Order complex of ideals in a commutative ring with identity

Nela Milošević; Zoran Z. Petrović

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 4, page 947-952
  • ISSN: 0011-4642

Abstract

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Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere.

How to cite

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Milošević, Nela, and Petrović, Zoran Z.. "Order complex of ideals in a commutative ring with identity." Czechoslovak Mathematical Journal 65.4 (2015): 947-952. <http://eudml.org/doc/276043>.

@article{Milošević2015,
abstract = {Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere.},
author = {Milošević, Nela, Petrović, Zoran Z.},
journal = {Czechoslovak Mathematical Journal},
keywords = {ideal; commutative ring; order complex; homotopy type},
language = {eng},
number = {4},
pages = {947-952},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {Order complex of ideals in a commutative ring with identity},
url = {http://eudml.org/doc/276043},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Milošević, Nela
AU - Petrović, Zoran Z.
TI - Order complex of ideals in a commutative ring with identity
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 4
SP - 947
EP - 952
AB - Order complex is an important object associated to a partially ordered set. Following a suggestion from V. A. Vassiliev (1994), we investigate an order complex associated to the partially ordered set of nontrivial ideals in a commutative ring with identity. We determine the homotopy type of the geometric realization for the order complex associated to a general commutative ring with identity. We show that this complex is contractible except for semilocal rings with trivial Jacobson radical when it is homotopy equivalent to a sphere.
LA - eng
KW - ideal; commutative ring; order complex; homotopy type
UR - http://eudml.org/doc/276043
ER -

References

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  1. Clark, E., Ehrenborg, R., 10.1007/s00026-012-0127-8, Ann. Comb. 16 (2012), 215-232. (2012) Zbl1302.06003MR2927604DOI10.1007/s00026-012-0127-8
  2. Hatcher, A., Algebraic Topology, Cambridge University Press Cambridge (2002). (2002) Zbl1044.55001MR1867354
  3. Hersh, P., Shareshian, J., 10.1007/s11083-006-9053-x, Order 23 (2006), 339-342. (2006) Zbl1118.06002MR2309698DOI10.1007/s11083-006-9053-x
  4. Kozlov, D., Combinatorial Algebraic Topology, Algorithms and Computation in Mathematics 21 Springer, Berlin (2008). (2008) Zbl1157.57300MR2361455
  5. Margolis, S. W., Saliola, F., Steinberg, B., 10.4171/JEMS/579, J. Eur. Math. Soc. 17 (2015), 3037-3080. (2015) MR3429159DOI10.4171/JEMS/579
  6. Meshulam, R., 10.1007/s11083-008-9086-4, Order 25 (2008), 153-155. (2008) Zbl1159.06006MR2425950DOI10.1007/s11083-008-9086-4
  7. Munkres, J. R., Elements of Algebraic Topology, Advanced Book Program Addison-Wesley Publishing Company, Menlo Park, California (1984). (1984) Zbl0673.55001MR0755006
  8. Patassini, M., 10.1016/j.jalgebra.2011.05.042, J. Algebra 343 (2011), 37-77. (2011) MR2824544DOI10.1016/j.jalgebra.2011.05.042
  9. Shareshian, J., Woodroofe, R., Order complexes of coset posets of finite groups are not contractible, (to appear) in Adv. Math. 
  10. Shelton, B., Splitting Algebras II: The Cohomology Algebra, (to appear) in arXiv:1208. 2202. 
  11. Vassiliev, V. A., Topology of discriminants and their complements, Proc. of the International Congress of Mathematicians, ICM'94, 1994, Zürich, Switzerland. Vol. I, II S. D. Chatterji Birkhäuser Basel (1995), 209-226. (1995) Zbl0852.55003MR1403923
  12. Wachs, M. L., Poset topology: tools and applications, Geometric Combinatorics E. Miller et al. IAS/Park City Math. Ser. 13 American Mathematical Society; Princeton: Institute for Advanced Studies, Providence (2007), 497-615. (2007) Zbl1135.06001MR2383132

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