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dc.contributor.advisorHuneke, Craig
dc.contributor.authorKummini, Neelakandhan Manoj
dc.date.accessioned2008-09-15T04:36:20Z
dc.date.available2008-09-15T04:36:20Z
dc.date.issued2008-08-19
dc.date.submitted2008
dc.identifier.otherhttp://dissertations2.umi.com/ku:2656
dc.identifier.urihttp://hdl.handle.net/1808/4199
dc.description.abstractIn 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.
dc.format.extent112 pages
dc.language.isoEN
dc.publisherUniversity of Kansas
dc.rightsThis item is protected by copyright and unless otherwise specified the copyright of this thesis/dissertation is held by the author.
dc.subjectMathematics
dc.subjectCommutative algebra
dc.subjectHomological invariants
dc.subjectFree resolutions
dc.titleHomological Invariants of Monomial and Binomial Ideals
dc.typeDissertation
dc.contributor.cmtememberAlexander, Perry
dc.contributor.cmtememberBayer, Margaret
dc.contributor.cmtememberKatz, Daniel
dc.contributor.cmtememberMartin, Jeremy
dc.thesis.degreeDisciplineMathematics
dc.thesis.degreeLevelPH.D.
kusw.oastatusna
kusw.oapolicyThis item does not meet KU Open Access policy criteria.
kusw.bibid6857209
dc.rights.accessrightsopenAccess


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