Quasi-polynomial Growth of Betti Sequences Over Complete Intersection Rings

This thesis is separated into 3 chapters. Chapter 1 is an introduction which provides standard definitions and notations that shall be used throughout the remainder of the document. In particular, it outlines the classical notions of complete intersection rings and Betti-sequences, as well as summarizes some related classical results and recent previous work done towards the main result of the thesis. Chapter two formalizes the definition of an invariant, called the quadratic codimension of a commutative ring, which has appeared in the literature previously under various notations, but not this nomenclature. Here we investigate the behavior of this invariant and collect several results relating the invariant to other properties of commutative rings. In Chapter three, homological methods are introduced, and the homological perspective through which the proof of the main result will take place is presented. In particular, we discuss the functorality of the homotopy Lie algebra π(R) of a commutative ring and the maps induced on these objects by homomorphisms of local rings. The chapter contains a proof of the existence of a graded subalgbera L of the Yoneda algebra ExtR*(k,k) in a local commutative ring with finite global dimension and with the graded piece L1 of dimension related to quadratic codimension. The chapter closes with an application to Betti sequences over complete intersection rings; a new bound on the maximum degree of the difference of the polynomials governing the even and odd subsequences of the Betti sequence for every finitely generated module is presented