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Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Schemes over F1 and Zeta Functions
, 2009
"... We determine the real counting function N(q) (q ∈ [1, ∞)) for the hypothetical “curve” C = Spec Z over F1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial F1schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. ..."
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We determine the real counting function N(q) (q ∈ [1, ∞)) for the hypothetical “curve” C = Spec Z over F1, whose corresponding zeta function is the complete Riemann zeta function. Then, we develop a theory of functorial F1schemes which reconciles the previous attempts by C. Soulé and A. Deitmar. Our construction fits with the geometry of monoids of K. Kato, is no longer limited to toric varieties and it covers the case of schemes associated to Chevalley groups. Finally we show, using the monoid of adèle classes over an arbitrary global field, how to apply our functorial theory of Moschemes to interpret conceptually the spectral realization of zeros of Lfunctions.
Lectures On Graded Differential Algebras And Noncommutative Geometry
, 2000
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. LPTORSAY 99/100 Keywords : graded differential alge ..."
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. LPTORSAY 99/100 Keywords : graded differential algebras; categories of algebras; bimodules; noncommutative differential calculus; noncommutative symplectic geometry. MSC classification: 16D20, 18B99, 51P05, 53A99, 81Q99. To be published in the Proceedings of the Workshop on Noncommutative Differential Geometry and its Application to Physics, ShonanKokusaimura, Japan, May 31  June 4, 1999. 1 Unit'e Mixte de Recherche du Centre National de la Recherche Scientifique  UMR 8627 1 Contents 1 Introduction 3 2 Graded differential algebras 10 3 Examples related to Lie algebras 15 4 Examples related to associative algebras 20 5 Categories of algebras 26 6 First order differential calculi 31 7 Higher order differential calculi 36 8 Diagonal an...