computation and algebraic computation 14H10 Families, moduli of curves (algebraic)

### Citations:

Zbl 0253.16003; Zbl 0429.13008; Zbl 0826.16001; Zbl 0706.16001

### Software:

BERGMAN; Macaulay2; DGAlgebras
Full Text:

### References:

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