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Projective resolutions and the global dimension of subhereditary orders. (English) Zbl 0686.16004

Let \(\Lambda\) be an order over a complete d.v.r. such that there exists a hereditary overorder \(\Gamma\) with Rad \(\Gamma\) \(\subset \Lambda\). For an indecomposable \(\Lambda\)-representation M with minimal projective resolution P, it is shown that the reduction \(\bar P\) of P modulo Rad \(\Gamma\) (replacing the \(P_ i\) by \(\bar P_ i:=P_ i/(Rad \Gamma)P_ i)\) decomposes into complexes \(\bar P_ j\), where \(P_ j\) is a minimal projective resolution of some irreducible \(\Gamma\)-representation \(G_ j\), and a minimal projective resolution of \(\bar M.\) From this fact, a number of corollaries is derived. In particular, a simple proof of the result in K. W. Roggenkamp’s paper [Math. Z. 160, 63-67 (1978; Zbl 0383.16002)] is obtained, and some interesting estimates for gld \(\Lambda\), especially for Bäckström or generalized Bäckström orders \(\Lambda\), are established.
Reviewer: W.Rump

MSC:

16H05 Separable algebras (e.g., quaternion algebras, Azumaya algebras, etc.)
16E10 Homological dimension in associative algebras
16D40 Free, projective, and flat modules and ideals in associative algebras
16Gxx Representation theory of associative rings and algebras

Citations:

Zbl 0383.16002
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References:

[1] M. Auslander andK. W. Roggenkamp, A characterization of orders of finite lattice type. Invent. Math.17, 79-84 (1972). · Zbl 0248.12012 · doi:10.1007/BF01390025
[2] E.Kirkman and J.Kuzmanovich, Global dimensions of a class of tiled orders. To appear in J. Algebra. · Zbl 0684.16001
[3] C. M. Ringel andK. W. Roggenkamp, Diagrammatic methods in the representation theory of orders. J. Algebra60, 11-42 (1979). · Zbl 0438.16021 · doi:10.1016/0021-8693(79)90106-6
[4] K. W. Roggenkamp, Some examples of orders of global dimension two. Math. Z.154, 225-238 (1977). · Zbl 0335.16010 · doi:10.1007/BF01214321
[5] K. W. Roggenkamp, Orders of global dimension two. Math. Z.160, 63-67 (1978). · Zbl 0383.16002 · doi:10.1007/BF01182330
[6] K. W. Roggenkamp, Auslander-Reiten species for socle determined categories of hereditary algebras and for generalized Bäckström orders. Mitt. Math. Sem. Gießen159, 1-98 (1983). · Zbl 0513.16003
[7] K. W. Roggenkamp, Corrigendum to Auslander-Reiten species for socle determined categories of hereditary algebras and for generalized Bäckström orders. Mitt. Math. Sem. Gießen175, 43-44 (1986). · Zbl 0599.16005
[8] K. W.Roggenkamp, Lattices over subhereditary orders and socle projective modules. To appear in J. Algebra. · Zbl 0665.16004
[9] S.Ruppmann, Über die globale Dimension von Ordnungen: eine hinreichende Bedingung. Diplomarbeit, Stuttgart 1977.
[10] R. Tarsy, Global dimension of orders. Trans. Amer. Math. Soc.151, 335-340 (1970). · Zbl 0221.16003 · doi:10.1090/S0002-9947-1970-0268226-3
[11] P.Wannenwetsch, Über die globale Dimension von Ordnungen ?: notwendige Bedingungen für gld ??2. Diplomarbeit, Stuttgart 1977.
[12] A.Wiedemann, Some remarks on subhereditary, Bäckström and Auslander orders. To appear in Comm. Algebra. · Zbl 0675.16005
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