Ringed space

About: Ringed space is a research topic. Over the lifetime, 93 publications have been published within this topic receiving 5874 citations.

Éléments de géométrie algébrique

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The spectrum of prime ideals in tensor triangulated categories

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.

On the cyclic homology of ringed spaces and schemes.

Abstract: We prove that the cyclic homology of a scheme with an ample line bundle coincides with the cyclic homology of its category of algebraic vector bundles. As a byproduct of the proof, we obtain a new construction of the Chern character of a perfect complex on a ringed space.

Model category structures on chain complexes of sheaves

Abstract: In this paper, we try to realize the unbounded derived category of an abelian category as the homotopy category of a Quillen model structure on the category of unbounded chain complexes. We construct such a model structure based on injective resolutions for an arbitrary Grothendieck category, as has apparently also been done by Morel. In particular, this works for sheaves on a ringed space, and for quasi-coherent sheaves on a quasi-compact, quasi-separated scheme. However, this injective model structure is not well suited to studying the derived tensor product, so we investigate other model structures. The most successful of these is the flat model structure on complexes of sheaves over a ringed space. This is based on flat resolutions, and is compatible with the tensor product. As a corollary, we get model categories of differential graded algebras of sheaves and differential graded modules over a given differential graded algebra of sheaves. This is the author's first attempt to understand sheaves, so comments from those more experienced with the subject are welcome.