Weight Filtrations and Derived Motivic Measures

Abstract

Let $k$ be a field admitting resolution of singularities. We lift a number of motivic measures such as the Gillet--Soul\'{e} measure and the compactly supported $\mathbb{A}^1$-Euler characteristic to derived motivic measures in the sense of Campbell--Wolfson--Zakharevich, answering various questions in the literature. The obstruction to such lifts is that the Gillet--Soul\'{e} weight complex of a variety is built from data that is a priori functorial only after passing to a homotopy category. We remove this obstruction by showing that the collection of all weight complexes of $X$ assembles into a canonical weakly constant pro-object in a Waldhausen category of simplicial smooth projective varieties. On the way, we prove a statement of independent interest: 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 and generalization of the existence of the Gillet--Soul\'{e} weight filtration to both the unstable and stable motivic homotopy category. Lastly, we organize all of the various maps of spectra into a single homotopy commutative diagram out of $K(\mathcal{V}_k)$, the Zakharevich $K$-theory of varieties.