Cancellation Theorem

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

2010 Mathematics Subject Classification: 14F42, 19E15

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