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arXiv · 2110.09203

Twisted K-theory in motivic homotopy theory

Elden Elmanto·Denis Nardin·Maria Yakerson
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Abstract

In this paper, we study twisted algebraic $K$-theory from a motivic viewpoint. For a smooth variety $X$ over a field of characteristic zero and an Azumaya algebra $\mathcal{A}$ over $X$, we construct the $\mathcal{A}$-twisted motivic spectral sequence, by computing the slices of the motivic twisted algebraic $K$-theory spectrum as a twisted form of motivic cohomology. This generalizes previous results due to Kahn-Levine where $\mathcal{A}$ is assumed to be pulled back from a base field. Our methods use interaction between the slice filtration and birational geometry. Along the way, we prove a representability result, expressing the motivic space of twisted $K$-theory as an extension of the twisted Grassmannian by the sheaf of "twisted integers". This leads to a proof of cdh descent and Milnor excision for twisted homotopy $K$-theory.

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BibTeXRIS

Elden Elmanto, Denis Nardin, Maria Yakerson. 2021-10-18. Twisted K-theory in motivic homotopy theory. https://arxiv.org/abs/2110.09203

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