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

for each integer [math]\displaystyle{ m \ge 0 }[/math], the map

[math]\displaystyle{ \operatorname{H}^m(Z_{\text{ét}}; \mathcal{F}) \to \operatorname{H}^{m+2r}_Z(X_{\text{ét}}; \mathcal{F}(r)) }[/math]

is bijective, where the map is induced by cup product with [math]\displaystyle{ c_r(Z) }[/math].

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