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Journal Article

K-Motives and Koszul Duality

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Eberhardt,  Jens Niklas
Max Planck Institute for Mathematics, Max Planck Society;

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arXiv:1909.11151.pdf
(Preprint), 271KB

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Citation

Eberhardt, J. N. (in press). K-Motives and Koszul Duality. Bulletin of the London Mathematical Society, Published Online - Print pending. doi:10.1112/blms.12691.


Cite as: http://hdl.handle.net/21.11116/0000-000A-AF79-A
Abstract
We construct an ungraded version of Beilinson-Ginzburg-Soergel's Koszul duality for Langlands dual flag varieties, inspired by Beilinson's construction of rational motivic cohomology in terms of $K$-theory. For this, we introduce and study categories $\operatorname{DK}_{\mathcal{S}}(X)$ of $\mathcal{S}$-constructible $K$-motivic sheaves on varieties $X$ with an affine stratification $\mathcal{S}$. We show that there is a natural and geometric functor, called Beilinson realisation, from $\mathcal{S}$-constructible mixed sheaves $\operatorname{D}^{mix}_{\mathcal{S}}(X)$ to $\operatorname{DK}_{\mathcal{S}}(X)$. We then show that Koszul duality intertwines the Betti realisation and Beilinson realisation functors and descends to an equivalence of constructible sheaves and constructible $K$-motivic sheaves on Langlands dual flag varieties.