There are at least two Alexander Rosenbergs (and one Jonathan Rosenberg) in mathematics:
Alexander L. Rosenberg (or Sasha Rosenberg) who was a professor until 2012 at Kansas State University.
Sasha Rosenberg had defended in 1973 dissertation at Moscow State University studying Tannakian reconstruction theorems, mainly using methods of functional analysis. His main goals were in representation theory. Until leaving Soviet Union around 1987, Rosenberg worked in applied mathematics; on the side however he was developing some methods in representation theory which included functional analysis and categorical and ring/module theoretic methods, and the noncommutative localization in particular. A main construction, presented at a conference at Baikal (1981), was a new spectrum of a ring, so-called left spectrum, later generalized to a spectrum of an abelian category, used to prove (in the quasicompact case) the Gabriel-Rosenberg reconstruction theorem for commutative schemes; more spectral constructions followed. This work outgrown into a wide ongoing work on foundations of noncommutative algebraic geometry, including a natural definition of noncommutative scheme. More recently he proposed a new framework for nonabelian derived functors, including a new version of algebraic K-theory. Other related entries in Lab include Q-category, neighborhood of a topologizing subcategory, sheaf on a noncommutative space, spectral cookbook.
Rosenberg’s coauthors in pure mathematics works are Valery Lunts and Maxim Kontsevich. One should be warned that most of the newly released articles of Rosenberg are not at the arXiv but at the Max Planck Bonn preprint server. Rosenberg’s available papers and online resources include:
3 volume work on noncommutative geometry: Geometry of Noncommutative ‘Spaces’ and Schemes, pdf; Homological algebra of noncommutative ‘spaces’ I., pdf (note: longer version than another MPI preprint with the same title mentioned before); Noncommutative ‘Spaces’ and ‘Stacks’, pdf
video of msri lecture Noncommutative schemes and spaces (Feb 2000) can be found at msri
A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf
A. L. Rosenberg, Almost quotient categories, sheaves and localizations, 181 p. Seminar on supermanifolds 25, University of Stockholm, D. Leites editor, 1988 (in Russian; partial remake in English exists)
A. L. Rosenberg, Non-commutative affine semischemes and schemes, Seminar on supermanifolds 26, Dept. Math., U. Stockholm (1988)
A. L. Rosenberg, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9
A. L. Rosenberg, Reconstruction of groups, Selecta Math. N.S. 9:1 (2003) doi (lab remark: this paper is on a generalization of Tannaka–Krein and not of the Gabriel–Rosenberg kind of reconstruction)
A. L. Rosenberg, The left spectrum, the Levitzki radical, and noncommutative schemes, Proc. Nat. Acad. Sci. U.S.A. 87 (1990), no. 21, 8583–8586.
A. L. Rosenberg, Noncommutative local algebra, Geom. Funct. Anal. 4 (1994), no. 5, 545–585.
A. L. Rosenberg, The existence of fiber functors, The Gelfand Mathematical Seminars, 1996–1999, 145–154, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000.
A. L. Rosenberg, 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, 257–274, 1998; MR99d:18011; and Max Planck Bonn preprint Reconstruction of Schemes, MPIM1996-108 (1996).
A. L. Rosenberg, Noncommutative schemes, Compos. Math. 112 (1998) 93–125 (doi)
V. A. Lunts, A. L. Rosenberg, Differential operators on noncommutative rings, Selecta Math. (N.S.) 3 (1997), no. 3, 335–359 (doi); sequel: Localization for quantum groups, Selecta Math. (N.S.) 5 (1999), no. 1, pp. 123–159 (doi).
V. A. Lunts, A. L. Rosenberg, Differential calculus in noncommutative algebraic geometry I. D-calculus on noncommutative rings, MPI 1996-53 pdf, II. D-Calculus in the braided case. The localization of quantized enveloping algebras, MPI 1996-76 pdf
M. Kontsevich, A. Rosenberg, Noncommutative smooth spaces, The Gelfand Mathematical Seminars, 1996–1999, 85–108, Gelfand Math. Sem., Birkhäuser Boston, Boston, MA, 2000; (arXiv:math/9812158)
A. Rosenberg, Homological algebra of noncommutative ‘spaces’ I, 199 pages, preprint Max Planck, Bonn: MPIM2008-91.