In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of

étale cohomology In mathematics, the étale cohomology groups of an algebraic variety or scheme are algebraic analogues of the usual cohomology groups with finite coefficients of a topological space, introduced by Grothendieck in order to prove the Weil conjecture ...

. It states:A version of the theorem is stated at given *a

regular scheme In algebraic geometry, a regular scheme is a locally Noetherian scheme whose local rings are regular everywhere. Every smooth scheme is regular, and every regular scheme of finite type over a perfect field is smooth.. For an example of a regul ...

''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

a locally constant étale sheaf with finite stalks and values in

\mathbb/n\mathbb

, for each integer

m \ge 0

, the map :

\operatorname^m(Z_; \mathcal) \to \operatorname^_Z(X_; \mathcal(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 Gabber (; ) is a style of electronic dance music and a subgenre of hardcore techno, as well as the surrounding subculture. The music is more commonly referred to as Hardcore, which is characterised by fast beats, distorted & heavier kickdrums, ...

in general.

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 Algebraic geometry {{algebraic-geometry-stub

See also

References