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Generic and Cogeneric Monomial Ideals

✍ Scribed by Ezra Miller; Bernd Sturmfels; Kohji Yanagawa

Publisher
Elsevier Science
Year
2000
Tongue
English
Weight
346 KB
Volume
29
Category
Article
ISSN
0747-7171

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✦ Synopsis

Monomial ideals which are generic with respect to either their generators or irreducible components have minimal free resolutions encoded by simplicial complexes. There are numerous equivalent ways to say that a monomial ideal is generic or cogeneric. For a generic monomial ideal, the associated primes satisfy a saturated chain condition, and the Cohen-Macaulay property implies shellability for both the Scarf complex and the Stanley-Reisner complex. Reverse lexicographic initial ideals of generic lattice ideals are generic. Cohen-Macaulayness for cogeneric ideals is characterized combinatorially; in the cogeneric case, the Cohen-Macaulay type is greater than or equal to the number of irreducible components. Methods of proof include Alexander duality and Stanley's theory of local h-vectors.

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