## Cancellation theorem

### Summary

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 $$\underline{Hom}({\bold Z}_{tr}(X),{\bold Z}_{tr}(Y))\rightarrow \underline{Hom}({\bold Z}_{tr}(X)\otimes_{tr}{\bold G}_m,{\bold Z}_{tr}(Y)\otimes_{tr}{\bold G}_m)$$ is a (weak) ${\bold A}^1$-homotopy equivalence. As a corollary we deduce that the Tate motive is quasi-invertible in the triangulated categories of motives over perfect fields.

14F42, 19E15