nLab
Gabriel-Rosenberg theorem

Gabriel-Rosenberg reconstruction theorem: an algebraic scheme can be reconstructed up to an isomorphism from the abelian category of quasicoherent sheaves over that scheme. If a scheme is quasicompact quasiseparated, then $X$ is the ‘geometric center’ of the category of quasicoherent sheaves ${\mathrm{Qcoh}}_X$. The geometric center is a construction obtained as a composition of two functors: first Rosenberg’s spectrum of an abelian category as a topological space with a stack of local categories and then a certain construction of a center, which will yield a topological space with a sheaf of local commutative unital rings.

History and versions

This theorem has been, under some restrictions proved, in Pierre Gabriel’s thesis and subsequent article Des catégories abéliennes using the notion of indecomposable spectrum (or Gabriel's spectrum of indecomposables?). The case of general quasicompact schemes is proved in Max Planck 1996 preprint (pdf) of Alexander Rosenberg published in 1998 (a short sketch is also in the 1998 paper Noncommutative schemes of Rosenberg) using his construction of a spectrum of an abelian category generalizing straightforwardly his 1982 construction of a left spectrum of a ring). The quasicompactness condition has been noted in this construction after a counterexample exhibited by Ofer Gabber. Few years later (see 2003 MPI preprint), Rosenberg got rid of quasicompactness condition inventing yet another notion of a spectrum.

Literature

• Pierre Gabriel, Des catégories abéliennes, Bulletin de la Société Mathématique de France, 90 (1962), p. 323-448 (numdam)

• 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, pdf (1996).

• A. L. Rosenberg, Spectra of noncommutative spaces, MPIM2003-110 ps dvi (2003)

• A. L. Rosenberg, Noncommutative schemes, Compos. Math. 112 (1998) 93–125 (doi)

• 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 algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9

• related MO question: what is the center of QCohX

category: algebraic geometry, noncommutative geometry