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CANCELLED Pullback maps for the Rost-Schmid complex

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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 codimension. Time permitting, I will sketch how this implies the following consequence, obtained in joint work with Maria Yakerson: given a pointed motivic space $\mathcal X$, its zeroth $\mathbb{P}1$-stable homotopy sheaf is given by $\underline{\pi}3((\mathbb{P}1)^{\wedge 3} \wedge \mathcal{X}){-3}$.




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