etale topology of local schemes

Question

My apologies if this is too elementary, but since I have seen similar questions here I offer it.

After years doing almost exclusively classical complex geometry, using analysis and topology, I am trying to learn schemes, starting from the “red book”, which I have browsed for years, but vowing now not to skip anything. After spending 6 enjoyable months on chapter I, varieties, which I already “knew”, I am up to section II.2, the definition of preschemes, and became puzzled by essentially the first example, “Ex. F bis”. Mumford says the punctured local scheme spec(C[[X1,...,Xn]]) - m, where m is the unique maximal ideal, has "topological properties identical to those of the ordinary 2n-1 sphere", with reference to “chapter 8”.

Since when n=1 this scheme has a one point underlying topological space I assumed this was a misprint, until I thought about it in the context of “K-valued points” discussed later. For n=1 this scheme, spec(C((X))), apparently admits maps of every degree n, coming from the field extensions C((X^1/n)) of fractional Laurent series, which mimic finite covering spaces of the circle. Further googling turns up statements that the “etale cohomology” of this scheme resembles topological cohomology of the 2n-1 sphere, where etale cohomology is apparently ordinary derived functor sheaf cohomology, applied to the category of all etale maps rather than that of just open immersions.

I have found comparison theorems for etale cohomology, e.g. in answers to questions on this site, that the etale cohomology of a variety over the complex numbers, at least for finite coefficient groups, is the ordinary topological cohomology of the underlying topological space, but I have not found comparison theorems that I can see apply to the local analytic case in the example above.

So I have this question, and a guess as to what such theorems might say.

Suppose we are given a polynomial, or a convergent power series f in n+1 complex variables X0,...,Xn, which defines an isolated singularity at the origin, and we mod out the power series ring by the corresponding principal ideal (f), obtaining the analytic local ring R of the singularity. Then I believe the C algebra structure of this ring determines the singularity up to local analytic isomorphism.hence the algebraic invariants of that ring should also determine local topological properties of the singularity.

In particular, if we intersect the variety defined by f with a small 2n+1 sphere centered at the origin, we get a smooth 2n-1 real dimensional manifold K, the boundary of the Milnor fiber of the singularity. Then the topological properties of the punctured local scheme S = spec(R) - m, where m is the maximal ideal, should reflect those of the manifold K. In Mumford’s example, we have f = X0, R = C[[X1,...,Xn]], and K = the 2n-1 sphere in the hyperplane X0 =0.

In Singularities of Complex Hypersurfaces, Milnor proves this manifold K is n-2 connected at least when f is a polynomial defining an isolated singularity, and I ask if this is reflected in the etale cohomology of the scheme S. In particular

1) does the etale fundamental group of S vanish for n ≥ 3?

2) does the etale cohomology of S vanish in degrees 1 through n-2?

3) does the “top” degree etale cohomology of S, in degree 2n-1, not vanish?

If so where can one find such local comparison theorems? What about results for isolated complete intersection singularities, or more general punctured local spectra?"

Answer 1

Let me describe how to answer the etale cohomology questions for a slightly different ring, which is the ring of algebraic elements in the ring of formal power series, also known as the etale local ring / Henselization of the algebraic local ring (although these definitions become different in the greater generality of non-excellent rings). It should be possible to generalize these to the formal power series ring also.

Let $X$ be a variety over $\mathbb C$ and $P$ a point of $X$. Let $R$ be the etale local ring of $X$ at $P$ and $S = \operatorname{Spec} R- P$. We will see that the etale cohomology of $S$, with coefficients in any torsion ring (hence also with $\ell$-adic coefficients), matches the singular cohomology of a punctured topological neighborhood of $X$. So everything known about the singular cohomology transfers to the etale setting, without a new proof. (Although a new proof would be useful to handle the characteristic $p$ case, for instance.)

The etale local ring $R$ of $X$ at $P$ is the forward limit of the rings of functions on all affine etale neighborhoods of $X$ in $R$, so $\operatorname{Spec} R$ is the inverse limit of these neighborhoods, and $S$ is the inverse limit of the punctured neighborhoods. Hence the cohomology of $S$ with coefficients in a torsion ring $\Lambda$ is the inverse limit of the etale cohomology groups of these neighborhoods. This is exactly the stalk cohomology of the sheaf $R j_* \Lambda$ at the point $P$, where $j: X- P \to X$ is the open immersion. This gives an exact sequence $H^*(X,j_!\Lambda) \to H^*(X-p,\Lambda) \to H^*(S,\Lambda)$ where the cohomology of $j_! \Lambda$ is just the cohomology of $X$ relative to $p$.

It follows from Artin's comparison theorem for singular varieties that the first two cohomology groups in the exact sequence are the same as the usual singular cohomology groups. In singular cohomology, this exact sequence is the same as the Mayer-Vietoris exact sequence, and the third term is the singular cohomology of a punctured topological neighborhood.

Answer 2

As an addition to Will Sawin's excellent answer: I was wondering about the same thing when I was writing my PhD thesis, and I managed to prove that over $\mathbb{C}$, the etale homotopy type (as defined by Artin and Mazur) of the "algebraic Milnor fiber" (either using henselization or completion) agrees with the homotopy type of the classical Milnor fiber up to profinite completion. In particular, the scheme you mention does not just share the properties of the sphere, it in some precise sense "is" the sphere.

The result can be found in Chapter 4 of my thesis. This is really nothing fancy: it is deduced from the comparison theorems for the fundamental group and cohomology from SGA1 and SGA4 mentioned by Will and a formal argument due to Artin and Mazur, which is probably why I never decided to publish that part. For the comparison of henselization and completion, you need a bit more work, but there are stronger results of this kind in the literature, due to Fujiwara and Gabber.

Answer 3

I can only address your question (1), to which I believe the answer is yes.

Your ring $R$ is a complete intersection local ring of dimension $n$, and hence if $n\ge 3$, then by Stacks 0BPD, it satisfies purity - ie, any finite etale cover of $S = \text{Spec }R - \{m\}$ extends to $\text{Spec }R$. But $R$ is also strictly henselian, so it has trivial etale fundamental group, so the extension of any etale cover of $S$ to $R$ is trivial, so $S$ only has trivial covers, hence $S$ has trivial etale fundamental group if $n\ge 3$.