Theorem of absolute purity explained

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,

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,

l{F} a locally constant étale sheaf with finite stalks and values in Z /n Z ,for each integer m \ge 0 , the map\operatorname{H} ^{m(Z} _{ét} ; l{F}) \to \operatorname{H} ^{m +2 r} _{Z(X} _{ét} ; l{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.

See also

• purity (algebraic geometry)

References

• 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

Notes and References

1. A version of the theorem is stated at Déglise. Frédéric. Fasel. Jean. Jin. Fangzhou. Khan. Adeel. 2019-02-06. Borel isomorphism and absolute purity. 1902.02055. math.AG.