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Commutative Algebra in Action: Betti Numbers and Combinatorics

Poursoltani Zarandi, Milad | 2024

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  1. Type of Document: Ph.D. Dissertation
  2. Language: Farsi
  3. Document No: 57267 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Pournaki, Mohammad Reza; Maimani, Hamid Reza; Parsaei Majd, Leila
  7. Abstract:
  8. In this thesis, we give necessary and sufficient conditions for a simplicial com- plex with small codimension to satisfy the Serre’s condition (Sr) as well as the CMt property. We also give a connection between being (Sr) and being CMt. Also, we focus on the dimension of dual modules of local cohomology of Stanley– Reisner rings to obtain a new vector which contains important information on the Serre’s condition (Sr) and the CMt property as well as the depth of Stanley–Reisner rings. We prove some results in this regard including lower bounds for the depth of Stanley–Reisner rings. Further, we give a characterization of (d − 1)-dimensional simplicial complexes with codimension two which are (Sd−3) but they are not Cohen– Macaulay. By using this characterization, we obtain a condition to equality of pro- jective dimension of the Stanley–Reisner rings and the arithmetical rank of their Stanley–Reisner ideals. Moreover, our characterization allows us to compute the h-vectors and give a negative answer to a known question regarding these vectors.
    We also focus on the gain graphs which are essential generalizations of the signed graphs. They are indeed directed graphs whose edges are labeled by the elements of a group rather than just ±1. In this paper, we prove the existence of matroids arising from these graphs. Further, we study the structure of the obtained matroids as well as their relation with the generalized Hamming weights, the regularity of the ideal of circuits, and the codes.
  9. Keywords:
  10. Cohen-Macualay Modules ; Simplicial Complex ; Stanley-Reisner Ideal ; Height Function ; Signed Graph ; Hamming Weight ; Matroids Theory ; Arithmetic Algorithms ; Cohen–Macaulay ُSimplicial Complex ; Generalized Hamming Weight

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