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Full-Text Articles in Physical Sciences and Mathematics
Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo
Lower Bounds On Betti Numbers, Adam Boocher, Eloisa Grifo
Department of Mathematics: Faculty Publications
We survey recent results on bounds for Betti numbers of modules over polynomial rings, with an emphasis on lower bounds. Along the way, we give a gentle introduction to free resolutions and Betti numbers, and discuss some of the reasons why one would study these.
Betti Sequences Over Local Rings And Connected Sums Of Gorenstein Rings, Zheng Yang
Betti Sequences Over Local Rings And Connected Sums Of Gorenstein Rings, Zheng Yang
Department of Mathematics: Dissertations, Theses, and Student Research
This thesis consists of two parts:
1) Polynomial growth of Betti sequences over local rings (Chapter 2),
2) Connected sums of Gorenstein rings (Chapter 3).
Chapter 1 gives an introduction for the two topics discussed in this thesis.
The first part of the thesis deals with modules over complete intersections using free resolutions. The asymptotic patterns of the Betti sequences of the finitely generated modules over a local ring R reflect and affect the singularity of R. Given a commutative noetherian local ring and an integer c, sufficient conditions and necessary conditions are obtained for all Betti sequences …