Theorem of absolute purity
In algebraic geometry, the theorem of absolute purity is an important theorem in the theory of étale cohomology. It states: given
- a regular scheme X over some base scheme,
- a closed immersion of a regular scheme of pure codimension r,
- an integer n that is invertible on the base scheme,
- a locally constant étale sheaf with finite stalks and values in,
is bijective, where the map is induced by cup product with.
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.