Pullbacks for the Rost-Schmid complex

Abstract: Let $F$ be a "strictly homotopy invariant" Nisnevich sheaf of abelian groups on the site of smooth varieties over a perfect field $k$. By work of Morel and Colliot-Thélène--Hoobler--Kahn, the cohomology of $F$ may be computed using a fairly explicit "Rost-Schmid" complex. However, given a morphism $f: X \to Y$ of smooth varieties, it is in general (in particular if f is not flat, e.g. a closed immersion) unclear how to compute the pullback map $f^*: H^*(Y, F) \to H^*(X,F)$ in terms of the Rost-Schmid complex. I will explain how to compute the pullback of a cycle with support $Z$ such that $f^{-1}(Z)$ has the expected dimension. Time permitting, I will sketch how this implies the following consequence, obtained in joint work with Maria Yakerson: given a pointed motivic space $X$, its zeroth $P^1$-stable homotopy sheaf is given by $\pi_3(\Sigma^3_{P^1} X)_{-3}$.

algebraic geometryalgebraic topology

Audience: researchers in the topic

MoVid-20

Series comments: Description: A one day conference on motivic homotopy theory

In order to register for the conference, please send an email to Denis Nardin (denis DOT nardin AT ur DOT de) with subject "MoVid-20"