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Theses and Dissertations

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Browsing Theses and Dissertations by Author "Achter, Jeffrey D., committee member"

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    A simplicial homotopy group model for K2 of a ring
    (Colorado State University. Libraries, 2010) Whitfield, JaDon Saeed, author; Duflot, Jeanne, advisor; Miranda, Rick, committee member; Achter, Jeffrey D., committee member; Gelfand, Martin Paul, committee member
    We construct an isomorphism between the group K2(R) from classical, algebraic K-Theory for a ring R and a simplicial homotopy group constructed using simplicial homotopy theory based on that same ring R. First I describe the basic aspects of simplicial homotopy theory. Special attention is paid to the use of category theory, which will be applied to the construction of a simplicial set. K-Theory for K0(R), K1(R) and K2(R) is then described before we set to work describing explicitly the nature of isomorphisms for K0(R) and K1(R) based on previous work. After introducing some theory related to K-Theory, some considerations and corrections on previous work motivate more new theory that helps the isomorphism with K2(R). Such theory is developed, mainly with regards to finitely generated projective modules over R and then elementary matrices with entries from R, culminating in the description of the Steinberg Relations that are central to the understanding of K2(R) in terms of homotopy classes. We then use new considerations on the previous work to show that a map whose image is constructed through this article is an isomorphism since it is the composition of isomorphisms.
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    Artin-Schreier curves
    (Colorado State University. Libraries, 2010) Farnell, Shawn, author; Pries, Rachel, advisor; Achter, Jeffrey D., committee member; Peterson, Christopher Scott, 1963-, committee member; Gelfand, Martin Paul, committee member
    Let k be an algebraically closed field of characteristic p where p is a prime number. The main focus of this work is on properties of Artin-Schreier curves. In particular, we study two invariants of the p-torsion of the Jacobian of these curves: the p-rank and the a-number. In the main result, we demonstrate a family of Artin-Schreier curves for which the a-number is constant. We also give a result concerning the existence of deformations of Artin-Schreier curves with varying p-rank.