Skip to main content
SpringerLink
Log in

Order complex of ideals in a commutative ring with identity

  • Published:
Czechoslovak Mathematical Journal Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. E. Clark, R. Ehrenborg: The Frobenius complex. Ann. Comb. 16 (2012), 215–232.

    Article  MATH  MathSciNet  Google Scholar 

  2. A. Hatcher: Algebraic Topology. Cambridge University Press, Cambridge, 2002.

    MATH  Google Scholar 

  3. P. Hersh, J. Shareshian: Chains of modular elements and lattice connectivity. Order 23 (2006), 339–342.

    Article  MATH  MathSciNet  Google Scholar 

  4. D. Kozlov: Combinatorial Algebraic Topology. Algorithms and Computation in Mathematics 21, Springer, Berlin, 2008.

    MATH  Google Scholar 

  5. S. W. Margolis, F. Saliola, B. Steinberg: Combinatorial topology and the global dimension of algebras arising in combinatorics. J. Eur. Math. Soc. 17 (2015), 3037–3080.

    Article  MathSciNet  Google Scholar 

  6. R. Meshulam: On the homological dimension of lattices. Order 25 (2008), 153–155.

    Article  MATH  MathSciNet  Google Scholar 

  7. J. R. Munkres: Elements of Algebraic Topology. Advanced Book Program, Addison-Wesley Publishing Company, Menlo Park, California, 1984.

    Google Scholar 

  8. M. Patassini: On the (non-)contractibility of the order complex of the coset poset of an alternating group. J. Algebra 343 (2011), 37–77.

    Article  MATH  MathSciNet  Google Scholar 

  9. J. Shareshian, R. Woodroofe: Order complexes of coset posets of finite groups are not contractible. To appear in Adv. Math.

  10. B. Shelton: Splitting Algebras II: The Cohomology Algebra. To appear in arXiv:1208. 2202.

  11. V. A. Vassiliev: Topology of discriminants and their complements. Proc. of the International Congress of Mathematicians, ICM’94, 1994, Zürich, Switzerland. Vol. I, II (S. D. Chatterji, ed.). Birkhäuser, Basel, 1995, pp. 209–226.

    Google Scholar 

  12. M. L. Wachs: Poset topology: tools and applications. Geometric Combinatorics (E. Miller et al., eds.). IAS/Park City Math. Ser. 13, American Mathematical Society; Princeton: Institute for Advanced Studies, Providence, 2007, pp. 497–615.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Faculty for Information Systems and Technologies, Donja Gorica bb, University of Donja Gorica, 81000, Podgorica, Montenegro

    Nela Milošević

  2. Faculty of Mathematics, University of Belgrade, Studentski trg 16, 11000, Belgrade, Serbia

    Zoran Z. Petrović

Authors
  1. Nela Milošević
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Zoran Z. Petrović
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nela Milošević.

Additional information

The second author is partially supported by Ministry of Education, Science and Technological Development of Republic of Serbia Project #174032.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Milošević, N., Petrović, Z.Z. Order complex of ideals in a commutative ring with identity. Czech Math J 65, 947–952 (2015). https://doi.org/10.1007/s10587-015-0219-9

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10587-015-0219-9

Keywords

MSC 2010

Search

Navigation