2024/01/12 by Nanavaty, Anubhav · 1 citation
#Algebraic Geometry (math.AG) #Algebraic Topology (math.AT) #FOS: Mathematics
paper · doi:10.48550/arxiv.2401.06879
Let k be a field admitting resolution of singularities. We lift a number of motivic measures, such as the Gillet-Soulé measure and the compactly supported \mathbbA1-Euler characteristic, to derived motivic measures in the sense of Campbell-Wolfson-Zakharevich, answering various questions in the literature. We do so by generalizing the construction of the Gillet-Soulé weight complex to show that it is well-defined up to a certain notion of weak equivalence in the category of simplicial smooth projective varieties. For a k-variety X, the collection of all Gillet-Soulé weight complexes of X form a 'weakly constant' pro-object of simplicial varieties, and under mild assumptions, the K-theory of a Waldhausen category is equivalent to the K-theory of its weakly constant pro-objects. This leads us to a new proof of the existence of the Gillet-Soulé weight filtration, along with the weight filtration on both the stable and unstable homotopy type of a variety over k. We show these constructions provide the aforementioned derived motivic measures, or maps of spectra, out of K(Vark), the Zakharevich K-theory of varieties.