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Title:On the Strong Direct Summand Conjecture
Author(s):McCullough, Jason
Doctoral Committee Chair(s):Griffith, Phillip A.
Department / Program:Mathematics
Discipline:Mathematics
Degree Granting Institution:University of Illinois at Urbana-Champaign
Degree:Ph.D.
Genre:Dissertation
Subject(s):Theoretical Mathematics
Abstract: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.
Issue Date:2009
Type:Text
Language:English
Description:59 p.
Thesis (Ph.D.)--University of Illinois at Urbana-Champaign, 2009.
URI:http://hdl.handle.net/2142/86931
Other Identifier(s):(MiAaPQ)AAI3392213
Date Available in IDEALS:2015-09-28
Date Deposited:2009


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