Grothendieck's Vanishing Cycles

Question

Suppose $S$ is the spectrum of a strict henselian ring $R$ which is also a discrete valuation ring (DVR), then $S$ consists of a closed point $s$ and a generic point $\eta$. We have a henselian trait, \begin{equation} \eta \rightarrow S \leftarrow s \end{equation} If $f:X \rightarrow S$ is a (flat) morphism, then Grothendieck studied the nearby cycle functor $R \Psi_f$ and vanishing cycle functor $R \Phi_f$. Suppose we have a fibration over \begin{equation} \pi: X \rightarrow C \end{equation} where $X$ is a variety over $k$ and $C$ is a curve over $k$. For a point $x$ in $C$, I have seen papers talking about the punctured henselian neighbourhood $S$ of $x$, which requires a local parameter. Is $S$ the strict henselization of the local ring $\mathcal{O}_{C,x}$? People thinks $S$ is like a small neighbourhood of $x$ in the analytic topology if there is an embedding $k \rightarrow \mathbb{C}$, could someone explain why more carefully?

Another question is does the definition of this punctured henselian neighbourhood of $x$ depends on the choice of a local parameter around $x$? If so, why?

Answer 1

The analogy with the classical picture is the following :

• A small disk around $x$ is the spectrum $\tilde{S}$ of a strict henselization $\mathcal{O}_{C,x}^{sh}$ of $\mathcal{O}_{C,x}$
• The point $x$ is the closed point of $\tilde{S}=\operatorname{Spec}\mathcal{O}_{C,x}^{sh}$ (hence a geometric point lying above $x$).
• The punctured disk around $x$ is the spectrum $\tilde{\eta}$ of the fraction field of $\mathcal{O}_{C,x}^{sh}$ (in other words, this is $\tilde{\eta}=\tilde{S}\setminus x$ : the disk without the point).

None of these depends of a local parameter, but it depends on the choice of a geometric point lying above $x$ (in other words, it depends on the choice of an algebraic closure of $\kappa(x)$). Because you are speaking of nearby cycles, these even depend on the choice of an algebraic closure $\overline{\eta}$ of $\tilde{\eta}$.

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Now I try to explain why this analogy is valid.

First, étale neighborhood are finer than Zariski one. Hence they are "closer" to the classical neighborhood. For example, the curve $y^2=x^3+x^2$ is irreducible and has a node at $x=(0,0)$. The local ring at $x$ for the Zariski topology remain integral (as a localization of an integral ring). However, the strict henselization (in other words, the local ring for the étale topology) is isomorphic the strict henselization of the local ring at $(0,0)$ of $k[u,v]/(uv)$. In other words, the étale topology sees the two branches at $(0,0)$. This is good because this is also the case for the classical topology.

Second, their categories of locally constant sheaves are (almost) the same :

• On a small disk, every locally constant sheaf is constant since the disk is contractible. The same is true for $\tilde{S}$ : every locally constant étale sheaf is constant.
• Obviously, the same is true for the center of the disk and the geometric point.
• On the punctured disk, the category of locally constant sheaf is the category of representation of $\pi_1(D^*)=\mathbb{Z}$. On $\tilde{\eta}$, the category of locally constant étale sheaf is the category of representation of $\operatorname{Gal}(\overline{\eta}/\tilde{\eta})$. Now this group may be very different from $\mathbb{Z}$. However, if the residue field has characteristic 0, this group is (isomorphic to) $\hat{\mathbb{Z}}$ the profinite completion of $\mathbb{Z}$. This is because every étale cover of $\tilde{\eta}$ is obtained by adding a $n$-th root the uniformizer (after picking one).

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Note however that in the classical picture, one need to choose a generator of $\pi_1(D^*)$. In the étale picture in characteristic 0, this is a bit more complicated. The isomorphism $\operatorname{Gal}(\overline{\eta}/\tilde{\eta})$ depends on the choice of the local parameter $\pi$ and the choice of primitive $n$-th root of $\pi$ for every $n$ (this is in fact $\hat{\mathbb{Z}}(1)$).