# Deformations of Tame Coverings

To say that I am a novice at deformation theory is to grossly overestimate my abilities in this area. I've come across the following theorem in a paper, and I'd like to know how far one is able to generalize it:

### Theorem

Let $R$ be a complete DVR, $X$ a proper smooth curve over $R$, and $D$ a simple divisor on $X$. Let $\bar f:\bar Y\rightarrow \bar X$ be a tame covering of $\bar X$ with branch locus contained in Supp($\bar D$). Then there exists a unique lifting of $\bar f$ to a D-tame covering $f:Y\rightarrow X$. In addition, $f$ is a tame covering of $X$, and the branch locus of $f$ is contained in Supp($D$).

### Notation

I'm not really sure how much of it is standard, so here's what I mean by the terms in the theorem:

A simple divisor on $X$ - a divisor that has no multiple components when base-changed to any geometric point of the base scheme.

Tame covering of integral varieties over a field ($f:\bar Y\rightarrow \bar X$ in our theorem) - what you think: the ramification indices in codim 1 are coprime to the characteristic.

D-tame covering of schemes over a DVR ($f:Y\rightarrow X$ in the theorem) - if for every (natural number) $k$, base change to $R/m^k$ is $D\times_{R}R/m^k$ -tame in the following sense:

D-tame covering of schemes over an Artin local ring, $\Lambda$ ($R/m^k$ in the previous notation) - a finite, flat morphism which is etale away from Supp($D$)., and such that if $x$ is in Supp($D$), and t is a local equation for D at $x$, then $f_\ast(O_Y)_x$ is a t-tame extension of $O_x(X)$. What's that? $A'$ in the following, plays the role of $O_x(X)$ here.

t-tame extensions of a ring $A'$ s.t. $A'/Nil(A')$ is a DVR with parameter $\check t$ (and assume (0) is primary in $Spec(A')$, meaning $Spec(A')$ has no imbedded components)- tame when base changed to $A'/tA'$ (where $t$ is some element going to $\check t$). $A'/tA'$ is an Artin ring. So what do I mean by tameness of extensions over Artin rings? Well:

A tame extension of an Artin ring $\Lambda$ - it is a (finite) product of $\Gamma_i$'s, such that for each $\Gamma_i$ it is free over $\Lambda$ of rank $e_if_i$; the extension of residue fields is $f_i$; $e_i$ is prime to the characterstic of the residue field of $\Lambda$; and $\Gamma_i$ contains a principal ideal $J$ such that $J^{e_i}$=0, and such that $\Gamma_i/J$ is free over $\Lambda$ of rank $f$.

### Question

Phew, that was a lot of work... So - other than finding the whole thing very confusing, here is a concrete set of questions:

1. Can this be generalized to $X$ a higher dimensional scheme?

2. What would be a good reference to read for this type of deformation theory?

3. Well - just about any insight and context you can give would be great (whether deformation theoretic or about the various generalizations of tameness here.)

This is a bit long winded, but if nothing else - it was good exercise for me to write this.

• "A simple divisor on X - a divisor that has no multiple components when base-changed to any geometric point of X.". Do you really mean that? – Kevin Buzzard Mar 6 '10 at 19:42
• Whoops, no. I'll edit that. – Randy Brown Mar 6 '10 at 19:44