There are two wellstudied basic correspondences between squarefree. Graduate texts in mathematics 227 university of oregon. Betti numbers of monomial ideals and shifted skew shapes. Pdf powers of squarefree monomial ideals and combinatorics. Some applications to rees algebras and combinatorial optimization are presented. We give an example of a gorenstein squarefree monomial ideal i such that s i 2 satisfies the serre condition s 2, but is not cohenmacaulay. The information carried by squarefree monomial ideals can be charac. This work leads to algebraic characterizations of perfect graphs independent of the strong perfect graph theorem. In this paper, we survey work on symbolic and ordinary powers of squarefree monomial ideals, and their combinatorial consequences in hypergraph theory and linear integer programming. Combinatorial commutative algebra duke mathematics department. We survey research relating algebraic properties of powers of squarefree monomial ideals to combinatorial structures. In this paper, we survey work on secant, symbolic and ordinary powers of squarefree monomial ideals, and their combinatorial consequences in hypergraph theory and linear integer programming. These are the squarefree monomial ideals, that is, the ideals. Abstract in this paper, we prove that the arithmetical rank of a squarefree monomial ideal i is equal to the projective dimension of ri in the following cases.
Powers of ideals are instrumental objects in commutative algebra. In addition, we discuss the equivalence between the conforticornuejols conjecture from linear programming and the question of when symbolic and ordinary powers of squarefree monomial ideals coincide. Arithmetical rank of squarefree monomial ideals of small arithmetic. Cohenmacaulay and s2 properties of the second power of. Bounding the socles of powers of squarefree monomial ideals.
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