Alexander L. Rosenberg

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Alexander Lwowitsch Rosenberg , Russian Александр Львович Розенберг (* 1946 ; † 2012 ), was a Soviet-American mathematician.

Rosenberg received his doctorate in 1973 from Lomonossow University under Yuri Manin . He left the Soviet Union around 1987, first went to Stockholm University and was most recently a professor at Kansas State University .

He dealt with functional analysis and representation theory (including contributions to the Tannaka duality ) and non-commutative algebraic geometry . A fundamental result is the reconstruction theorem of Pierre Gabriel (1962, proof under restrictive assumptions) and Rosenberg: apart from isomorphism, a commutative scheme is determined by the Abelian category of quasi-coherent sheaves on the scheme. For the proof, Rosenberg used his concept of the spectrum of an Abelian category as a topological space with a stack of local categories and his construction of a geometric center of this space, so that a topological space with a sheaf of local commutative rings with one element results. This concept of the spectrum of an Abelian category (Rosenberg's spectrum) in turn builds on his earlier construction of the left spectrum of a ring, which he first presented at a conference in Baikal in 1981. In 2003 he published a new version of his spectrum, so that the reconstruction theorem applies without a quasi-compactness condition, which he had to introduce due to a counterexample from Ofer Gabber . This led to his occupation with non-commutative algebraic geometry, including a definition of a non-commutative scheme. Most recently he gave a theory of non-Abelian derived functors including a new version of algebraic K-theory.

He is not to be confused with the mathematician Alex Rosenberg .

Fonts (selection)

  • Noncommutative Affine Semischemes and Schemes, "Seminar on Supermanifolds", Vol. 26, pp. 1-317. Reports of Dept. of Math, of Stockholm Univ. 1988
  • The left spectrum, the Levitzki radical, and noncommutative schemes, Proc. Nat. Acad. Sci. USA, Vol. 87, 1990, pp. 8583-8586. PMC 55001 (free full text)
  • The Spectrum of the Algebra of Skew Differential Operators and the Irreducible Representations of the Quantum Heisenberg Algebra. Commun. Math. Phys. Vol. 142, 1991, pp. 567-588, Project Euclid
  • The unitary irreducible representations of the quantum Heisenberg algebra, Comm. Math. Phys. Vol. 144, 1992, pp. 41-51, Project Euclid
  • Noncommutative local algebra, Geom. Funct. Anal., Vol. 4, 1994, pp. 545-585
  • Noncommutative algebraic geometry and representations of quantized algebras, Kluwer 1995, ISBN 0-7923-3575-9
  • with M. Kontsevich : Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996-1999, 85-108, Gelfand Math. Sem., Birkhäuser, Boston 2000, pp. 85-108, arxiv : math / 9812158
  • The spectrum of abelian categories and reconstructions of schemes, in: Rings, Hopf Algebras, and Brauer groups, Lectures Notes in Pure and Appl. Math. 197, Marcel Dekker, New York, 1998, pp. 257–274 (and preprint Max Planck Institute for Mathematics 1996-108)
  • Noncommutative schemes, Compositio Mathematica, Volume 112, 1998, pp. 93-125
  • Selected Papers on noncommutative geometry, New Prairie Press 2014, Online

Preprints at the Max Planck Institute for Mathematics Bonn:

  • Underlying spaces of noncommutative schemes, MPIM 2003-111,
  • with Kontsevich: Noncommutative spaces, preprint MPI-2004-35
  • with Kontsevich: Noncommutative spaces and flat descent, MPI-2004-36
  • with Kontsevich: Noncommutative stacks, MPI-2004-37
  • Homological algebra of noncommutative 'spaces' I, MPIM 2008-91.
  • Topics in noncommutative algebraic geometry, homological algebra and K-theory, MPIM Bonn 2008-57

Web links

Individual evidence

  1. ↑ Alexander L. Rosenberg in the Mathematics Genealogy Project (English)Template: MathGenealogyProject / Maintenance / id used
  2. ^ Gabriel-Rosenberg theorem , nLab
  3. ↑ Preprints MPIM . Also in his Selected Papers on noncommutative geometry, New Prairie Press 2014
personal data
SURNAME Rosenberg, Alexander L.
ALTERNATIVE NAMES Rosenberg, Alexander Lwowitsch
BRIEF DESCRIPTION Soviet-American mathematician
DATE OF BIRTH 1946
DATE OF DEATH 2012