Graded algebras of global dimension 3.

*(English)*Zbl 0633.16001Let \(A=k\oplus A_ 1\oplus A_ 2\oplus..\). be a finitely presented graded algebra over the algebraically closed field k of characteristic 0. The algebra A is here called regular if (i) every graded A-module has projective dimension at most d; (ii) A has polynomial growth; and (iii) A is Gorenstein, meaning that \(Ext^ q_ A(k,A)=0\) if \(q\neq d\), and \(Ext^ d_ A(k,A)\cong k\). When \(d=2\), A has the form k[x,y]/(f), where k[x,y] is the free algebra and f is either yx-cxy (0\(\neq c\in k)\) or \(yx- xy-x^ 2.\)

The bulk of this very interesting paper consists of a complete classification of the regular algebras of dimension 3. The results were obtained with the aid of computer calculations, and are too complicated to state here. A wealth of subsidiary examples and results is included, of which I mention two: It is shown (Theorem 1.16) that when A is a finite module over its centre, then (i) alone suffices to ensure that A is regular - this was for the authors one of the main reasons for undertaking the present study. Secondly, a generalized notion of skew polynomial ring is introduced, and it is proved that a regular algebra A of dimension 3 is a skew polynomial ring if and only if a certain invariant associated with A is infinite (Theorem 6.11).

The bulk of this very interesting paper consists of a complete classification of the regular algebras of dimension 3. The results were obtained with the aid of computer calculations, and are too complicated to state here. A wealth of subsidiary examples and results is included, of which I mention two: It is shown (Theorem 1.16) that when A is a finite module over its centre, then (i) alone suffices to ensure that A is regular - this was for the authors one of the main reasons for undertaking the present study. Secondly, a generalized notion of skew polynomial ring is introduced, and it is proved that a regular algebra A of dimension 3 is a skew polynomial ring if and only if a certain invariant associated with A is infinite (Theorem 6.11).

Reviewer: K.A.Brown

##### MSC:

16W50 | Graded rings and modules (associative rings and algebras) |

16E10 | Homological dimension in associative algebras |

16W60 | Valuations, completions, formal power series and related constructions (associative rings and algebras) |

16Exx | Homological methods in associative algebras |

16D70 | Structure and classification for modules, bimodules and ideals (except as in 16Gxx), direct sum decomposition and cancellation in associative algebras) |

##### Keywords:

finitely presented graded algebra; projective dimension; polynomial growth; regular algebras of dimension 3; skew polynomial ring
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\textit{M. Artin} and \textit{W. F. Schelter}, Adv. Math. 66, 171--216 (1987; Zbl 0633.16001)

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