Uniform annihilation of local cohomology and powers of ideals generated by quadratic sequences

K. N Raghavan, Purdue University

Abstract

This thesis consists of two independent chapters, whose titles are joined together to form the title of the thesis. Given ideals $I,J$ of a Noetherian ring R, and a finitely generated R-module M, Brodmann has defined as integer $s(I,J,M)$, which in the special case $J$ = $I$ equals the least value of j for which the local cohomology module $H\sbsp{I}{j}(M)$ is not finitely generated. By using the same tools as Brodmann, but taking a global approach, we show, under mild hypothesis on R, that the integer k in Brodmann's theorem may be chosen to be independent of $I,J$. As a corollary we obtain a generalization of a result of Hochster-Huneke about uniform annihilation of Koszul homology. The section on partial results includes a proof of the first non-trivial case of the "local-global principle for annihilation of local cohomology." The second chapter simplifies and extends the theory of d-sequences and weak d-sequences. Our main theorem about d-sequences gives an immediate proof that d-sequences generate ideals of linear type. Moreover, we exploit the idea of its simple proof to generalize results of Costa about sequences of linear type, and to prove Huneke's conjecture that weak d-sequences generate ideals of quadratic type. The proof of the latter suggests the definition of quadratic sequences, which are simpler to define and more general than weak d-sequences, yet amenable, as we demonstrate, to the whole theory of d-sequences and weak d-sequences. Let X be an ideal generated by a quadratic sequence of a ring R. The main result is that, for every natural number m, there exists a filtration of $R/X\sp m$ in which the quotients are cyclic modules defined by "related" ideal of the quadratic sequence. This gives, among other things, a lower bound on the asymptotic value of depth($R/X\sp m$). We show that certain ideals of small analytic deviation, namely those studied recently by Huckaba-Huneke, are generated by weak d-sequences. That these ideals are of quadratic type and other significant results follow. Also, an example is given to show that a Noetherian local ring in which every parameter ideal is of linear type need not be Buchsbaum.

Degree

Ph.D.

Advisors

Huneke, Purdue University.

Subject Area

Mathematics

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