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Tight Closure in Positive Characteristic

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Part of the Lecture Notes in Mathematics book series (LNM,volume 1999)

Abstract

In this chapter, p is a fixed prime number, and all rings are assumed to have characteristic p, unless explicitly mentioned otherwise. We review the notion of tight closure due toHochster and Huneke (as a general reference, we will use [59]). The main protagonist in this elegant theory is the p-th power Frobenius map. We will focus on five key properties of tight closure, which will enable us to prove, virtually effortlessly, several beautiful theorems. Via these five properties, we can give a more axiomatic treatment, which lends itself nicely to generalization, and especially to a similar theory in characteristic zero (see Chapters 6 and 7).

Keywords

  • Local Ring
  • Characteristic Zero
  • Noetherian Ring
  • Integral Closure
  • Discrete Valuation Ring

These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. Boutot, J.F.: Singularités rationelles et quotients par les groupes réductifs. Invent. Math. 88, 65–68 (1987) 78

    CrossRef  MATH  MathSciNet  Google Scholar 

  2. Brenner, H., Monsky, P.: Tight closure does not commute with localization (2007). ArXiv:0710.2913 70, 92

    Google Scholar 

  3. Briançon, J., Skoda, H.: Sur la clôture intégrale d’un idéal de germes de fonctions holomorphes en un point de Cn. C. R. Acad. Sci. Paris 278, 949–951 (1974) 75

    MATH  Google Scholar 

  4. Denef, J., Schoutens, H.: On the decidability of the existential theory of Fp[[t]]. Amer. Math. Soc. (2003) 103

    Google Scholar 

  5. Goldblatt, R.: Lectures on the hyperreals, Graduate Texts in Mathematics, vol. 188. Springer-Verlag, New York (1998). An introduction to nonstandard analysis 15

    Google Scholar 

  6. Henkin, L.: Some interconnections between modern algebra and mathematical logic. Trans. Amer. Math. Soc. 74, 410–427 (1953) 179, 187

    MATH  MathSciNet  Google Scholar 

  7. Hochster, M., Huneke, C.: Applications of the existence of big Cohen-Macaulay algebras. Adv. in Math. 113, 45–117 (1995) 93

    CrossRef  MATH  MathSciNet  Google Scholar 

  8. Hochster, M., Huneke, C.: Tight closure. In: Commutative Algebra, vol. 15, pp. 305–338 (1997) 3

    Google Scholar 

  9. Hodges, W.: Model Theory. Cambridge University Press, Cambridge (1993) 8, 11, 14, 15, 16

    CrossRef  MATH  Google Scholar 

  10. Olberding, B., Saydam, S., Shapiro, J.: Completions, valuations and ultrapowers of Noetherian domains. J. Pure Appl. Algebra 197(1-3), 213–237 (2005) 58

    CrossRef  MATH  MathSciNet  Google Scholar 

  11. Schoutens, H.: Bounds in cohomology. Israel J. Math. 116, 125–169 (2000) 51, 59, 62

    CrossRef  MATH  MathSciNet  Google Scholar 

  12. Schoutens, H.: Projective dimension and the singular locus. Comm. Algebra 31, 217–239 (2003) 46

    CrossRef  MATH  MathSciNet  Google Scholar 

  13. Schoutens, H.: Asymptotic homological conjectures in mixed characteristic. Pacific J. Math. 230, 427–468 (2007) 114, 149, 152, 155, 161

    CrossRef  MATH  MathSciNet  Google Scholar 

  14. Serre, J.: Corps Locaux. Hermann, Paris (1968) 171, 172

    Google Scholar 

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Correspondence to Hans Schoutens .

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Schoutens, H. (2010). Tight Closure in Positive Characteristic. In: The Use of Ultraproducts in Commutative Algebra. Lecture Notes in Mathematics(), vol 1999. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-13368-8_5

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