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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 three 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 later middle two statements are essentially referring to cohomology groups with $\mathbb{G}_m$ coefficients. Similar relations hold for $\pi_1(X)$ and $\pi_1(U)$ and can be found in SGA2.

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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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 three 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'III')

The later two statements are essentially referring to cohomology groups with $\mathbb{G}_m$ coefficients. Similar relations hold for $\pi_1(X)$ and $\pi_1(U)$ and can be found in SGA2.

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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# Algebraic data and purity associated to codimension greater than 2

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 three 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'

The later two statements are essentially referring to cohomology groups with $\mathbb{G}_m$ coefficients. Similar relations hold for $\pi_1(X)$ and $\pi_1(U)$ and can be found in SGA2.

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?