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Singularities in mixed characteristic. The perfectoid approach

  • Special Feature: The Takagi Lectures
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Japanese Journal of Mathematics Aims and scope

Abstract

The homological conjectures, which date back to Peskine, Szpiro and Hochster in the late 60’s, make fundamental predictions about syzygies and intersection problems in commutative algebra. They were settled long ago in the presence of a base field and led to tight closure theory, a powerful tool to investigate singularities in characteristic p.

Recently, perfectoid techniques coming from p-adic Hodge theory have allowed us to get rid of any base field; this solves the direct summand conjecture and establishes the existence and weak functoriality of big Cohen-Macaulay algebras, which solve in turn the homological conjectures in general. This also opens the way to the study of singularities in mixed characteristic.

We sketch a broad outline of this story, taking lastly a glimpse at ongoing work by L. Ma and K. Schwede, which shows how such a study could build a bridge between singularity theory in characteristic p and in characteristic 0.

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References

  1. Y. André, Le lemme d’Abhyankar perfectoïde, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 1–70.

    Article  MathSciNet  Google Scholar 

  2. Y. André, La conjecture du facteur direct, Publ. Math. Inst. Hautes Études Sci., 127 (2018), 71–93.

    Article  MathSciNet  Google Scholar 

  3. Y. André, Perfectoid spaces and the homological conjectures, In: Proc. Int. Cong. of Math.—2018, Rio de Janeiro, Vol. 1, 2018, pp. 249–260.

    Google Scholar 

  4. Y. André, Weak functoriality of Cohen-Macaulay algebras, preprint, arXiv:1801.10010.

  5. B. Bhatt, Almost direct summands, Nagoya Math. J., 214 (2014), 195–204.

    Article  MathSciNet  Google Scholar 

  6. B. Bhatt, S.B. Iyengar and L. Ma, Regular rings and perfect(oid) algebras, preprint, arXiv:1803.03229.

  7. W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge Stud. Adv. Math., 39, Cambridge Univ. Press, Cambridge, 1993.

    MATH  Google Scholar 

  8. E.G. Evans and P. Griffith, The syzygy problem, Ann. of Math. (2), 114 (1981), 323–333.

    Article  MathSciNet  Google Scholar 

  9. G. Faltings, Almost étale extensions, In: p-adic Cohomologies and Arithmetic Applications (II), Astérisque, 279, Soc. Math. France, Paris, 2002, pp. 185–270.

    MATH  Google Scholar 

  10. O. Gabber, Some remarks on big Cohen-Macaulay algebras, talk at MSRI, Feb. 2018.

    Google Scholar 

  11. O. Gabber and L. Ramero, Almost Ring Theory, Lecture Notes in Math., 1800, Springer-Verlag, 2003.

    Book  Google Scholar 

  12. N. Hara, A characterization of rational singularities in terms of injectivity of Frobenius maps, Amer. J. Math., 120 (1998), 981–996.

    Article  MathSciNet  Google Scholar 

  13. R.C. Heitmann, The direct summand conjecture in dimension three, Ann. of Math. (2), 156 (2002), 695–712.

    Article  MathSciNet  Google Scholar 

  14. R.C. Heitmann and L. Ma, Big Cohen-Macaulay algebras and the vanishing conjecture for maps of Tor in mixed characteristic, Algebra Number Theory, 12 (2018), 1659–1674.

    Article  MathSciNet  Google Scholar 

  15. M. Hochster, Contracted ideals from integral extensions of regular rings, Nagoya Math. J., 51 (1973), 25–43.

    Article  MathSciNet  Google Scholar 

  16. M. Hochster, Canonical elements in local cohomology modules and the direct summand conjecture, J. Algebra, 84 (1983), 503–553.

    Article  MathSciNet  Google Scholar 

  17. M. Hochster, Big Cohen-Macaulay algebras in dimension three via Heitmann’s theorem, J. Algebra, 254 (2002), 395–408.

    Article  MathSciNet  Google Scholar 

  18. M. Hochster, Homological conjectures, old and new, Illinois J. Math., 51 (2007), 151–169.

    Article  MathSciNet  Google Scholar 

  19. M. Hochster and C. Huneke, Tight closure, invariant theory, and the Briançon-Skoda theorem, J. Amer. Math. Soc., 3 (1990), 31–116.

    MathSciNet  MATH  Google Scholar 

  20. M. Hochster and C. Huneke, Infinite integral extensions and big Cohen-Macaulay algebras, Ann. of Math. (2), 135 (1992), 53–89.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  22. S. Ishii, Introduction to Singularities. 2nd ed., Springer-Verlag, 2018.

    Book  Google Scholar 

  23. K.S. Kedlaya and Liu, Relative p-adic Hodge Theory: Foundations, Astérisque, 371, Soc. Math. France, Paris, 2015.

    MATH  Google Scholar 

  24. L. Ma and K. Schwede, Perfectoid multiplier/test ideals in regular rings and bounds on symbolic powers, Invent. Math., 214 (2018), 913–955.

    Article  MathSciNet  Google Scholar 

  25. L. Ma and K. Schwede, Singularities in mixed characteristic via perfectoid big Cohen- Macaulay algebras, preprint, arXiv:1806.09567.

  26. V.B. Mehta and V. Srinivas, A characterization of rational singularities, Asian J. Math., 1 (1997), 249–271.

    Article  MathSciNet  Google Scholar 

  27. T. Ogoma, A note on the syzygy problem, Comm. Algebra, 17 (1989), 2061–2066.

    Article  MathSciNet  Google Scholar 

  28. C. Peskine and L. Szpiro, Dimension projective finie et cohomologie locale, Inst. Hautes Études Sci. Publ. Math., 42 (1973), 47–119.

    Article  Google Scholar 

  29. M. Raynaud and L. Gruson, Critères de platitude et de projectivité, Invent. Math., 13 (1971), 1–89.

    Article  MathSciNet  Google Scholar 

  30. P. Roberts, Le théorème d’intersection, C. R. Acad. Sci. Paris Sér. I Math., 304 (1987), 177–180.

    MathSciNet  MATH  Google Scholar 

  31. P. Scholze, Perfectoid spaces, Publ. Math. Inst. Hautes Études Sci., 116 (2012), 245–313.

    Article  MathSciNet  Google Scholar 

  32. P. Scholze, On torsion in the cohomology of locally symmetric varieties, Ann. of Math. (2), 182 (2015), 945–1066.

    Article  MathSciNet  Google Scholar 

  33. K. Shimomoto, An application of the almost purity theorem to the homological conjectures, J. Pure Appl. Algebra, 220 (2016), 621–632.

    Article  MathSciNet  Google Scholar 

  34. K. Shimomoto, Integral perfectoid big Cohen-Macaulay algebras via André’s theorem, Math. Ann., 372 (2018), 1167–1188.

    Article  MathSciNet  Google Scholar 

  35. K.E. Smith, F-rational rings have rational singularities, Amer. J. Math., 119 (1997), 159–180.

    Article  MathSciNet  Google Scholar 

  36. S. Takagi and K.-i. Watanabe, F-singularities: applications of characteristic p methods to singularity theory, Sugaku Expositions, 31 (2018), 1–42.

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Sorbonne-Université, Institut de Mathématiques de Jussieu, 4 Place Jussieu, 75005, Paris, France

    Yves André

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  1. Yves André
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Correspondence to Yves André.

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Communicated by: Takeshi Saito

This article is based on the 22nd Takagi Lectures that the author delivered at The University of Tokyo on November 17–18, 2018.

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André, Y. Singularities in mixed characteristic. The perfectoid approach. Jpn. J. Math. 14, 231–247 (2019). https://doi.org/10.1007/s11537-019-1840-4

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  • DOI: https://doi.org/10.1007/s11537-019-1840-4

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