Perfect complexes on Deligne-Mumford stacks and applications

Krishna, Amalendu (2009) Perfect complexes on Deligne-Mumford stacks and applications Journal of K-Theory, 4 (3). pp. 559-603. ISSN 1865-2433

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For a tame Deligne-Mumford stack X with the resolution property, we show that the Cartan-Eilenberg resolutions of unbounded complexes of quasicoherent sheaves are K-injective resolutions. This allows us to realize the derived category of quasi-coherent sheaves on X as a reflexive full subcategory of the derived category of OX-modules. We then use the results of Neeman and recent results of Kresch to establish the localization theorem of Thomason-Trobaugh for the K-theory of perfect complexes on stacks of above type which have coarse moduli schemes. As a byproduct, we get a generalization of Krause's result about the stable derived categories of schemes to such stacks.We prove Thomason's classification of thick triangulated tensor subcategories of D(perf / X). As the final application of our localization theorem, we show that the spectrum of D(perf / X) as defined by Balmer, is naturally isomorphic to the coarse moduli scheme of X, answering a question of Balmer for the tensor triangulated categories arising from Deligne-Mumford stacks.

Item Type:Article
Source:Copyright of this article belongs to Cambridge University Press.
Keywords:Algebraic Stacks; Algebraic K-Theory; Derived Category
ID Code:102462
Deposited On:09 Mar 2018 10:47
Last Modified:09 Mar 2018 10:47

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