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2006 | 4 | 2 | 250-259
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Squared cycles in monomial relations algebras

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Let $$\mathbb{K}$$ be an algebraically closed field. Consider a finite dimensional monomial relations algebra $$\Lambda = {{\mathbb{K}\Gamma } \mathord{\left/ {\vphantom {{\mathbb{K}\Gamma } I}} \right. \kern-\nulldelimiterspace} I}$$ of finite global dimension, where Γ is a quiver and I an admissible ideal generated by a set of paths from the path algebra $$\mathbb{K}\Gamma $$ . There are many modules over Λ which may be represented graphically by a tree with respect to a top element, of which the indecomposable projectives are the most natural example. These trees possess branches which correspond to right subpaths of cycles in the quiver. A pattern in the syzygies of a specific factor module of the corresponding indecomposable projective module is found, allowing us to conclude that the square of any cycle must lie in the ideal I.
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4
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2
Strony
250-259
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wydano
2006-06-01
online
2006-06-01
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Bibliografia
  • [1] D. Anick and E. Green: “On the homology of quotients of path algebras”, Comm. Algebra, Vol. 15(1,2), (1987), pp. 309–341.
  • [2] M. Auslander, I. Reiten and S. Smalø: Representation theory of Artin algebras, Cambridge Studies in Advanced Mathematics 36, Cambridge University Press, Cambridge, 1995.
  • [3] K. Bongartz and B. Huisgen-Zimmermann: “The geometry of uniserial representations of algebras II. Alternate viewpoints and uniqueness”, J. Pure Appl. Algebra, Vol. 157, (2001), pp. 23–32. http://dx.doi.org/10.1016/S0022-4049(00)00031-1
  • [4] K. Bongartz and B. Huisgen-Zimmermann: “Varieties of uniserial representations IV. Kinship to geometric quotients”, Trans. Am. Math. Soc., Vol. 353, (2001), pp. 2091–2113. http://dx.doi.org/10.1090/S0002-9947-01-02712-X
  • [5] W. D. Burgess: “The graded Cartan matrix and global dimension of 0-relations Algebras”, Proc. Edinburgh Math. Soc., Vol. 30(3), (1987), pp. 351–362.
  • [6] P. Gabriel: Auslander-Reiten seuquence and representation-finite algebras, Lect. Notes Math. 831, Springer-Verlag, New York, 1980, pp. 1–71.
  • [7] E. Green, D. Happel and D. Zacharia: “Projective resolutions over Artin algebras with zero relations”, Illnois J. Math., Vol. 29(1), (1985), pp. 180–190.
  • [8] B. Huisgen-Zimmermann: “The geometry of uniserial representations of finite dimensional algebras I”, J. Pure Appl. Algebra, Vol. 127, (1988), pp. 39–72.
  • [9] B. Huisgen-Zimmermann: “The geometry of uniserial representations of finite dimensional algebras III”, Trans. Am. Math. Soc., Vol. 348(12), (1996), pp. 4775–4812. http://dx.doi.org/10.1090/S0002-9947-96-01575-9
  • [10] B. Huisgen-Zimmermann: “Predicting syzygies of monomial relations algebras”, Manuscr. Math., Vol. 70, (1991), pp. 157–182.
  • [11] K. Igusa: “Notes on the no loops conjecture”, J. Pure Appl. Algebra, Vol. 69, (1990), pp. 161–176. http://dx.doi.org/10.1016/0022-4049(90)90040-O
  • [12] B. Jue: The uniserial geometry and homology of finite dimensional algebras, Thesis (Ph.D), University of California, Santa Barbara, 1999.
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Bibliografia
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bwmeta1.element.doi-10_2478_s11533-006-0010-0
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