Abstract
Using the resolution quiver for a connected Nakayama algebra, a fast algorithm is given to decide whether its global dimension is finite or not and whether it is Gorenstein or not. The latter strengthens a result of Ringel.
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References
M. Auslander, I. Reiten, and S.O. Smalø, Representation Theory of Artin Algebras, Cambridge Studies in Adv. Math. 36, Cambridge Univ. Press, Cambridge, 1995.
W.D. Burgess, K.R. Fuller, E.R. Voss, and B. Zimmermann-Huisgen, The Cartan matrix as an indicator of finite global dimension for Artinian rings, Proc. Amer. Math. Soc. 95 (1985), 157–165.
X.-W. Chen and Y. Ye, Retractions and Gorenstein homological properties, Algebr. Represent. Theory 17 (2014), 713–733.
W.H. Gustafson, Global dimension in serial rings, J. Algebra 97 (1985), 14–16.
K. Igusa and D. Zacharia, On the cyclic homology of monomial relation algebras, J. Algebra 151 (1992), 502–521.
D. Madsen, Projective dimensions and Nakayama algebras, In: Representations of algebras and related topics, Fields Inst. Commun. 45, Amer. Math. Soc., Providence, RI, 2005, 247–265.
C.M. Ringel, The Gorenstein projective modules for the Nakayama algebras. I, J. Algebra 385 (2013), 241–261.
D. Shen, A note on resolution quivers, J. Algebra Appl. 13 (2014), 1350120.
D. Shen, The singularity category of a Nakayama algebra, J. Algebra 429 (2015), 1–18.
G. Williams, Smith forms for adjacency matrices of circulant graphs, Linear Algebra Appl. 443 (2014), 21–33.
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Shen, D. A note on homological properties of Nakayama algebras. Arch. Math. 108, 251–261 (2017). https://doi.org/10.1007/s00013-016-1016-x
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DOI: https://doi.org/10.1007/s00013-016-1016-x