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Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs

In: Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics

Author

Listed:
  • Philippe Gimenez

    (Instituto de Investigación en Matemáticas de la Universidad de Valladolid (IMUVA), Facultad de Ciencias)

  • José Martínez-Bernal

    (Centro de Investigación y de Estudios Avanzados del IPN, Departamento de Matemáticas)

  • Aron Simis

    (Universidade Federal de Pernambuco, Departamento de Matemática)

  • Rafael H. Villarreal

    (Centro de Investigación y de Estudios Avanzados del IPN, Departamento de Matemáticas)

  • Carlos E. Vivares

    (Centro de Investigación y de Estudios Avanzados del IPN, Departamento de Matemáticas)

Abstract

In this paper we study irreducible representations and symbolic Rees algebras of monomial ideals. Then we examine edge ideals associated to vertex-weighted oriented graphs. These are digraphs having no oriented cycles of length two with weights on the vertices. For a monomial ideal with no embedded primes we classify the normality of its symbolic Rees algebra in terms of its primary components. If the primary components of a monomial ideal are normal, we present a simple procedure to compute its symbolic Rees algebra using Hilbert bases, and give necessary and sufficient conditions for the equality between its ordinary and symbolic powers. We give an effective characterization of the Cohen–Macaulay vertex-weighted oriented forests. For edge ideals of transitive weighted oriented graphs we show that Alexander duality holds. It is shown that edge ideals of weighted acyclic tournaments are Cohen–Macaulay and satisfy Alexander duality.

Suggested Citation

  • Philippe Gimenez & José Martínez-Bernal & Aron Simis & Rafael H. Villarreal & Carlos E. Vivares, 2018. "Symbolic Powers of Monomial Ideals and Cohen-Macaulay Vertex-Weighted Digraphs," Springer Books, in: Gert-Martin Greuel & Luis Narváez Macarro & Sebastià Xambó-Descamps (ed.), Singularities, Algebraic Geometry, Commutative Algebra, and Related Topics, pages 491-510, Springer.
    • Handle: RePEc:spr:sprchp:978-3-319-96827-8_21
    DOI: 10.1007/978-3-319-96827-8_21
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