Date of Award

12-2017

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics

Committee Chair

Giulio Caviglia

Committee Member 1

William J. Heinzer

Committee Member 2

Bernd Ulrich

Committee Member 3

Hans U. Walther

Abstract

This dissertation deals with questions concerning ideals and modules over graded or local noetherian commutative rings. The main results deal with the quantitative analysis of free resolutions, with particular emphasis on infinite ones, and with the structure of blowup algebras of certain determinantal ideals. In Chapter 3 we consider resolutions of ideals containing a regular sequence of powers of variables. Let S be a polynomial ring and R a complete intersection in S defined by pure powers of the variables. We prove that, in the Hilbert scheme parametrizing the closed subschemes of Proj(R) with a fixed Hilbert polynomial ℘, there exists a point whose saturated ideal I achieves the largest possible betti numbers in the finite resolution of R/I over S and, when the relations of R are quadratic, the largest possible betti numbers in the infinite resolution of R/I over R. This chapter is based on joint work with Giulio Caviglia. In Chapter 4 we investigate the infinite resolution of the residue field of a graded algebra R through a sequence of integers known as the ‘deviations’ of R. Deviations arise as the number of generators of certain DG algebra resolutions. We prove that deviations, among those of algebras with a fixed Hilbert series, are maximized by the lexsegment ideal. We also prove that the sequence of deviations grows exponentially for Golod rings and for certain Koszul algebras. This chapter is based on joint work with Adam Boocher, Alessio D’Al`ı, Elo´ısa Grifo, and Jonathan Monta˜no. In Chapter 5 we study how free resolutions and Tor modules change when passing from modules over a noetherian local ring R to some associated graded modules. Let M,N be finite R-modules equipped with filtrations that are stable with respect to the maximal ideal of R. We describe a tight numerical relationship between the bigraded Hilbert series of gr(TorR(M,N)) and the one of Torgr(R)(gr(M), gr(N)). In Chapter 6 we study the Rees ring and special fiber ring of the ideal of a rational normal scroll in projective space. We determine squarefree Gr¨obner bases of quadrics for the defining ideals of these blowup algebras. A notable consequence is the fact that the Rees ring and the special fiber ring of rational normal scrolls are Koszul algebras, i.e. the residue field has linear graded free resolutions over these rings.

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