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Title:  Adams Operations and the Dennis Trace Map 
Author(s):  Kantorovitz, Miriam Ruth 
Doctoral Committee Chair(s):  McCarthy, Randy 
Department / Program:  Mathematics 
Discipline:  Mathematics 
Degree Granting Institution:  University of Illinois at UrbanaChampaign 
Degree:  Ph.D. 
Genre:  Dissertation 
Subject(s):  Mathematics 
Abstract:  For a commutative algebra A, the algebraic Ktheory of A, K*(A), and the Hochschild homology of A, HH*(A), are graded rings, and the Dennis trace map D : K *(A) → HH*( A) is a graded ring map. Since Hochschild homology is a more user friendly theory than the algebraic Ktheory, one would like to use the Dennis trace map to study the algebraic Ktheory via Hochschild homology. For example, this idea was used by Geller and Weibel to give a counterexample to a conjecture of Beilinson and Soule on the vanishing of certain components of K*( A). To further study the algebraic Ktheory via the Dennis trace map, one would like to know what additional structure the Dennis trace map preserves. In the first part of this thesis we prove a conjecture of Loday, Geller and Weibel that rationally, the Dennis trace map preserves the Adams operations and the Hodge decomposition. In the second part of the thesis we give a tool for comparing the Adams operations on Ktheory with the ones on Hochschild homology in the non rational case. We do so by giving a formula for the Dennis trace map, as a map from a split version of the Sconstruction model for Ktheory to additive cyclic nerve model of Hochschild homology. The motivation to find such a formula is Grayson's explicit description of the Adams operations on the Sconstruction for Ktheory and McCarthy's explicit description of the Adams operations on the additive cyclic nerve complex for Hochschild homology. 
Issue Date:  1999 
Type:  Text 
Language:  English 
Description:  47 p. Thesis (Ph.D.)University of Illinois at UrbanaChampaign, 1999. 
URI:  http://hdl.handle.net/2142/86978 
Other Identifier(s):  (MiAaPQ)AAI9944906 
Date Available in IDEALS:  20150928 
Date Deposited:  1999 
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Dissertations and Theses  Mathematics

Graduate Dissertations and Theses at Illinois
Graduate Theses and Dissertations at Illinois