## A generalization of tight closure and multiplier ideals

HTML articles powered by AMS MathViewer

- by Nobuo Hara and Ken-ichi Yoshida PDF
- Trans. Amer. Math. Soc.
**355**(2003), 3143-3174 Request permission

## Abstract:

We introduce a new variant of tight closure associated to any fixed ideal $\mathfrak {a}$, which we call $\mathfrak {a}$-tight closure, and study various properties thereof. In our theory, the annihilator ideal $\tau (\mathfrak {a})$ of all $\mathfrak {a}$-tight closure relations, which is a generalization of the test ideal in the usual tight closure theory, plays a particularly important role. We prove the correspondence of the ideal $\tau (\mathfrak {a})$ and the multiplier ideal associated to $\mathfrak {a}$ (or, the adjoint of $\mathfrak {a}$ in Lipman’s sense) in normal $\mathbb {Q}$-Gorenstein rings reduced from characteristic zero to characteristic $p \gg 0$. Also, in fixed prime characteristic, we establish some properties of $\tau (\mathfrak {a})$ similar to those of multiplier ideals (e.g., a Briançon-Skoda-type theorem, subadditivity, etc.) with considerably simple proofs, and study the relationship between the ideal $\tau (\mathfrak {a})$ and the F-rationality of Rees algebras.## References

- Ian Aberbach, Mordechai Katzman, and Brian MacCrimmon,
*Weak F-regularity deforms in $\textbf {Q}$-Gorenstein rings*, J. Algebra**204**(1998), no. 1, 281–285. MR**1623973**, DOI 10.1006/jabr.1997.7369 - Ian M. Aberbach and Brian MacCrimmon,
*Some results on test elements*, Proc. Edinburgh Math. Soc. (2)**42**(1999), no. 3, 541–549. MR**1721770**, DOI 10.1017/S0013091500020502 - Jean-François Boutot,
*Singularités rationnelles et quotients par les groupes réductifs*, Invent. Math.**88**(1987), no. 1, 65–68 (French). MR**877006**, DOI 10.1007/BF01405091 - Henri Skoda and Joël Briançon,
*Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de $\textbf {C}^{n}$*, C. R. Acad. Sci. Paris Sér. A**278**(1974), 949–951 (French). MR**340642** - Winfried Bruns and Jürgen Herzog,
*Cohen-Macaulay rings*, Cambridge Studies in Advanced Mathematics, vol. 39, Cambridge University Press, Cambridge, 1993. MR**1251956** - Jean-Pierre Demailly, Lawrence Ein, and Robert Lazarsfeld,
*A subadditivity property of multiplier ideals*, Michigan Math. J.**48**(2000), 137–156. Dedicated to William Fulton on the occasion of his 60th birthday. MR**1786484**, DOI 10.1307/mmj/1030132712 - Lawrence Ein,
*Multiplier ideals, vanishing theorems and applications*, Algebraic geometry—Santa Cruz 1995, Proc. Sympos. Pure Math., vol. 62, Amer. Math. Soc., Providence, RI, 1997, pp. 203–219. MR**1492524**, DOI 10.1090/s0894-0347-97-00223-3 - Lawrence Ein, Robert Lazarsfeld, and Karen E. Smith,
*Uniform bounds and symbolic powers on smooth varieties*, Invent. Math.**144**(2001), no. 2, 241–252. MR**1826369**, DOI 10.1007/s002220100121 - Richard Fedder and Keiichi Watanabe,
*A characterization of $F$-regularity in terms of $F$-purity*, Commutative algebra (Berkeley, CA, 1987) Math. Sci. Res. Inst. Publ., vol. 15, Springer, New York, 1989, pp. 227–245. MR**1015520**, DOI 10.1007/978-1-4612-3660-3_{1}1 - Shiro Goto and Yasuhiro Shimoda,
*On the Rees algebras of Cohen-Macaulay local rings*, Commutative algebra (Fairfax, Va., 1979) Lecture Notes in Pure and Appl. Math., vol. 68, Dekker, New York, 1982, pp. 201–231. MR**655805** - Shiro Goto and Keiichi Watanabe,
*On graded rings. I*, J. Math. Soc. Japan**30**(1978), no. 2, 179–213. MR**494707**, DOI 10.2969/jmsj/03020179 - Nobuo Hara,
*A characterization of rational singularities in terms of injectivity of Frobenius maps*, Amer. J. Math.**120**(1998), no. 5, 981–996. MR**1646049**, DOI 10.1353/ajm.1998.0037 - Nobuo Hara,
*Geometric interpretation of tight closure and test ideals*, Trans. Amer. Math. Soc.**353**(2001), no. 5, 1885–1906. MR**1813597**, DOI 10.1090/S0002-9947-01-02695-2 - N. Hara and S. Takagi,
*Some remarks on a generalization of test ideals*, preprint. - Nobuo Hara and Kei-Ichi Watanabe,
*F-regular and F-pure rings vs. log terminal and log canonical singularities*, J. Algebraic Geom.**11**(2002), no. 2, 363–392. MR**1874118**, DOI 10.1090/S1056-3911-01-00306-X - N. Hara, K.-i. Watanabe, and K. Yoshida,
*F-rationality of Rees algebras*, J. Algebra**247**(2002), 153–190. - —,
*Rees algebras of F-regular type*, J. Algebra**247**(2002), 191–218. - M. Herrmann, E. Hyry, and T. Korb,
*On Rees algebras with a Gorenstein Veronese subring*, J. Algebra**200**(1998), no. 1, 279–311. MR**1603275**, DOI 10.1006/jabr.1997.7207 - Melvin Hochster,
*Cyclic purity versus purity in excellent Noetherian rings*, Trans. Amer. Math. Soc.**231**(1977), no. 2, 463–488. MR**463152**, DOI 10.1090/S0002-9947-1977-0463152-5 - Melvin Hochster,
*The tight integral closure of a set of ideals*, J. Algebra**230**(2000), no. 1, 184–203. MR**1774763**, DOI 10.1006/jabr.1999.7954 - Melvin Hochster and Craig Huneke,
*Tight closure and strong $F$-regularity*, Mém. Soc. Math. France (N.S.)**38**(1989), 119–133. Colloque en l’honneur de Pierre Samuel (Orsay, 1987). MR**1044348** - Melvin Hochster and Craig Huneke,
*Tight closure, invariant theory, and the Briançon-Skoda theorem*, J. Amer. Math. Soc.**3**(1990), no. 1, 31–116. MR**1017784**, DOI 10.1090/S0894-0347-1990-1017784-6 - Melvin Hochster and Craig Huneke,
*$F$-regularity, test elements, and smooth base change*, Trans. Amer. Math. Soc.**346**(1994), no. 1, 1–62. MR**1273534**, DOI 10.1090/S0002-9947-1994-1273534-X - —,
*Tight closure in equal characteristic zero*, to appear. - J. A. Howald,
*Multiplier ideals of monomial ideals*, Trans. Amer. Math. Soc.**353**(2001), no. 7, 2665–2671. MR**1828466**, DOI 10.1090/S0002-9947-01-02720-9 - Craig Huneke,
*Tight closure and its applications*, CBMS Regional Conference Series in Mathematics, vol. 88, Published for the Conference Board of the Mathematical Sciences, Washington, DC; by the American Mathematical Society, Providence, RI, 1996. With an appendix by Melvin Hochster. MR**1377268**, DOI 10.1016/0167-4889(95)00136-0 - Eero Hyry,
*Blow-up rings and rational singularities*, Manuscripta Math.**98**(1999), no. 3, 377–390. MR**1717540**, DOI 10.1007/s002290050147 - Eero Hyry,
*Coefficient ideals and the Cohen-Macaulay property of Rees algebras*, Proc. Amer. Math. Soc.**129**(2001), no. 5, 1299–1308. MR**1712905**, DOI 10.1090/S0002-9939-00-05673-2 - Yujiro Kawamata,
*Elementary contractions of algebraic $3$-folds*, Ann. of Math. (2)**119**(1984), no. 1, 95–110. MR**736561**, DOI 10.2307/2006964 - Ernst Kunz,
*On Noetherian rings of characteristic $p$*, Amer. J. Math.**98**(1976), no. 4, 999–1013. MR**432625**, DOI 10.2307/2374038 - R. Lazarsfeld,
*Positivity in Algebraic Geometry*, preprint. - Joseph Lipman,
*Adjoints of ideals in regular local rings*, Math. Res. Lett.**1**(1994), no. 6, 739–755. With an appendix by Steven Dale Cutkosky. MR**1306018**, DOI 10.4310/MRL.1994.v1.n6.a10 - B. MacCrimmon,
*Weak F-regularity is strong F-regularity for rings with isolated non-$\mathbb Q$-Gorenstein points*, Trans. Amer. Math. Soc., to appear. - Hideyuki Matsumura,
*Commutative ring theory*, Cambridge Studies in Advanced Mathematics, vol. 8, Cambridge University Press, Cambridge, 1986. Translated from the Japanese by M. Reid. MR**879273** - V. B. Mehta and V. Srinivas,
*A characterization of rational singularities*, Asian J. Math.**1**(1997), no. 2, 249–271. MR**1491985**, DOI 10.4310/AJM.1997.v1.n2.a4 - Alan Michael Nadel,
*Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature*, Ann. of Math. (2)**132**(1990), no. 3, 549–596. MR**1078269**, DOI 10.2307/1971429 - Anurag K. Singh,
*$F$-regularity does not deform*, Amer. J. Math.**121**(1999), no. 4, 919–929. MR**1704481**, DOI 10.1353/ajm.1999.0029 - Karen E. Smith,
*$F$-rational rings have rational singularities*, Amer. J. Math.**119**(1997), no. 1, 159–180. MR**1428062**, DOI 10.1353/ajm.1997.0007 - Karen E. Smith,
*The multiplier ideal is a universal test ideal*, Comm. Algebra**28**(2000), no. 12, 5915–5929. Special issue in honor of Robin Hartshorne. MR**1808611**, DOI 10.1080/00927870008827196 - S. Takagi,
*An interpretation of multiplier ideals via tight closure*, preprint. - Paolo Valabrega and Giuseppe Valla,
*Form rings and regular sequences*, Nagoya Math. J.**72**(1978), 93–101. MR**514892**, DOI 10.1017/S0027763000018225 - Adela Vraciu,
*Strong test ideals*, J. Pure Appl. Algebra**167**(2002), no. 2-3, 361–373. MR**1874550**, DOI 10.1016/S0022-4049(01)00039-1 - Keiichi Watanabe,
*$F$-regular and $F$-pure normal graded rings*, J. Pure Appl. Algebra**71**(1991), no. 2-3, 341–350. MR**1117644**, DOI 10.1016/0022-4049(91)90157-W - Lori J. Williams,
*Uniform stability of kernels of Koszul cohomology indexed by the Frobenius endomorphism*, J. Algebra**172**(1995), no. 3, 721–743. MR**1324179**, DOI 10.1006/jabr.1995.1067

## Additional Information

**Nobuo Hara**- Affiliation: Mathematical Institute, Tohoku University, Sendai 980–8578, Japan
- Email: hara@math.tohoku.ac.jp
**Ken-ichi Yoshida**- Affiliation: Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya 464–8602, Japan
- MR Author ID: 359418
- Email: yoshida@math.nagoya-u.ac.jp
- Received by editor(s): August 20, 2002
- Received by editor(s) in revised form: December 19, 2002
- Published electronically: April 11, 2003
- Additional Notes: Both authors are partially supported by a Grant-in-Aid for Scientific Research, Japan
- © Copyright 2003 American Mathematical Society
- Journal: Trans. Amer. Math. Soc.
**355**(2003), 3143-3174 - MSC (2000): Primary 13A35, 14B05
- DOI: https://doi.org/10.1090/S0002-9947-03-03285-9
- MathSciNet review: 1974679