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Cancellation theorem. (English) Zbl 1202.14022
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.

##### MSC:
 14F42 Motivic cohomology; motivic homotopy theory 19E15 Algebraic cycles and motivic cohomology ($$K$$-theoretic aspects)
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