Stanley-Reisner theory provides a link between commutative algebra and combinatorics. We will define simplicial complexes and give the Stanley-Reisner correspondence between square-free monomial ideals in a polynomial ring and simplicial complexes. We will discuss the Alexander dual of simplicial complexes and of square-free monomial ideals. Finally, we present various criterions for Cohen-Macaulayness of a square-free monomial ideal quotient in terms of combinatorial and geometrical information on the corresponding simplicial complex, such as shellability, pureness, and homology. Speaker(s): Dawei Shen (UM)
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