On the Strong Direct Summand Conjecture

In this thesis, our aim is the study the Vanishing of Maps of Tor Conjecture of Hochster and Huneke. We mainly focus on an equivalent characterization called the Strong Direct Summand Conjecture, due to N. Ranganathan. Our results are separated into three chapters. In Chapter 3, we prove special cases of the Strong Direct Summand Conjecture in mixed characteristic using knowledge about splittings in lower dimensions. In particular, we show that the vanishing of the first Ext module allows us to lift a splitting in lower dimension and prove the Strong Direct Summand Conjecture. In Chapter 4, we study the related Strong Monomial Conjecture. Extending work of Dutta, we give several reformulations of the Strong Monomial Conjecture and prove the Strong Monomial Conjecture for systems of parameters of a certain form. In Chapter 5, we present a generalization of the Vanishing Maps of Tor Conjecture and prove the equicharacteristic case. We then give a shorter proof of the Strong Direct Summand Conjecture.

Name thesis:degree_name

Ph.D.

Level thesis:degree_level

Dissertation

Discipline thesis:degree_discipline

Mathematics

Grantor

University of Illinois at Urbana-Champaign

Year dc:date

2015

Author dc:creator

• McCullough, Jason

Contributors dc:contributor

• Griffith, Phillip A.

• Theoretical Mathematics

Language dc:language

eng

Identifier

(MiAaPQ)AAI3392213

OAI identifier oai:identifier

oai:www.ideals.illinois.edu:2142/86931