Let ${X \subset \mathbb{P}^r_{\mathbb{C}}}$ be a smooth projective variety, and let ${H}$ be a generic hyperplane. For generic enough ${H}$, the intersection ${X \cap H}$ is itself a smooth projective variety of dimension one less. The Lefschetz hyperplane theorem asserts that the map

$\displaystyle H \cap X \rightarrow X$

induces an isomorphism on ${\pi_1}$, if ${\dim X \geq 3}$.

We might be interested in analog over any field, possibly of characteristic ${p}$. Here ${\pi_1}$ has to be replaced with its étale analog, but otherwise it is a theorem of Grothendieck that ${H \cap X \rightarrow X}$ still induces an isomorphism on ${\pi_1}$, under the same hypotheses. This is one of the main results of SGA 2, and it uses the local cohomology machinery developed there. One of my goals in the next few posts is to understand some of the ideas that go into Grothendieck’s argument.

More generally, suppose ${Y \subset X}$ is a subvariety. To say that ${\pi_1(Y) \simeq \pi_1(X)}$ (always in the étale sense) is to say that there is an equivalence of categories

$\displaystyle \mathrm{Et}(X) \simeq \mathrm{Et}(Y)$

between étale covers of ${X}$ and étale covers of ${Y}$. How might one prove such a result? Grothendieck’s strategy is to attack this problem in three stages:

1. Compare ${\mathrm{Et}(X)}$ to ${\mathrm{Et}(U)}$, where ${U \supset Y}$ is a neighborhood of ${Y}$ in ${X}$.
2. Compare ${\mathrm{Et}(U)}$ to ${\mathrm{Et}(\hat{X}_Y)}$, where ${\hat{X}_Y}$ is the formal completion of ${X}$ along ${Y}$ (i.e., the inductive limit of the infinitesimal thickenings of ${Y}$).
3. Compare ${\mathrm{Et}( \hat{X})_Y)}$ and ${\mathrm{Et}(Y)}$.

In other words, to go from ${Y}$ to ${X}$, one first passes to the formal completion along ${Y}$, then to an open neighborhood, and then to all of ${Y}$. The third step is the easiest: it is the topological invariance of the étale site. The second step is technical. In this post, we’ll only say something about the first step.

The idea behind the first step is that, if ${Y}$ is not too small, the passage from ${U}$ to ${X}$ will involve adding only subvarieties of codimension ${\geq 2}$, and these will unaffect the category of étale covers. There are various “purity” theorems to this effect.

The goal of  this post is to sketch Grothendieck’s proof of the following result of Zariski and Nagata.

Theorem 10 (Purity in dimension two) Let ${(A, \mathfrak{m})}$ be a regular local ring of dimension ${2}$, and let ${X = \mathrm{Spec} A}$. Then the map

$\displaystyle \mathrm{Et}(X) \rightarrow \mathrm{Et}(X \setminus \left\{\mathfrak{m}\right\})$

is an equivalence of categories.

In other words, “puncturing” the spectrum of a regular local ring does not affect the fundamental group. (more…)