External perfect-field Tate cancellation

PDF p. 7: Lemma2.9: smooth-point supported purity

Let $k$ be perfect. For any $K,L\in D{M}_{-}^{\mathrm{e}\mathrm{f}\mathrm{f}}(k)$, tensoring with $\mathbb{Z}(1)$ induces the natural bijection $\mathrm{Hom}(K,L)\stackrel{\sim }{\to }\mathrm{Hom}(K(1),L(1))\mathrm{.}$ The effective category and integral coefficients are the classical category used by the completed Gysin input. This is valid in every characteristic without a resolution-of-singularities hypothesis.

Given: A perfect field and objects of the stated effective motive category.

Goal: Import precisely the full-faithfulness needed for coefficient cancellation.

[F1] Voevodsky2010 Corollary4.10 supplies exactly this natural map and bijection.

Import [F1]. Its deep proof is not locally reconstructed. MVW Theorem16.25's printed proof assumes resolution of singularities, while Remark16.26 explicitly points to the perfect-field theorem; the present import uses the actual published perfect-field result rather than silently discarding the earlier hypothesis.