2021/10/18 by Elmanto, Elden, Nardin, Denis, Yakerson, Maria
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics #K-Theory and Homology (math.KT)
paper · doi:10.48550/arxiv.2110.09203
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 A over X, we construct the 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 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.