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Consider the following statement: Let $X$ be a smooth and geometrically integral variety over a field $k$ and let $U$ be any open subset of $X$ whose complement is of codimension greater or equal to $2$. Then the following statements hold

$k[X] \cong k[U]$ (clear)

$Pic(X) \cong Pic(U)$ (restriction of codimension $1$ irreducibles)

$Br(X) \cong Br(U)$ (Grothendieck purity for the Brauer group, see 'Le groupe de Brauer III')

$\pi_1^{et}(X) \cong \pi_1^{et}(U)$ (SGA1 Corollary X.3.3)

The middle two statements are essentially referring to cohomology groups with $\mathbb{G}_m$ coefficients.

The question is (albeit rather vague) whether there is some reason why one would expect a lot of the algebraic data associated to a scheme to be in some sense encoded in lower dimension. This is in line with various purity statements such as that in etale cohomology with coefficients in locally constant sheaves of $\mathbb{Z}/n\mathbb{Z}$ modules as can be found in Milne's book. Do many other statements of this type exist?

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I just found this nice question, and the following more recent one is somewhat related:… – Lars Apr 28 '10 at 19:21

2 Answers 2

The explanation why a lot of the algebraic data is encoded in lower dimension might reside partially in the following theorem.

Let X/k be smooth and irreducible over the field k. Let $F \subset X $ be a closed subvariety with $codim_X F\geq c$. Then the restriction maps in étale cohomology

$ H^i(X,\mu_n) \to H^i(X-F,\mu_n ) $

are injective for i<2c and isomorphisms for i<2c-1. For c=2 these are more or less the results you mention. (There is a little nit-picking here: for i=2 you get results for the cohomological Brauer group, Br'(X) in Grothendieck's notation) These results are due to Mike Artin and nicely summarized by Colliot-Thélène in the Proceedings of the Conference on K-Theory and Algebraic Geometry in Santa-Barbara (1992) (Corollary 3.4.2. page23)"K+theory

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Nice -- is that related to Lefschetz hyperplane theorem? – Ilya Nikokoshev Nov 15 '09 at 9:29
He znayu o Lefschetze. – Georges Elencwajg Nov 15 '09 at 10:30

If the differences between the spaces starts in codimension 2, then the nerves of the chech coverings (for any reasonable topos you choose) are different in dimensions 2 and up.

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