A Characteristic p Analog of Multiplier Ideals and Applications^#

^#Communicated by S. Goto.

We present two applications of a characteristic p analog of multiplier ideals, which is a generalization of the test ideal in the theory of tight closure. Namely, we give alternative proofs to Smith's result on base-point-freeness of adjoint bundles in characteristic p > 0 and results on uniform behavior of symbolic powers in a regular local ring due to Ein, Lazarsfeld and Smith, and Hochster and Huneke.

Key Words:

• Multiplier ideal
• Test ideal
• Tight closure
• Uniform bound

1991 Mathematics Subject Classification:

• Primary 13A35
• Secondary 13A10, 13H05, 14Q20

ACKNOWLEDGMENTS

The author thanks Professors Shiro Goto, Robert Lazarsfeld, and Ken-ichi Yoshida for their helpful comments about Theorems 2.1 and 2.12. He also thanks MSRI for the support and hospitality during his stay in the fall of 2002. The author was partially supported by Grant-in-Aid for Scientific Research, Japan.

Notes

^aWe say that R is weakly F-regular if every ideal I of R is tightly closed, that is, I* = I. We say that R is F-regular if all of its local rings are weakly F-regular. The following implications are known (Hochster and Huneke, 1990): regular ↠ F-regular ↠ weakly F-regular. Also, for an excellent ℚ-Gorenstein ring, F-regularity and weak F-regularity are equivalent to each other.

^bA local ring (R, 𝔪) of characteristic p > 0 is said to be F-rational if every ideal generated by a system of parameters of R is tightly closed (Fedder and Watanabe, 1987). In general, we say that R is F-rational if all of its local rings are F-rational.

^cThe author learned from Lazarsfeld about a proof without using vanishing theorems under a stronger assumption that ℒ is very ample.

^# Communicated by S. Goto.