 
Summary: FACTORIAL EXTENSIONS OF REGULAR LOCAL RINGS
AND INVARIANTS OF FINITE GROUPS
Luchezar L. Avramov and Adam Borek
Dedicated to the memory of Maurice Auslander
Introduction
It is often useful to approach the singularity of a noetherian commutative local domain
S from a regular local subring P over which S is nite as a module and residually trivial
(that is, the induced extension of residue elds is trivial). Such subrings exist in a variety of
situations: in commutative algebra they are given by Cohen's structure theory for complete
local rings; in analytic geometry they are in place by denition; in algebraic geometry they
are sometimes produced by Noether normalization.
If S is normal (here meaning noetherian and integrally closed) and divisorially unrami
ed over P , then S = P by the classical purity property of the branch locus, cf. [2]. Thus,
when studying nite normal local extensions of a regular local ring P a natural next step
is to allow a controlled amount of ramication in codimension one.
An early example appears in [15], where Serre uses purity to show that if P ,! S is
ramied in codimension one only at principal primes of S, and is generically Galois in the
sense that the induced extension of fraction elds P 0 ,! S 0 is Galois, then Gal(S 0 jP 0 ) is
generated by generalized re
ections (an automorphism h of S is a generalized re
ection if
(h id)(S) (x) for some nonunit x 2 S).
