The spectrum of an abelian category (or Rosenberg’s spectrum) is a set attached to a svelte (equivalent to small) abelian category generalizing the left spectrum of a noncommutative ring. It can be equipped with natural Zariski topology and a stack of local abelian categories, similar to the picture of locally ringed spaces. It has been invented by A. L. Rosenberg in 1980-s.
Let $A$ be a svelte abelian category. One first defines a preorder $\succ$ on the class of nonzero objects in $A$ by setting $M\succ N$ if $N$ is a subquotient of the direct sum of finitely many copies of $M$. In other words, $M\succ N$ if there exists a positive number $k$ and a subobject $U$ of a direct sum $\oplus_{i = 1}^k M$ of $k$ copies of $M$ and an epimorphism from $U$ to $N$.
As a set, spectrum $\mathbf{Spec}\,A$ of an abelian category $A$, can be described as a set of equivalence classes of topologizing subcategories $[M]$ generated by objects $M$ in $A$ such that $L\succ M$ for any nonzero subobject $L$ of $M$, equipped with a specialization preorder.
If $A = Qcoh(X)$ where $X$ is the abelian category of quasicoherent sheaves of $\mathcal{O}_X$ modules on a quasiseparated? quasicompact scheme, then the scheme $X$ can be reconstructed up to isomorphisms in two steps. First one constructs the spectrum $\mathbf{Spec} Qcoh(X)$ as a topological space with a stack of local abelian categories. Then one does fibrewise the construction of a center of an abelian category, that is replacing each fibre (which is an abelian category) by the commutative ring of endotransformations of the identity functor and sheafifies. The Rosenberg’s version of the Gabriel-Rosenberg reconstruction of commutative schemes states that the result is isomorphic to the original scheme. One can generalize to arbitrary schemes possessing a cover by affines whose inclusions have a direct image functor, but then one needs to use a different spectrum.
Alexander Rosenberg, Geometry of Noncommutative ‘Spaces’ and Schemes, MPIM 2011-68, pdf
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A. L. Rosenberg, Topics in noncommutative algebraic geometry, homological algebra and K-theory, preprint MPIM Bonn 2008-57 pdf
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).
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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, Noncommutative algebraic geometry and representations of quantized algebras, MIA 330, Kluwer Academic Publishers Group, Dordrecht, 1995. xii+315 pp. ISBN: 0-7923-3575-9
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