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The preprojective algebra of a tame hereditary Artin algebra. (English) Zbl 0612.16015
Let \(\Lambda\) be a finite dimensional hereditary k-algebra and TrD the inverse of the Auslander-Reiten translation DTr. In the paper, the preprojective algebra \(\Pi\) (\(\Lambda)\) of \(\Lambda\) is the Z-graded algebra \(\Pi (\Lambda)=\oplus^{\infty}_{n=0}\Pi_ n(\Lambda)\), where \(\Pi_ n(\Lambda)=Hom_{\Lambda}(\Lambda,(TrD)^ n(\Lambda))\) and multiplication is given by \(u_ r\cdot u_ s=(TrD)^ r(u_ r)\circ u_ s\). It is shown that, if \(\Lambda\) is tame, then \(\Pi\) (\(\Lambda)\) is a prime finitely generated k-algebra, satisfies a polynomial identity, is left and right Noetherian of Krull dimension and global dimension two, and has zero Jacobson radical. The homogeneous prime spectrum of \(\Pi\) (\(\Lambda)\) is also described. In the proofs, the functorial methods are essentially applied. If \(\Lambda\) is wild, the homological properties of \(\Pi\) (\(\Lambda)\) are described by D. Baer [Lect. Notes Math. 1177, 1-12 (1986)].

MSC:
16Gxx Representation theory of associative rings and algebras
16Rxx Rings with polynomial identity
16P60 Chain conditions on annihilators and summands: Goldie-type conditions
16P40 Noetherian rings and modules (associative rings and algebras)
16B50 Category-theoretic methods and results in associative algebras (except as in 16D90)
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References:
[1] DOI: 10.1080/00927877508822046 · Zbl 0331.16027 · doi:10.1080/00927877508822046
[2] Auslander M., Representation theory of algebras pp 389– (1978)
[3] DOI: 10.1090/S0002-9947-1979-0530043-2 · doi:10.1090/S0002-9947-1979-0530043-2
[4] DOI: 10.1007/BFb0075255 · doi:10.1007/BFb0075255
[5] DOI: 10.1016/0021-8693(75)90125-8 · Zbl 0332.16014 · doi:10.1016/0021-8693(75)90125-8
[6] Dlab V., Mem. Amer. Math. Soc 173 (1976)
[7] Dlab V., Representation Theory pp 329– (1978)
[8] DOI: 10.1007/BFb0088465 · doi:10.1007/BFb0088465
[9] Dlab V., Proc. Amer. Math. Soc 83 pp 228– (1981)
[10] Dlab, V. 1983. The regular representations of the tame hereditary algebras. Séminaire d’ Algèbre P. Dubreil et M. P. Malliavin. 1983. Vol. 1029, pp.120–133. Springer-Verlag. Lecture Notes in Mathematics
[11] DOI: 10.1016/0021-8693(85)90123-1 · Zbl 0554.16001 · doi:10.1016/0021-8693(85)90123-1
[12] Gabriel P., Calculus of fractions and homotopy theory (1967) · Zbl 0186.56802
[13] Gabriel P., Bull. Soc. math. France 90 pp 323– (1967)
[14] Gabriel, P. and Rentschler, R. 1967.Sur la dimension des anneaux et ensembles ordonnes, Vol. 265, A712–A715. Paris: C. R. Acad. Sci. · Zbl 0155.36201
[15] DOI: 10.1007/BFb0089778 · doi:10.1007/BFb0089778
[16] Geigle W., Dissertation Paderborn 831 (1984)
[17] DOI: 10.1007/BF01171701 · Zbl 0593.16022 · doi:10.1007/BF01171701
[18] DOI: 10.1007/BFb0075263 · doi:10.1007/BFb0075263
[19] Gelfand I.M., Punct. Anal. Appl 13 pp 157– (1979) · Zbl 0437.16020 · doi:10.1007/BF01077482
[20] DOI: 10.1016/0021-8693(74)90081-7 · Zbl 0285.16018 · doi:10.1016/0021-8693(74)90081-7
[21] Grothendieck A., Tohuku math. J 10 pp 119– (1957)
[22] Krause G.R., Growth of algebras and Gelfand-Kirillov dimension (1985) · Zbl 0564.16001
[23] DOI: 10.1007/BFb0103748 · doi:10.1007/BFb0103748
[24] DOI: 10.1080/00927878408823020 · Zbl 0532.16019 · doi:10.1080/00927878408823020
[25] DOI: 10.1007/BFb0075266 · doi:10.1007/BFb0075266
[26] Nastasescu C., Graded ring theory (1982)
[27] Oystaeyen F. Van, Noncommutative algebraic geometry 887 (1982)
[28] Oystaeyen F. Van, Bull. Soc. Math. Belg 887 (1985)
[29] Popescu N., Abelian categories with applications to rings and modules (1973) · Zbl 0271.18006
[30] Procesi C., Rings with polynomial identities (1973) · Zbl 0262.16018
[31] DOI: 10.1007/BF02566682 · Zbl 0444.16018 · doi:10.1007/BF02566682
[32] DOI: 10.1016/0021-8693(76)90184-8 · Zbl 0338.16011 · doi:10.1016/0021-8693(76)90184-8
[33] Ringel C.M., Ist. Naz. Atta Mat. Symp. Math 23 pp 321– (1979)
[34] Rowen L.H., Polynomial identities in ring theory (1980) · Zbl 0461.16001
[35] DOI: 10.2307/1969915 · Zbl 0067.16201 · doi:10.2307/1969915
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