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:A version of the theorem is stated at {{cite arXiv|last=Déglise|first=Frédéric|last2=Fasel|first2=Jean|last3=Jin|first3=Fangzhou|last4=Khan|first4=Adeel|date=2019-02-06|title=Borel isomorphism and absolute purity|eprint=1902.02055|class=math.AG}} given

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

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

:$\backslash operatorname\{H\}^m(Z\_\{\backslash text\{ét\}\};\; \backslash mathcal\{F\})\; \backslash to\; \backslash operatorname\{H\}^\{m+2r\}\_Z(X\_\{\backslash text\{ét\}\};\; \backslash mathcal\{F\}(r))$

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.