# Squared cycles in monomial relations algebras

Open Mathematics (2006)

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

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topBrian 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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