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,
• ${\displaystyle i:Z\to X}$ a closed immersion of a regular scheme of pure codimension r,
• an integer n that is invertible on the base scheme,
• ${\displaystyle {\mathcal {F}}}$ a locally constant étale sheaf with finite stalks and values in ${\displaystyle \mathbb {Z} /n\mathbb {Z} }$,

for each integer ${\displaystyle m\geq 0}$, the map

${\displaystyle \operatorname {H} ^{m}(Z_{\text{ét}};{\mathcal {F}})\to \operatorname {H} _{Z}^{m+2r}(X_{\text{ét}};{\mathcal {F}}(r))}$

is bijective, where the map is induced by cup product with ${\displaystyle c_{r}(Z)}$.

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

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". 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