2026-09-19T15:01:17Zhttps://dspace.mit.edu/server/oai/requestoai:dspace.mit.edu:1721.1/836922022-01-13T07:54:04Zcom_1721.1_7582com_1721.1_7581col_1721.1_131022
Jacob Lurie.
Clausen, Dustin (Dustin Tate)
Massachusetts Institute of Technology. Department of Mathematics.
Massachusetts Institute of Technology. Department of Mathematics
2014-01-09T19:45:30Z
2014-01-09T19:45:30Z
2013
http://hdl.handle.net/1721.1/83692
864023117
Thesis (Ph. D.)--Massachusetts Institute of Technology, Department of Mathematics, 2013.
Cataloged from PDF version of thesis.
Includes bibliographical references (pages 37-38).
Let X be a regular arithmetic curve or point (meaning a regular separated scheme of finite type over Z which is connected and of Krull dimension </= 1). We define a compactly-supported variant Kc(X) of the algebraic K-theory spectrum K(X), and establish the basic functoriality of Kc. Briefly, K, behaves as if it were dual to K. Then we give this duality some grounding: for every prime t invertible on X, we define a natural l-adic pairing between Kc(X) and K(X). This pairing is of an explicit homotopy-theoretic nature, and reflects a simple relation between spheres, tori, and real vector spaces. Surprisingly, it has the following two properties: first (a consequence of work of Rezk), when one tries to compute it the e-adic logarithm inevitably appears; and second, it can be used to give a new description of the global Artin map, one which makes the Artin reciprocity law manifest.
by Dustin Clausen.
Ph.D.
38 pages
eng
Massachusetts Institute of Technology
M.I.T. theses are protected by
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http://dspace.mit.edu/handle/1721.1/7582
Mathematics.
Arithmetic duality in algebraic K-theory
Thesis
application/pdf
Publication
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unknown
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<Abstract>Let X be a regular arithmetic curve or point (meaning a regular separated scheme of finite type over Z which is connected and of Krull dimension </= 1). We define a compactly-supported variant Kc(X) of the algebraic K-theory spectrum K(X), and establish the basic functoriality of Kc. Briefly, K, behaves as if it were dual to K. Then we give this duality some grounding: for every prime t invertible on X, we define a natural l-adic pairing between Kc(X) and K(X). This pairing is of an explicit homotopy-theoretic nature, and reflects a simple relation between spheres, tori, and real vector spaces. Surprisingly, it has the following two properties: first (a consequence of work of Rezk), when one tries to compute it the e-adic logarithm inevitably appears; and second, it can be used to give a new description of the global Artin map, one which makes the Artin reciprocity law manifest.</Abstract>
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