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.

1. The discriminant argument

The purity theorem is valid more generally for a regular local ring of dimension ${\geq 2}$, or a complete intersection of dimension ${\geq 3}$. Let’s start with the present case, and let’s start with a weaker question of surjectivity: any étale cover of ${X' \stackrel{\mathrm{def}}{=}X \setminus \left\{\mathfrak{m}\right\}}$ extends to an étale cover of ${X}$. Suppose given ${p: Z' \rightarrow X'}$ which is finite étale. Alternatively, the data of ${Z'}$ is equivalent to that of a vector bundle of algebras ${\mathcal{A}'}$ over ${X'}$, such that after base-change to any geometric point ${\mathrm{Spec} k \rightarrow X'}$ the associated algebra ${\mathcal{A}' \otimes_{X'} k}$ is a product of copies of ${k}$. We’d like to extend this to ${X}$.

The first thing we need to do, to prove the purity theorem, is to extend ${Z' \rightarrow X' }$ to an étale cover of ${X}$, is to extend it as a finite flat cover, and then check that it is étale. Actually, that is all we need to do. If we can find a finite flat cover ${Z \rightarrow X}$ (corresponding to a vector bundle ${\mathcal{A}}$ over ${X}$ of algebras) which extends ${Z' \rightarrow X'}$, it is automatically étale.

The justification is that étaleness can be tested on geometric points (once you know flatness), and so étaleness is equivalent to the invertibility of the discriminant ${\Delta}$ of the form ${\mathrm{Tr}: \mathcal{A} \otimes \mathcal{A} \rightarrow \mathcal{O}_X}$. The locus of étaleness of ${Z \rightarrow X}$ is thus precisely the complement of one equation: it’s ${\mathrm{Spec} A[\Delta^{-1}]}$. However, since ${A}$ has dimension two, ${\Delta}$ cannot vanish only on ${\mathfrak{m}}$: in other words, since ${Z \rightarrow X}$ is étale outside of the point ${\left\{\mathfrak{m}\right\}}$, it is forced to be étale everywhere.

2. Extending a vector bundle

So, in effect, we have reduced our problem to extending the vector bundle (of algebras) ${\mathcal{A}'}$ over the punctured spectrum ${X'}$ to all of ${X = \mathrm{Spec} A}$. A natural candidate for this is just to take the push-forward

$\displaystyle \mathcal{A} = i_* \mathcal{A}', \quad \text{for} \ i: X' \rightarrow X.$

Let’s pretend that we knew this was a coherent sheaf; we don’t yet. Once we know that it is a coherent sheaf, the claim is that it is automatically locally free. In other words, we have to show that (where $\mathrm{pd}$ is projective dimension)

$\displaystyle \mathrm{pd} \Gamma(X, \mathcal{A}) = 0,$

or that ${\mathrm{pd} \Gamma(X, \mathcal{A})}$ is a free ${A}$-module. The strategy is to show this using the Auslander-Buchsbaum formula relating projective dimension to depth, and Serre’s criterion to bound below the depth. (Another way is to use a theorem of Horrocks that any vector bundle on the punctured spectrum of a two-dimensional regular local ring is trivial, and to use the fact that $i_*( \mathcal{O}_{X'}) \simeq \mathcal{O}_X$.)

Since (by étaleness) ${\mathcal{A}'}$ is a sheaf of normal domains, so is ${\mathcal{A}}$. In particular,

$\displaystyle \Gamma(X, \mathcal{A})$

is a normal domain of dimension two, finite over the local ring ${A}$. By Serre’s criterion, we find that

$\displaystyle \mathrm{depth} \Gamma(X, \mathcal{A}) = 2$

(e.g., we can calculate at any closed point)—equivalently, the depth of ${\mathfrak{m} \subset A}$ on the ${A}$-module ${\Gamma(X, \mathcal{A})}$ is two. The Auslander-Buchsbaum formula gives

$\displaystyle \mathrm{pd} \Gamma(X, \mathcal{A}) + \mathrm{depth} \Gamma(X, \mathcal{A}) = \dim A = 2,$

as ${A}$ is regular. From this, we get that ${\mathrm{pd} \Gamma(X, \mathcal{A})}$ is forced to be zero. In particular, ${\mathcal{A}}$ is a vector bundle of algebras. We can make this argument once we know that ${\mathcal{A}}$ is coherent.

3. Full faithfulness

Let’s next describe full faithfulness of the pull-back functor ${\mathrm{Et}(X) \rightarrow \mathrm{Et}(X')}$. In fact, we have a more general claim on the level of vector bundles; recall that an étale cover corresponds to a special type of vector bundle with an algebra structure.

Proposition 11 Let ${(A, \mathfrak{m})}$ be a regular local ring of dimension ${\geq 2}$. Let ${X = \mathrm{Spec} A, X' = \mathrm{Spec} A \setminus \left\{\mathfrak{m}\right\}}$ as above. Then the pull-back functor

$\displaystyle \mathrm{Vect}(X) \rightarrow \mathrm{Vect}(X')$

is fully faithful.

In other words, we have to show that if ${\mathcal{E}, \mathcal{F}}$ are vector bundles over ${X}$, then

$\displaystyle \hom(\mathcal{E}, \mathcal{F}) \rightarrow \hom(\mathcal{E}|_{X'}, \mathcal{F}|_{X'})$

is an isomorphism. This is a special case of a more general claim: if ${\mathcal{G}}$ is a vector bundle on ${X}$, then

$\displaystyle \mathcal{G}\rightarrow i_* (\mathcal{G}|_{X'})$

is an isomorphism of sheaves on ${X}$. (In other words, we take ${\mathcal{G} = \underline{\hom}(\mathcal{E}, \mathcal{F})}$.)

It follows that we have encountered, in trying to prove the purity theorem, two natural questions:

1. Given a sheaf ${\mathcal{G} }$ on ${X}$, when is the morphism ${\mathcal{G} \rightarrow i_* ( \mathcal{G}|_{X'})}$ an isomorphism?
2. Given a sheaf ${\mathcal{F}}$ on ${X'}$, when is ${i_* \mathcal{F}}$ a coherent sheaf on ${X}$ (rather than merely quasi-coherent)?

The claim is that local cohomology allows us to give answers to both questions, and that these answers are in the affirmative in this particular case. The key fact is that, given any topological space ${X}$ and a closed subset ${Z \subset X}$, ${U = X \setminus Z}$, we have an exact sequence

$\displaystyle 0 \rightarrow \underline{H}_Z^0(\mathcal{F}) \rightarrow \mathcal{F} \rightarrow j_* j^* \mathcal{F} \rightarrow \underline{H}_Z^1(\mathcal{F})$

where ${\underline{H}_Z^i}$ are versions of the local-cohomology functors ${H^i_Z}$ which take values in sheaves and $j: U \to X$. In our particular case, we are going to get, for any ${A}$-module ${M}$, an exact sequence

$\displaystyle 0 \rightarrow H^0_{\mathfrak{m}}(M) \rightarrow M \rightarrow \Gamma( X', M) \rightarrow H^1_{\mathfrak{m}}(M).$

Using this exact sequence, we will get criterion for when ${\Gamma(X', M)}$ is finitely generated, and when the map ${M \rightarrow \Gamma(X', M)}$ is an isomorphism.

I should mention in this case (dimension two) that one has more. For a regular local ring $A$ of dimension two, there is an equivalence of categories between vector bundles on $\mathrm{Spec} A$ and vector bundles on the punctured spectrum, by a theorem of Horrocks; consequently one doesn’t have to invoke anything more to get the purity result. I’ll try to get back to this in a later post.