Theorem of absolute purity - Misplaced Pages

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,
${\mathcal {F}}$ 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.

References

A version of the theorem is stated at Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel (2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 .

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

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