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Lim Ulrich sequences and Lech’s conjecture

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Abstract

The long standing Lech’s conjecture in commutative algebra states that for a flat local extension \((R,\mathfrak {m})\rightarrow (S,\mathfrak {n})\) of Noetherian local rings, we have an inequality on the Hilbert–Samuel multiplicities: \(e(R)\le e(S)\). In general the conjecture is wide open when \(\dim R>3\), even in equal characteristic. In this paper, we prove Lech’s conjecture in all dimensions, provided \((R,\mathfrak {m})\) is a standard graded ring over a perfect field localized at the homogeneous maximal ideal. We introduce the notions of lim Ulrich and weakly lim Ulrich sequences. Roughly speaking these are sequences of finitely generated modules that are not necessarily Cohen–Macaulay, but asymptotically behave like Ulrich modules. We prove that the existence of these sequences imply Lech’s conjecture. Though the existence of Ulrich modules is known in very limited cases, we construct weakly lim Ulrich sequences for all standard graded domains over perfect fields of positive characteristic.

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Notes

  1. In [23], we are not assuming R is the completion of a finite type algebra therefore we choose a complete regular local ring A inside R and descend data to the Henselization of the localization of a polynomial ring, while here R is finite type (in fact standard graded) over k so we can run the same argument over R, the counter-example then descends to the Henselization of \(R_\mathfrak {m}\) and thus to a pointed étale extension of \(R_\mathfrak {m}\).

References

  1. Bhatt, B., Hochster, M., Ma, L.: Lim Cohen–Macaulay sequences. (in preparation)

  2. Brennan, J.P., Herzog, J., Ulrich, B.: Maximally generated Cohen–Macaulay modules. Math. Scand. 61(2), 181–203 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  3. Bruns, W., Herzog, J.: Cohen–Macaulay Rings. Cambridge Studies in Advanced Mathematics, vol. 39. Cambridge University Press, Cambridge (1993). (1251956)

  4. Bruns, W., Römer, T., Wiebe, A.: Initial algebras of determinantal rings, Cohen–Macaulay and Ulrich ideals. Michigan Math. J. 53(1), 71–81 (2005). (2125534)

    Article  MathSciNet  MATH  Google Scholar 

  5. Eisenbud, D., Erman, D.: Categorified duality in Boij–Söderberg theory and invariants of free complexes. J. Eur. Math. Soc. (JEMS) 19(9), 2657–2695 (2017). (3692884)

    Article  MathSciNet  MATH  Google Scholar 

  6. Eisenbud, D., Schreyer, F.-O.: Betti numbers of graded modules and cohomology of vector bundles. J. Am. Math. Soc. 22(3), 859–888 (2009). (2505303)

    Article  MathSciNet  MATH  Google Scholar 

  7. Eisenbud, D., Schreyer, F.-O., Weyman, J.: Resultants and Chow forms via exterior syzygies. J. Am. Math. Soc. 16(3), 537–579 (2003). (1969204)

    Article  MathSciNet  MATH  Google Scholar 

  8. Goto, S., Watanabe, K.: On graded rings I. J. Math. Soc. Jpn. 30(2), 179–213 (1978)

    Article  MATH  Google Scholar 

  9. Hanes, D.: Special conditions on maximal Cohen–Macaulay modules, and applications to the theory of multiplicities. Thesis, University of Michigan (1999)

  10. Hanes, D.: Length approximations for independently generated ideals. J. Algebra 237(2), 708–718 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. Hanes, D.: On the Cohen–Macaulay modules of graded subrings. Trans. Am. Math. Soc. 357(2), 735–756 (2005). (2095629)

    Article  MathSciNet  MATH  Google Scholar 

  12. Herzog, B.: Kodaira–Spencer Maps in Local Algebra. Springer, Berlin (1994)

    Book  MATH  Google Scholar 

  13. Herzog, J., Ulrich, B., Backelin, J.: Linear maximal Cohen–Macaualy modules over strict complete intersections. J. Pure Appl. Algebra 71(2–3), 187–202 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  14. Hochster, M.: Cohen–Macaulay modules. In: Conference on Commutative Algebra (Univ. Kansas, Lawrence, Kan., 1972). Lecture Notes in Math., vol. 311, pp. 120–152 (1973). (0340251)

  15. Hochster, M.: Homological conjectures and lim Cohen–Macaulay sequences. In: Homological and Computational Methods in Commutative Algebra, Springer INdAM Ser., vol. 20, pp. 173–197. Springer, Cham (2017). (3751886)

  16. Hochster, M., Huneke, C.: Tight closure, invariant theory, and the Briançon–Skoda theorem. J. Am. Math. Soc. 3(1), 31–116 (1990)

    MATH  Google Scholar 

  17. Huneke, C., Swanson, I.: Integral Closure of Ideals, Rings, and Modules. London Mathematical Society Lecture Note Series, vol. 336. Cambridge University Press, Cambridge (2006). (2266432)

  18. Iyengar, S., Ma, L., Walker, M.: Multiplicities and Betti numbers in local algebra via lim Ulrich points. arXiv:2104.10140 (To appear in Algebra Number Theory)

  19. Iyengar, S., Ma, L., Walker, M.: Lim Ulrich sequences and the Boij–Söderberg cones. (in preparation)

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

    Article  MathSciNet  MATH  Google Scholar 

  21. Lech, C.: Inequalities related to certain couples of local rings. Acta. Math 112, 69–89 (1964)

    Article  MathSciNet  MATH  Google Scholar 

  22. Ma, L.: The Frobenius endomorphism and multiplicities. University of Michigan, Thesis (2014)

  23. Ma, L.: Lech’s conjecture in dimension three. Adv. Math. 322, 940–970 (2017). (3720812)

    Article  MathSciNet  MATH  Google Scholar 

  24. Matsumura, H.: Commutative Ring Theory. Cambridge Univ. Press, Cambridge (1986)

    MATH  Google Scholar 

  25. Meng, C.: Strongly Lech-independent ideals and Lech’s conjecture. arXiv:2112.09849

  26. Nagata, M.: Local Rings. Wiley, New York (1962)

    MATH  Google Scholar 

  27. Saccochi, R.C.: Ulrich Schur Bundles in Prime Characteristic p. Thesis (Ph.D.), University of Illinois at Chicago, ProQuest LLC, Ann Arbor, MI (2020). (4257373)

  28. Serre, J.-P.: Algèbre locale. Multiplicités, Cours au Collège de France, 1957–1958, rédigé par Pierre Gabriel. Seconde édition, 1965. Lecture Notes in Mathematics, vol. 11, pp. vii+188 pp. Springer, Berlin (1965)

  29. Ulrich, B.: Gorenstein rings and modules with high numbers of generators. Math. Z. 188(1), 23–32 (1984). (767359)

    Article  MathSciNet  MATH  Google Scholar 

  30. Yhee, F.: Ulrich modules and weakly lim Ulrich sequences do not always exist. arXiv:2104.05766

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Acknowledgements

Many ideas of this manuscript originate from [1], I would like to thank Bhargav Bhatt and Mel Hochster for initiating this collaboration. In particular, I thank Mel Hochster for valuable discussions on various weak notions of lim Cohen–Macaulay sequences. I would also like to thank David Eisenbud, Ray Heitmann, Srikanth Iyengar, Bernd Ulrich and Mark Walker for their comments on this manuscript. Finally, I would like to thank the anonymous referees for their very detailed comments and suggestions.

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  1. Department of Mathematics, Purdue University, West Lafayette, IN, 47907, USA

    Linquan Ma

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The author is partially supported by NSF Grant DMS #1901672, NSF FRG Grant #1952366, and a fellowship from the Sloan Foundation.

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Ma, L. Lim Ulrich sequences and Lech’s conjecture. Invent. math. 231, 407–429 (2023). https://doi.org/10.1007/s00222-022-01149-2

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