Cancellation Theorem

  • Vladimir Voevodsky

    School of Mathematics Institute for Advanced Study Princeton NJ 08540 USA


We give a direct proof of the fact that for any schemes of finite type X,YX, Y over a Noetherian scheme SS the natural map of presheaves with transfers Hom(Ztr(X),Ztr(Y))Hom(Ztr(X)trGm,Ztr(Y)trGm)\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) A1{\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.