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Non-commutative Gröbner bases and Anick’s resolution. (English) Zbl 0951.16020

Dräxler, P. (ed.) et al., Computational methods for representations of groups and algebras. Proceedings of the Euroconference in Essen, Germany, April 1-5, 1997. Basel: Birkhäuser. Prog. Math. 173, 139-159 (1999).
The article is an introduction to the theory of Gröbner bases for noncommutative algebras which are graded algebras of the type \(K\langle\{X\}\rangle/I\), \(\{X\}\) is a set of well ordered unknowns.
The algebras which are monomial play a special role. To every algebra \(A\) one can associate a monomial algebra \(B\) such that \(B\) and \(A\) have the same set of normal words, that is words that cannot be written as a linear combination of smaller words. The obstruction words (tips) are the minimal between the non-normal.
A Gröbner basis (reduced) is the set \(\{f_i-u_i\}\) such that \(f_i\) is an obstruction and \(\{f_i-u_i\}\in I\), with \(u_i\) a linear combination of normal words.
For each algebra \(A\), as above, the authors define the Ufnarovski quiver \({\mathcal U}(A)\). This quiver is used to describe the growth of the algebra and its paths describe almost all normal words. Another quiver is described which can be used to describe the Poincaré series and also the set of \(n\)-chains. The set of \(n\)-chains in its turn is used to describe an \(A\)-projective resolution of \(K\) over \(A\), which is Anick’s resolution.
Finally they describe two softwares: the Bergman package and ANICK(C\(_{+,+}\)) which was used to implement Anick’s resolution, computing Betti numbers, etc.….
For the entire collection see [Zbl 0920.00022].

MSC:

16Z05 Computational aspects of associative rings (general theory)
16W50 Graded rings and modules (associative rings and algebras)
16E05 Syzygies, resolutions, complexes in associative algebras
16S15 Finite generation, finite presentability, normal forms (diamond lemma, term-rewriting)
16P90 Growth rate, Gelfand-Kirillov dimension

Software:

BERGMAN; ANICK
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