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Graded algebras of global dimension 3. (English) Zbl 0633.16001
Let $$A=k\oplus A_ 1\oplus A_ 2\oplus..$$. 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).
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)
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##### References:
 [1] Anick, D, On the homology of algebras, Trans. amer. math. soc., 296, 641-659, (1986) · Zbl 0598.16028 [2] Auslander, M, On the dimension of modules and algebras, III, Nagoya math. J., 9, 67-77, (1955) · Zbl 0067.27103 [3] Brown, K.A; Hajarnavis, C.R, Homologically homogeneous rings, Trans. amer. math. soc., 281, 197-208, (1985) · Zbl 0531.16019 [4] Brown, K.A; Hajarnavis, C.R; MacEachern, A.B, Noetherian rings of finite global dimension integral over their centers, (), 349-371 · Zbl 0485.16008 [5] Bergman, G.M, The diamond lemma for ring theory, Advances in math., 29, 178-218, (1978) · Zbl 0326.16019 [6] Clebsch, A, Theorie der binären algebraischen formen, (1872), Teubner Leibzig · JFM 04.0047.02 [7] Cohn, P.M, () [8] Hartshorne, R, Residues and duality, () · Zbl 0196.24301 [9] Irving, R.S, Prime ideals of ore extensions over commutative rings, II, J. algebra, 58, 399-423, (1979) · Zbl 0411.16025 [10] Lemaire, J.-M, Algèbres connexes et homologie des espaces de lacets, () · Zbl 0293.55004 [11] Mumford, D; Suominen, K, Introduction to the theory of moduli, () · Zbl 0242.14004 [12] Năstăcescu, C; Van Oystaeyen, F, Graded ring theory, (1982), North-Holland Amsterdam · Zbl 0494.16001 [13] Ramras, R, Maximal orders over regular rings of dimension two, Trans. amer. math. soc., 142, 457-474, (1969) · Zbl 0186.07101 [14] Vasconcelos, W.V, On quasi-local regular algebras, (), 11-22 [15] Walker, R, Local rings and normalizing sets of elements, Proc. London math. soc., 24, 3, 27-45, (1972) · Zbl 0224.16004 [16] Weber, H, () [17] Weyl, H, The classical groups, (1946), Princeton Univ. Press Princeton, N. J · JFM 65.0058.02
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