Showing papers on "Ringed space published in 2005"
TL;DR: In this paper, the authors define the spectrum of a tensor triangulated category K as the set of so-called prime ideals, endowed with a suitable topology, and establish in complete generality a classification of thick ⊗-ideal subcategories in terms of arbitrary unions of closed subsets with quasicompact complements.
Abstract: We define the spectrum of a tensor triangulated category K as the set of so-called prime ideals, endowed with a suitable topology. In this very generality, the spectrum is the universal space in which one can define supports for objects of K. This construction is functorial with respect to all tensor triangulated functors. Several elementary properties of schemes hold for such spaces, e.g. the existence of generic points or some quasi-compactness. Locally trivial morphisms are proved to be nilpotent. We establish in complete generality a classification of thick ⊗-ideal subcategories in terms of arbitrary unions of closed subsets with quasi-compact complements (Thomason’s theorem for schemes, mutatis mutandis). We also equip this spectrum with a sheaf of rings, turning it into a locally ringed space. We compute examples and show that our spectrum unifies the schemes of algebraic geometry and the support varieties of modular representation theory.
320 citations
TL;DR: In this article, it was shown that the category of sheaves of modules on a variety/manifold/ringed space is an abelian category with enough injectives by the construction of an explicit injective resolution.
Abstract: Several important cases of vector bundles with extra structure (such as Higgs bundles and triples) may be regarded as examples of twisted representations of a finite quiver in the category of sheaves of modules on a variety/manifold/ringed space. It is shown that the category of such representations is an abelian category with enough injectives by the construction of an explicit injective resolution. Using this explicit resolution, a long exact sequence is found that computes the Ext groups in this new category in terms of the Ext groups in the old category. The quiver formulation is directly reflected in the form of the long exact sequence. It is also shown that under suitable circumstances, the Ext groups are isomorphic to certain hypercohomology groups.
100 citations
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TL;DR: In this paper, it was shown that the deformations of the abelian category Mod(O) of sheaves of O-modules are obtained from algebroid prestacks, as introduced by Kontsevich.
Abstract: For a ringed space (X,O), we show that the deformations of the abelian category Mod(O) of sheaves of O-modules are obtained from algebroid prestacks, as introduced by Kontsevich. In case X is a quasi-compact separated scheme the same is true for Qch(O), the category of quasi-coherent sheaves on X. It follows in particular that there is a deformation equivalence between Mod(O) and Qch(O).
21 citations