Loading...

On Some Algebraic Structures of the Stanley-Reisner Rings Attached to Simplicial Complexes

Khoshnevis, Mona | 2018

828 Viewed
  1. Type of Document: M.Sc. Thesis
  2. Language: Farsi
  3. Document No: 50482 (02)
  4. University: Sharif University of Technology
  5. Department: Mathematical Sciences
  6. Advisor(s): Jafari, Amir
  7. Abstract:
  8. In this thesis, we study the f-vectors and h-vectors of simplicial complexes and we state and prove a theorem of Kruskal and Katona that characterizes the f-vectors of simplicial complexes. We then define the Stanley-Reisner ring A associated to a simplicial complex, and state certain connections between f and h-vectors of this simplicial complex with Hilbert function of A, and show that if A is a Cohen-Macaulay ring then the h-vector of the simplicial complex in an O-sequence. conversely any O-sequence, equivalently, the f-vector of any multicomplex is the h-vector of a simplicial complex. Finally it is shown that if a simplicial complex arises from a triangulation of the sphere and A is Cohen-Macaulay, then the upper bound conjecture is valid in this case. We also study balanced shellable simplicial complexes, and the values of h are defined and studied. Then we show the existence of a homogeneous system of parameters and construct it for a Cohen-Macaulay balanced simplicial complexes, and it is proved that the h-multivectors of a balanced Cohen-Macaulay simplicial complex is the f-multivector of an especial multicomplex. as an especial case, We conclude that the h-vectors of completely balanced Cohen-Macaulay simplicial complex is f-vector of a simplicial complex that it's 1-skeleton is m-colorable
  9. Keywords:
  10. Stanley-Reisner Ideal ; Simplicial Complex ; Cohen-Macualay Modules ; Upper Bound Conjecture ; Balanced Simplicial Complex ; Kruskal-Katona Theorem ; Modules Peeling

 Digital Object List

 Bookmark