# 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:[1] 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 finie stalks and values in ,

for each integer , the map

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.

## See also

## References

- Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel (2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 [math.AG].

- Fujiwara, K.: A proof of the absolute purity conjecture (after Gabber). Algebraic geometry 2000, Azumino (Hotaka), pp. 153–183, Adv. Stud. Pure Math. 36, Math. Soc. Japan, Tokyo, 2002
- R. W. Thomason, Absolute cohomological purity, Bull. Soc. Math. France 112 (1984), no. 3, 397–406. MR 794741

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