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    Homological Invariants of Monomial and Binomial Ideals

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    umi-ku-2656_1.pdf (665.1Kb)
    Issue Date
    2008-08-19
    Author
    Kummini, Neelakandhan Manoj
    Publisher
    University of Kansas
    Format
    112 pages
    Type
    Dissertation
    Degree Level
    PH.D.
    Discipline
    Mathematics
    Rights
    This item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
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    Abstract
    In this dissertation, we study numerical invariants of minimal graded free resolutions of homogeneous ideals in a polynomial ring R. Chapters 2, 3 and 4 deal with homological invariants of edge ideals of bipartite graphs. First, in Chapter 2, we relate regularity and depth of bipartite edge ideals to combinatorial invariants of the graphs. Chapter 3 discusses arithmetic rank, and shows that some classes of Cohen-Macaulay bipartite edge ideals define set-theoretic complete intersections. It is known, due to G. Lyubeznik, that arithmetic rank of a square-free monomial ideal I is at least the projective dimension of R/I. As an application of the results in Chapter 2, we show in Chapter 4 that the multiplicity conjectures of J. Herzog, C. Huneke and H. Srinivasan hold for bipartite edge ideals, and that if the conjectured bounds hold with equality, then the ideals are Cohen-Macaulay and has a pure resolution. Chapter 5 describes joint work with G. Caviglia, showing that any upper bound for projective dimension of an ideal supported on N monomials counted with multiplicity is at least 2N/2. We give the example of a binomial ideal, whose projective dimension grows exponentially with respect to the number of monomials appearing in a set of generators. Finally, in Chapter 6, we study Alexander duality, giving an alternate proof of a theorem of K. Yanagawa which states that for a square-free monomial ideal I, R/I has Serre's property (Si) if and only if its Alexander dual has a linear resolution up to homological degree i. Further, if R/I has property (S2) , then it is locally connected in codimension 1.
    URI
    http://hdl.handle.net/1808/4199
    Collections
    • Dissertations [4660]
    • Mathematics Dissertations and Theses [179]

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    785-864-8983
    KU Libraries
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    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
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    Contact KU ScholarWorks
    785-864-8983
    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    785-864-8983

    KU Libraries
    1425 Jayhawk Blvd
    Lawrence, KS 66045
    Image Credits
     

     

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