Cancellation theorem (Q612999)

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 \[ \underline{Hom}({\mathbf Z}_{tr}(X),{\mathbf Z}_{tr}(Y))\rightarrow \underline{Hom}({\mathbf Z}_{tr}(X)\otimes_{tr}{\mathbf G}_m,{\mathbf Z}_{tr}(Y)\otimes_{tr}{\mathbf G}_m) \] 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.