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Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang
Polynomial Growth Of Betti Sequences Over Local Rings, Luchezar L. Avramov, Alexandra Seceleanu, Zheng Yang
Department of Mathematics: Faculty Publications
We study sequences of Betti numbers (βRi (M)) of finite modules M over a complete intersection local ring, R. It is known that for every M the subsequence with even, respectively, odd indices i is eventually given by some polynomial in i. We prove that these polynomials agree for all R-modules if the ideal I☐ generated by the quadratic relations of the associated graded ring of R satisfies height I☐ ≥ codim R − 1, and that the converse holds when R is homogeneous and when codim R ≤ 4. Avramov, …
The Derived Category Of A Locally Complete Intersection Ring, Joshua Pollitz
The Derived Category Of A Locally Complete Intersection Ring, Joshua Pollitz
Department of Mathematics: Dissertations, Theses, and Student Research
Let R be a commutative noetherian ring. A well-known theorem in commutative algebra states that R is regular if and only if every complex with finitely generated homology is a perfect complex. This homological and derived category characterization of a regular ring yields important ring theoretic information; for example, this characterization solved the well-known ``localization problem" for regular local rings. The main result of this thesis is establishing an analogous characterization for when R is locally a complete intersection. Namely, R is locally a complete intersection if and only if each nontrivial complex with finitely generated homology can build a …
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 …