Cancellation Theorem

Abstract

We give a direct proof of the fact that for any schemes of finite type $X,Y$ over a Noetherian scheme $S$ the natural map of presheaves with transfers $\underline{Hom}({\mathbf{Z}}_{tr}(X),{\mathbf{Z}}_{tr}(Y))\to \underline{Hom}({\mathbf{Z}}_{tr}(X){\otimes }_{tr}{\mathbf{G}}_m,{\mathbf{Z}}_{tr}(Y){\otimes }_{tr}{\mathbf{G}}_m)$ is a (weak) ${\mathbf{A}}^1$-homotopy equivalence. As a corollary we deduce that the Tate motive is quasi-invertible in the triangulated categories of motives over perfect fields.