Squared cycles in monomial relations algebras

Brian Jue

Open Mathematics (2006)

  • Volume: 4, Issue: 2, page 250-259
  • ISSN: 2391-5455

Abstract

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Let 𝕂 be an algebraically closed field. Consider a finite dimensional monomial relations algebra Λ = 𝕂 Γ 𝕂 Γ I 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 𝕂 Γ . 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.

How to cite

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Brian Jue. "Squared cycles in monomial relations algebras." Open Mathematics 4.2 (2006): 250-259. <http://eudml.org/doc/268956>.

@article{BrianJue2006,
abstract = {Let \[\mathbb \{K\}\] be an algebraically closed field. Consider a finite dimensional monomial relations algebra \[\Lambda = \{\{\mathbb \{K\}\Gamma \} \mathord \{\left\bad. \{\vphantom\{\{\mathbb \{K\}\Gamma \} I\}\} \right. \hspace\{0.0pt\}\} I\}\] 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.},
author = {Brian Jue},
journal = {Open Mathematics},
keywords = {16E05; 16E10; 16G10; 16G20},
language = {eng},
number = {2},
pages = {250-259},
title = {Squared cycles in monomial relations algebras},
url = {http://eudml.org/doc/268956},
volume = {4},
year = {2006},
}

TY - JOUR
AU - Brian Jue
TI - Squared cycles in monomial relations algebras
JO - Open Mathematics
PY - 2006
VL - 4
IS - 2
SP - 250
EP - 259
AB - Let \[\mathbb {K}\] be an algebraically closed field. Consider a finite dimensional monomial relations algebra \[\Lambda = {{\mathbb {K}\Gamma } \mathord {\left\bad. {\vphantom{{\mathbb {K}\Gamma } I}} \right. \hspace{0.0pt}} I}\] 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.
LA - eng
KW - 16E05; 16E10; 16G10; 16G20
UR - http://eudml.org/doc/268956
ER -

References

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  8. [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. [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 Zbl0862.16008
  10. [10] B. Huisgen-Zimmermann: “Predicting syzygies of monomial relations algebras”, Manuscr. Math., Vol. 70, (1991), pp. 157–182. Zbl0723.16003
  11. [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. [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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