 
Summary: arXiv:math.AC/0208067
v2
25
Nov
2002
THE FSIGNATURE AND STRONG FREGULARITY
IAN M. ABERBACH AND GRAHAM J. LEUSCHKE
Abstract. We show that the F signature of a local ring of characteristic p, dened by
Huneke and Leuschke, is positive if and only if the ring is strongly F regular.
In [7], Huneke and Leuschke dene the F signature of an F nite local ring of prime
characteristic with perfect residue eld. The F signature, denoted s(R), is an asymptotic
measure of the proportion of Rfree direct summands in a directsum decomposition of R 1=p e
,
the ring of p e th roots of R. This proportion seems to give subtle information on the nature
of the singularity dening R. For example, the F signature of any of the twodimensional
quotient singularities (A n ), (D n ), (E 6 ), (E 7 ), (E 8 ) is the reciprocal of the order of the group
G dening the singularity [7, Example 18]. The main theorem of [7] on F signatures is as
follows.
Theorem 0.1. [7, Theorem 11] Let (R; m; k) be a reduced complete F nite Cohen{Macaulay
local ring containing a eld of prime characteristic p. Assume that k is perfect. Then
