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

• 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 $\mathbb {Z} /n\mathbb {Z}$,

for each integer $m\geq 0$, the map

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

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

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