Theorem of absolute purity

In algebraic geometry, the theorem of absolute (cohomological) purity is an important theorem in the theory of étale cohomology. It states:[1] given

  • a regular scheme X over some base scheme,
  • a closed immersion of a regular scheme of pure codimension r,
  • an integer n that is invertible on the base scheme,
  • a locally constant étale sheaf with finie stalks and values in ,

for each integer , the map

is bijective, where the map is induced by cup product with .

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


  1. Déglise, Frédéric; Fasel, Jean; Jin, Fangzhou; Khan, Adeel (2019-02-06). "Borel isomorphism and absolute purity". arXiv:1902.02055 [math.AG].
  • 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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