Betti sequences over local rings and connected sums of Gorenstein rings

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 of finitely generated modules to be eventually polynomial of degree less than c. When c less than or equal to 3, this property characterizes hypersurface sections of local complete intersections with multiplicity 2b. This is joint work with Avramov and Seceleanu. The second part of the thesis studies a construction on the set of Gorenstein local rings, known as their connected sum. Given a Gorenstein ring, one would like to know whether it can be decomposed as a connected sum and if so, what are its components. We give a concrete description in the case of Gorenstein Artin local algebra over a field. We further investigate conditions on the decomposability of some classes of Gorenstein Artin rings. This is joint work with Hariharan and Celikbas. Adviser: Luchezar Avramo