Date of Award

4-30-2018

Degree Type

Dissertation

Degree Name

Doctor of Philosophy (PhD)

Department

Mathematics and Statistics

First Advisor

Yongwei Yao

Second Advisor

Florian Enescu

Third Advisor

Zhongshan Li

Fourth Advisor

Mariana Montiel

Abstract

This dissertation investigates the existence of surjective and split surjective maps between modules. A classic result in this direction is Serre's Splitting Theorem, which gives a lower bound on the maximum rank of a free summand of a finitely generated projective module. Here, the underlying ring is assumed to be commutative and Noetherian with a finite-dimensional maximal spectrum. De Stefani, Polstra, and Yao generalize this theorem by removing the projective condition. Bass extends Serre's Splitting Theorem by considering a module-finite algebra over the ring and replacing the Noetherian condition on the ring with a Noetherian condition on the maximal spectrum. We generalize all of these results by replacing the free summand with a direct sum of copies of a finitely presented module. This requires us to replace rank with a more abstract notion that we call splitting capacity. We generalize Serre, De Stefani-Polstra-Yao, and Bass in a second way by studying the number of summands isomorphic to one module that can appear in a quotient of another module. This is related to the notion of surjective capacity. In the case of finitely generated modules over a Dedekind domain, we show that we can even characterize when a surjective or splitting capacity is equal to a fixed nonnegative integer. In a separate case, we can guarantee when there exists a split surjective map of a special form. This allows us to extend cancellation theorems by Bass and De Stefani-Polstra-Yao, which provide criteria for when two modules with isomorphic direct-sum complements in a larger module are isomorphic. The complement in each of these theorems is assumed to be finitely generated and projective, although each theorem easily reduces to the case in which the complement is a rank-one free module. We show that we can replace the rank-one free module with a finitely presented homothetic module. As a consequence, our work reveals a cancellation property shared by ideals of finite-dimensional Noetherian normal domains and canonical modules of finite-dimensional Cohen-Macaulay rings.

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