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Local cohomology and numerical semigroup rings
Let R be a Noetherian ring, I an ideal of R, and M a finitely generated R-module. The i-th local cohomology module of M with respect to I, HiI (M), is of considerable interest, as it has many applications in commutative algebra and algebraic geometry. An important open question on local cohomology is whether local cohomology modules have a finite number of associated primes. In chapter 2 we answer this question in the affirmative for certain special cases; these include the first local cohomology module which is not I-cofinite, for all modules over a four-dimensional UFD, and for first syzygy modules over a ring which is equidimensional, unmixed, and admits a canonical module. ^ In chapters 3 and 4 we apply facts about local cohomology, as well as other methods, to examine the associated graded ring of R = k[ &sqbl0;ta1,&ldots;,tan &sqbr0;m , where m is the homogeneous maximal ideal. In chapter 3 we study the Cohen-Macaulay property. We give necessary and sufficient conditions for the associated graded ring of R to be Cohen-Macaulay in the case where the embedding dimension is three and sufficient conditions for larger embedding dimension. We also give sufficient conditions for the associated graded ring to be not Cohen-Macaulay for larger embedding dimension. In chapter 4 we study the Buchsbaum property. We give sufficient conditions for the associated graded ring of R to be Buchsbaum in embedding dimension three. We also give sufficient conditions for the associated graded ring of R to be not Buchsbaum. Lastly, we investigate necessary conditions in embedding dimension three for the associated graded ring to be Buchsbaum. ^ In chapter 5 we describe the topics in chapters 3 and 4 in the more general setting of one-dimensional analytically irreducible local domains and conjecture whether the results in these chapters are valid in this context. We also give some interesting examples of rings whose associated graded rings exhibit various properties using the results in chapters 3 and 4, and give a conjecture. Lastly, we give a counterexample to a conjecture of Corso, et al. and restate their conjecture, adding an additional hypothesis. ^
Sapko, Victoria Ann, "Local cohomology and numerical semigroup rings" (2001). ETD collection for University of Nebraska - Lincoln. AAI3009736.