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Homological Conjectures and Lim Cohen-Macaulay Sequences

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Book cover Homological and Computational Methods in Commutative Algebra

Part of the book series: Springer INdAM Series ((SINDAMS,volume 20))

Abstract

We discuss the new notion of a lim Cohen-Macaulay sequence of modules over a local ring, and also a somewhat weaker notion, as well as the theory of content for local cohomology modules. We relate both to the problem of proving the direct summand conjecture and other homological conjectures without using almost ring theory and perfectoid space theory, and we also indicate some other open problems whose solution would yield a new proof of the direct summand conjecture.

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References

  1. Y. André, La lemme d’Abhyankar perfectoide, preprint, 55 pp. arXiv:1609.00320 [math.AG] 31 Aug 2016

    Google Scholar 

  2. Y. André, La conjecture du facteur direct, preprint, 15 pp. arXiv:1609.00345 [math.AG] 1 Sept 2016.

    Google Scholar 

  3. M. Auslander, Modules over unramified regular local rings, in Proceedings of the International Congress of Mathematics in Stockholm 1962 (Almqvist and Wiksells, Uppsala, 1963), pp. 230–234

    Google Scholar 

  4. L. Avramov, H.-B. Foxby, B. Herzog, Structure of local homomorphisms. J. Algebra 164, 124–145 (1994)

    Google Scholar 

  5. J. Bartijn, J.R. Strooker, Modifications monomiales, in Paul Dubreil and Marie-Paule Malliavin Algebra Seminar, 35th Year (Paris, 1982). Lecture Notes in Mathematics, vol. 1029 (Springer, Berlin, 1983), pp. 192–217

    Google Scholar 

  6. H. Bass, On the ubiquity of Gorenstein rings. Math. Z. 82, 8–28 (1963)

    Google Scholar 

  7. P. Baum, W. Fulton, R. MacPherson, Riemann-Roch for singular varieties. Publ. Math. I.H.E.S. 45, 101–145 (1975)

    Google Scholar 

  8. P. Berthelot, Altérations de variétes algébriques [d’après A. J. de Jong](French). Alterations of algebraic varieties (following A. J. de Jong) Séminaire Bourbaki, Vol. 1995/96. Astérisque No. 241, Exp. No. 815, 5, 273–311 (1997)

    Google Scholar 

  9. B. Bhatt, Derived splinters in positive characteristic. Compos. Math. 148, 1757–1786 (2012)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  11. B. Bhatt, On the non-existence of small Cohen-Macaulay algebras. J. Algebra 411, 1–11 (2014)

    Google Scholar 

  12. B. Bhatt, On the direct summand conjecture and its derived variant, preprint, 10 pp. arXiv:1608.08882v1[math.AG] 31 Aug 2016

    Google Scholar 

  13. B. Bhatt, M. Hochster, L. Ma, Lim Cohen-Macaulay sequences of modules, preliminary preprint, 38 pp. (2016)

    Google Scholar 

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

    Google Scholar 

  15. M.K. Brown, C. Miller, P. Thompson, M.E. Walker, Cyclic adams operations. J. Pure Appl. Algebra, 221(7), 1589–1613 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  16. W. Bruns, “Jede” endliche freie Auflösung ist freie Auflösung eines von drei Elementen erzeugten Ideals. J. Algebra 39, 429–439 (1976)

    Google Scholar 

  17. W. Bruns, The Eisenbud-Evans generalized principal ideal theorem and determinantal ideals. Proc. Am. Math. Soc. 83, 19–24 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  18. W. Bruns, Tight closure. Bull. Am. Math. Soc. 33, 447–458 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  19. W. Bruns, J. Herzog, Cohen-Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39 (Cambridge University Press, Cambridge, 1993)

    Google Scholar 

  20. A.J. de Jong, Smoothness, semistability, and alterations. Publ. Math. I.H.E.S. 83 51–93 (1996)

    Google Scholar 

  21. G.D. Dietz, A characterization of closure operators that induce big Cohen-Macaulay modules. Proc. Am. Math. Soc. 138, 3849–3862 (2010)

    Google Scholar 

  22. S.P. Dutta, Frobenius and multiplicites. J. Algebra 85 424–448 (1983)

    Google Scholar 

  23. S.P. Dutta, Generalized intersection multiplicities of modules. Trans. Am. Math. Soc. 276, 657–669 (1983)

    Google Scholar 

  24. S.P. Dutta, On the canonical element conjecture. Trans. Am. Math. Soc. 299, 803–811 (1987)

    Google Scholar 

  25. S.P. Dutta, A special case of positivity. Proc. Am. Math. Soc. 103, 344–346 (1988)

    Google Scholar 

  26. S.P. Dutta, A note on Chow groups and intersection multiplicity of modules. J. Algebra 161, 186–198 (1993)

    Google Scholar 

  27. S.P. Dutta, A theorem on smoothness–Bass-Quillen, Chow groups and intersection multiplicity of Serre. Trans. Am. Math. Soc. 352, 1635–1645 (2000)

    Google Scholar 

  28. S.P. Dutta, A special case of positivity. II. Proc. Am. Math. Soc. 133, 1891–1896 (2005)

    Google Scholar 

  29. S.P. Dutta, M. Hochster, J.E. McLaughlin, Modules of finite projective dimension with negative intersection multiplicities. Invent. Math. 79, 253–291 (1985)

    Google Scholar 

  30. G. Evans, P. Griffith, The syzygy problem. Ann. Math. 114, 323–333 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  31. G. Faltings, Almost étale extensions. Ast’erisque 279, 185–270 (2002)

    MATH  Google Scholar 

  32. W. Fulton, Intersection Theory (Springer, Berlin, 1984)

    Book  MATH  Google Scholar 

  33. O. Gabber, L. Ramero, Almost Ring Theory. Lecture Notes in Mathematics, vol. 1800 (Springer, Berlin, 2003)

    Google Scholar 

  34. H. Gillet, C. Soulé, K-théorie et nullité des multiplicités d’intersection. C. R. Acad. Sci. Paris Sér. I no. 3 t. 300 71–74 (1985)

    Google Scholar 

  35. R.C. Heitmann, A counterexample to the rigidity conjecture for rings. Bull. Am. Math. Soc. New Series 29, 94–97 (1993)

    Google Scholar 

  36. R.C. Heitmann, Extensions of plus closure. J. Algebra 238, 801–826 (2001)

    Google Scholar 

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

    Google Scholar 

  38. R.C. Heitmann, L. Ma, Big Cohen-Macaulay Algebras and the vanishing conjecture for maps of Tor in mixed characteristic (2017, preprint), arXiv:1703.08281v1 [math.AC]

    Google Scholar 

  39. M. Hochster, Cohen-Macaulay modules, in Conference on Commutative Algebra, Lecture Notes in Mathematics, vol. 311 (Springer, Berlin/New York, 1973), pp. 120–152

    Book  Google Scholar 

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

    Google Scholar 

  41. M. Hochster, Topics in the Homological Theory of Modules Over Commutative Rings. C.B.M.S. Regional Conference Series in Mathematics, vol. 24 (American Mathematical Society, Providence, RI, 1975)

    Google Scholar 

  42. M. Hochster, Big Cohen-Macaulay modules and algebras and embeddability in rings of Witt vectors. Queen’s Pap. Pure Appl. Math. 42, 106–195 (1975)

    MathSciNet  MATH  Google Scholar 

  43. M. Hochster, Cyclic purity versus purity in excellent Noetherian rings. Trans. Am. Math. Soc. 231, 463–488 (1977)

    Google Scholar 

  44. M. Hochster, Parameter-like sequences and extensions of tight closure, in Commutative Ring Theory and Applications, Lecture Notes in Pure and Applied Mathematics, vol. 231 (Marcel Dekker, New York, 2003), pp. 267–287

    MATH  Google Scholar 

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

    Google Scholar 

  46. M. Hochster, Solid closure, in Commutative Algebra: Syzygies, Multiplicities and Birational Algebra. Contemporary Mathematics, vol. 159 (American Mathematical Society, Providence, RI, 1994), pp. 103–172

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  50. M. Hochster, C. Huneke, Tight closure of parameter ideals and splitting in module-finite extensions. J. Algebr. Geom. 3, 599–670 (1994)

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  52. M. Hochster, C. Huneke, Quasilength, latent regular sequences, and content of local cohomology. J. Algebra 322 3170–3193 (2009)

    Google Scholar 

  53. M. Hochster, J.L. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay. Adv. Math. 13, 115–175 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  54. M. Hochster, J. Vélez, Diamond closure, in Algebra, Arithmetic and Geometry with Applications (Springer, Berlin, 2004), pp. 511–523

    Book  MATH  Google Scholar 

  55. M. Hochster, Y. Yao, Splitting theorems for the Frobenius endomorphism and strong F-regularity, in preparation

    Google Scholar 

  56. M. Hochster, W. Zhang, Content of local cohomology, parameter ideals, and robust algebras, Trans. Amer. Math. Soc., to appear. arXiv:1609.07017 [math.AC]

    Google Scholar 

  57. C. Huneke, Tight closure and its applications, in Proceeding of the CBMS Conference Held at Fargo, North Dakota, July 1995. C.B.M.S. Regional Conference Series (American Mathematical Society, Providence, RI, 1996)

    Google Scholar 

  58. C. Lech, Note on multiplicities of ideals. Ark. Mat. 4, 63–86 (1960)

    Google Scholar 

  59. L. Ma, Lech’s conjecture in dimension three (2016, preprint), arXiv:1609.00095v2 [math.AC]

    Google Scholar 

  60. L. Ma, The vanishing conjecture for maps of Tor and derived splinters. J. Eur. Math. Soc., to appear

    Google Scholar 

  61. L. Ma, The Vanishing Conjecture for Maps of Tor, Splinters, and Derived Splinters. Lecture Notes for the UIC Homological Conjectures Workshop (November, 18–20, 2016) available at http://kftucker.people.uic.edu/homconj/assets/vanishingoftor-lecture.pdf.

  62. H. Matsumura, Commutative Algebra (Benjamin, New York, 1970)

    MATH  Google Scholar 

  63. C. Peskine, L. Szpiro, Dimension projective finie et cohomologie locale. Publ. Math. I.H.E.S. (Paris) 42, 323–395 (1973)

    Google Scholar 

  64. C. Peskine, L. Szpiro, Syzygies et multiplicités. C.R. Acad. Sci. Paris Sér. A 278, 1421–1424 (1974)

    Google Scholar 

  65. N. Ranganathan, Splitting in module-finite extension rings and the vanishing conjecture for maps of Tor, Thesis, University of Michigan, 2000

    Google Scholar 

  66. R. Rebhuhn-Glanz, Closure operations that induce big Cohen-Macaulay modules and algebras, and classification of singularities, Thesis, University of Michigan, 2016

    Google Scholar 

  67. R. Rebhuhn-Glanz, Closure operations that induce big Cohen-Macaulay modules and classification of singularities, preprint, arXiv:1512.06843

    Google Scholar 

  68. R. Rebhuhn-Glanz, Closure operations that induce big Cohen-Macaulay algebras, preprint, arXiv:1512.07862

    Google Scholar 

  69. P. Roberts, Two applications of dualizing complexes over local rings. Ann. Sci. Éc. Norm. Sup. 9, 103–106 (1976)

    Google Scholar 

  70. P. Roberts, Cohen-Macaulay complexes and an analytic proof of the new intersection conjecture, J. Algebra 66, 225–230 (1980)

    Google Scholar 

  71. P. Roberts, The vanishing of intersection multiplicities of perfect complexes. Bull. Am. Math. Soc. 13, 127–130 (1985)

    Google Scholar 

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

    Google Scholar 

  73. P. Roberts, Intersection theorems, in Commutative Algebra. Mathematical Sciences Research Institute Publications, vol. 15 (Springer, New York-Berlin-Heidelberg, 1989), pp. 417–436

    Google Scholar 

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

    Google Scholar 

  75. J.-P. Serre, Algèbre locale ⋅ Multiplicités. Springer-Verlag Lecture Notes in Mathematics, vol. 11, 2nd edn. (Springer, Berlin, 1965)

    Google Scholar 

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Acknowledgements

The author is indebted both to the referee and to Linquan Ma for their valuable comments, corrections, and suggested improvements.

The author was partially supported by grants from the National Science Foundation (DMS–0901145 and DMS–1401384).

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Authors and Affiliations

  1. Mathematics Department, University of Michigan, East Hall, 530 Church St., Ann Arbor, MI, 48019-1043, USA

    Melvin Hochster

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  1. Melvin Hochster
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Correspondence to Melvin Hochster .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Genova, Genova, Italy

    Aldo Conca

  2. Department of Mathematics, San Francisco State University, San Francisco, California, USA

    Joseph Gubeladze

  3. FB Mathematik/Informatik, Universität Osnabrück, Osnabrück, Germany

    Tim Römer

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Hochster, M. (2017). Homological Conjectures and Lim Cohen-Macaulay Sequences. In: Conca, A., Gubeladze, J., Römer, T. (eds) Homological and Computational Methods in Commutative Algebra. Springer INdAM Series, vol 20. Springer, Cham. https://doi.org/10.1007/978-3-319-61943-9_11

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