Suppose that $(R, m) \subseteq (S,n)$ is a finite extension of normal local domains that is:
- etale on the punctured spectrum
- not flat / etale at the origin
- and such that the residue fields $R/m = S/n$ are equal.
Question: Does anyone know of such examples where $S$, when viewed as an $R$-module, has more than 1 free $R$-summand?
Note, that if we remove the second condition, then we can have lots of summands (since $S$ will be free if flat, and flat + etale in codim 1 implies etale by one of the standard purity of the branch locus theorems).
Likewise if we remove the residue field condition, then it's very easy to construct such examples (do an extension with 1 summand and then extend the residue field).