Squared cycles in monomial relations algebras

• Autorzy: Brian Jue
• Języki publikacji: EN

Abstrakty

EN

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.

Słowa kluczowe

EN

Representation theory; homological dimension; syzygies; 16E05; 16E10; 16G10; 16G20

• Wydawca: De Gruyter Open
• Czasopismo: Open Mathematics
• Rocznik: 2006
• Tom: 4
• Numer: 2
• Strony: 250-259

Daty

wydano

2006-06-01

online

2006-06-01

Twórcy

autor

Brian Jue

• California State University, Stanislaus

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.

Identyfikatory

DOI

10.2478/s11533-006-0010-0