Theorem of absolute purity

Short description: Mathematical theorem

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,
• ${ℱ}$ a locally constant étale sheaf with finite stalks and values in $\mathbb{\mathbb{Z}}/n\mathbb{\mathbb{Z}}$,

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

${\mathrm{H}}^m(Z_{\text{ét}};{ℱ})\to {\mathrm{H}}_Z^{m+2r}(X_{\text{ét}};{ℱ}(r))$

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