Theorem of absolute purity

In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states: given for each integer $$m \ge 0$$, the map
 * a regular scheme X over some base scheme,
 * $$i: Z \to X$$ a closed immersion of a regular scheme of pure codimension r,
 * an integer n that is invertible on the base scheme,
 * $$\mathcal{F}$$ a locally constant étale sheaf with finite stalks and values in $$\mathbb{Z}/n\mathbb{Z}$$,
 * $$\operatorname{H}^m(Z_{\text{ét}}; \mathcal{F}) \to \operatorname{H}^{m+2r}_Z(X_{\text{ét}}; \mathcal{F}(r))$$

is bijective, where the map is induced by cup product with $$c_r(Z)$$.

The theorem was introduced in SGA 5 Exposé I, § 3.1.4. as an open problem. Later, Thomason proved it for large n and Gabber in general.