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?"