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The ai-invariants of powers of ideals

Funds:  Supported by the Fundamental Research Funds for the Central Universities(AHY150200).
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https://doi.org/10.3969/j.issn.0253-2778.2020.03.013
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  • Author Bio:

    TIAN Shixin, male, born in 1995, Master. Research field: Algebra. E-mail: tsx@mail.ustc.edu.cn

  • Corresponding author: SHEN Yihuang
  • Received Date: 06 January 2020
  • Accepted Date: 28 March 2020
  • Rev Recd Date: 28 March 2020
  • Publish Date: 31 March 2020
    Abstract

  • Abstract

    Inspired by the recent work of Lu and O’Rourke, we study the ai-invariants of (symbolic) powers of some graded ideals. When I and J are two graded ideals in two distinct polynomial rings R and S over a common field K. We study the ai-invariants of the powers of the fiber product via the corresponding conditions on I and J. When IΔ is the Stanley-Reisner ideal of a k-dimensional complex Δ with k≥2. We investigate the ai-invariants of the symbolic powers of IΔ.

    Abstract

    Inspired by the recent work of Lu and O’Rourke, we study the ai-invariants of (symbolic) powers of some graded ideals. When I and J are two graded ideals in two distinct polynomial rings R and S over a common field K. We study the ai-invariants of the powers of the fiber product via the corresponding conditions on I and J. When IΔ is the Stanley-Reisner ideal of a k-dimensional complex Δ with k≥2. We investigate the ai-invariants of the symbolic powers of IΔ.
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    • References(19)
  • [1]
    HOA L T, TRUNG N V. Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals[J]. Journal of Algebra, 2009, 322: 4219-4227.
    [2]
    HOA L T, TRUNG N V. Partial Castelnuovo-Mumford regularities of sums and intersections of powers of monomial ideals[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 2010, 149: 229-246.
    [3]
    IYENGAR S B, LEUSCHKE G J, LEYKIN A, et al. Twenty-four hours of local cohomology[M]. Providence: American Mathematical Society, 2007.
    [4]
    LU D C. Geometric regularity of powers of two-dimensional squarefree monomial ideals[J]. (2018.8.22)[2020.4.20]. https://arxiv.org/pdf/1808.07266.pdf. Accepted by Journal of Algebraic Com- binatorics.
    [5]
    MILLER E, STURMFELS B. Combinatorial commutative algebra[M]. New York: Springer-Verlag, 2005.
    [6]
    MINH N C, TRUNG N V. Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals[J]. Journal of Algebra, 2009, 322: 4219-4227.
    [7]
    MUNKRES J R. Elements of algebraic topology[M]. Menlo Park: Addison-Wesley Publishing Com- pany, 1984.
    [8]
    NASSEH S, SATHER-WAGSTAFF S. Vanishing of Ext and Tor over fiber products[J]. Proceedings of the American Mathematical Society, 2017, 145: 4661-4674.
    [9]
    GOTO S, WATANABE K. On graded rings.I[J]. Journal of the Mathematical Society of Japan, 1978, 30: 179-213.
    [10]
    BRUNS W, HERZOG J. Cohen-Macaulay rings[M]. Cambridge: Cambridge University Press, 1993.
    [11]
    HERZOG J, HOA L T, TRUNG N V. Asymptotic linear bounds for the Castelnuovo-Mumford reg- ularity[J]. Transactions of the American Mathematical Society. 2002, 354: 1793-1809.
    [12]
    EISENBUD D. Commutative algebra[M]. New York: Springer-Verlag, 1995.
    [13]
    HERZOG J, HIBI T. Monomial ideals[M]. London: Springer-Verlag London, Ltd., 2011.
    [14]
    DAO H, DE STEFANI A, GRIFO E, et al. Symbolic powers of ideals[C]. Cham: Springer. 2018, 222: 387-432.
    [15]
    BRODMANN M P, SHARP R Y. Local cohomology[M]. Cambridge: Cambridge University Press, 2013.
    [16]
    NGUYEN H D, VU T. Homological invariants of powers of fiber products[J]. Acta Mathematica Vietnamica. 2019, 44: 617-638.
    [17]
    TAKAYAMA Y. Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals[J]. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie Nouvelle Serie, 2005, 48(96): 327-344.
    [18]
    O’ROURKE J L. Local Cohomology and Degree Complexes of Monomial Ideals[OL]. (2019.11.14)[2020.4.22]. https://arxiv.org/pdf/1910.14140.pdf.
    [19]
    HERZOG J, HOA L T, TRUNG N V. The stable set of associated prime ideals of a polymatroidal ideal[J]. Journal of Algebraic Combinatorics. An International Journal, 2013, 37: 289-312.
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    [1]
    HOA L T, TRUNG N V. Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals[J]. Journal of Algebra, 2009, 322: 4219-4227.
    [2]
    HOA L T, TRUNG N V. Partial Castelnuovo-Mumford regularities of sums and intersections of powers of monomial ideals[J]. Mathematical Proceedings of the Cambridge Philosophical Society, 2010, 149: 229-246.
    [3]
    IYENGAR S B, LEUSCHKE G J, LEYKIN A, et al. Twenty-four hours of local cohomology[M]. Providence: American Mathematical Society, 2007.
    [4]
    LU D C. Geometric regularity of powers of two-dimensional squarefree monomial ideals[J]. (2018.8.22)[2020.4.20]. https://arxiv.org/pdf/1808.07266.pdf. Accepted by Journal of Algebraic Com- binatorics.
    [5]
    MILLER E, STURMFELS B. Combinatorial commutative algebra[M]. New York: Springer-Verlag, 2005.
    [6]
    MINH N C, TRUNG N V. Cohen-Macaulayness of powers of two-dimensional squarefree monomial ideals[J]. Journal of Algebra, 2009, 322: 4219-4227.
    [7]
    MUNKRES J R. Elements of algebraic topology[M]. Menlo Park: Addison-Wesley Publishing Com- pany, 1984.
    [8]
    NASSEH S, SATHER-WAGSTAFF S. Vanishing of Ext and Tor over fiber products[J]. Proceedings of the American Mathematical Society, 2017, 145: 4661-4674.
    [9]
    GOTO S, WATANABE K. On graded rings.I[J]. Journal of the Mathematical Society of Japan, 1978, 30: 179-213.
    [10]
    BRUNS W, HERZOG J. Cohen-Macaulay rings[M]. Cambridge: Cambridge University Press, 1993.
    [11]
    HERZOG J, HOA L T, TRUNG N V. Asymptotic linear bounds for the Castelnuovo-Mumford reg- ularity[J]. Transactions of the American Mathematical Society. 2002, 354: 1793-1809.
    [12]
    EISENBUD D. Commutative algebra[M]. New York: Springer-Verlag, 1995.
    [13]
    HERZOG J, HIBI T. Monomial ideals[M]. London: Springer-Verlag London, Ltd., 2011.
    [14]
    DAO H, DE STEFANI A, GRIFO E, et al. Symbolic powers of ideals[C]. Cham: Springer. 2018, 222: 387-432.
    [15]
    BRODMANN M P, SHARP R Y. Local cohomology[M]. Cambridge: Cambridge University Press, 2013.
    [16]
    NGUYEN H D, VU T. Homological invariants of powers of fiber products[J]. Acta Mathematica Vietnamica. 2019, 44: 617-638.
    [17]
    TAKAYAMA Y. Combinatorial characterizations of generalized Cohen-Macaulay monomial ideals[J]. Bulletin Mathematique de la Societe des Sciences Mathematiques de Roumanie Nouvelle Serie, 2005, 48(96): 327-344.
    [18]
    O’ROURKE J L. Local Cohomology and Degree Complexes of Monomial Ideals[OL]. (2019.11.14)[2020.4.22]. https://arxiv.org/pdf/1910.14140.pdf.
    [19]
    HERZOG J, HOA L T, TRUNG N V. The stable set of associated prime ideals of a polymatroidal ideal[J]. Journal of Algebraic Combinatorics. An International Journal, 2013, 37: 289-312.
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