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arXiv:math.AC/0208067 THE F-SIGNATURE AND STRONG F-REGULARITY
 

Summary: arXiv:math.AC/0208067
v2
25
Nov
2002
THE F-SIGNATURE AND STRONG F-REGULARITY
IAN M. ABERBACH AND GRAHAM J. LEUSCHKE
Abstract. We show that the F -signature of a local ring of characteristic p, de ned by
Huneke and Leuschke, is positive if and only if the ring is strongly F -regular.
In [7], Huneke and Leuschke de ne 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 R-free direct summands in a direct-sum 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 de ning R. For example, the F -signature of any of the two-dimensional
quotient singularities (A n ), (D n ), (E 6 ), (E 7 ), (E 8 ) is the reciprocal of the order of the group
G de ning 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

  

Source: Aberbach, Ian - Department of Mathematics, University of Missouri-Columbia

 

Collections: Mathematics