Algebra & Number Theory

When are permutation invariants Cohen–Macaulay over all fields?

Ben Blum-Smith and Sophie Marques

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Abstract

We prove that the polynomial invariants of a permutation group are Cohen–Macaulay for any choice of coefficient field if and only if the group is generated by transpositions, double transpositions, and 3-cycles. This unites and generalizes several previously known results. The “if” direction of the argument uses Stanley–Reisner theory and a recent result of Christian Lange in orbifold theory. The “only if” direction uses a local-global result based on a theorem of Raynaud to reduce the problem to an analysis of inertia groups, and a combinatorial argument to identify inertia groups that obstruct Cohen–Macaulayness.

Article information

Source
Algebra Number Theory, Volume 12, Number 7 (2018), 1787-1821.

Dates
Received: 26 February 2018
Revised: 16 May 2018
Accepted: 17 June 2018
First available in Project Euclid: 9 November 2018

Permanent link to this document
https://projecteuclid.org/euclid.ant/1541732442

Digital Object Identifier
doi:10.2140/ant.2018.12.1787

Mathematical Reviews number (MathSciNet)
MR3871511

Zentralblatt MATH identifier
06976303

Subjects
Primary: 13A50: Actions of groups on commutative rings; invariant theory [See also 14L24]
Secondary: 05E40: Combinatorial aspects of commutative algebra

Keywords
invariant theory modular invariant theory henselization Stanley–Reisner Cohen–Macaulay commutative ring finite group

Citation

Blum-Smith, Ben; Marques, Sophie. When are permutation invariants Cohen–Macaulay over all fields?. Algebra Number Theory 12 (2018), no. 7, 1787--1821. doi:10.2140/ant.2018.12.1787. https://projecteuclid.org/euclid.ant/1541732442


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References

  • B. Blum-Smith, Two inquiries about finite groups and well-behaved quotients, Ph.D. thesis, New York University, 2017, https://search.proquest.com/docview/1938647220.
  • N. Bourbaki, Éléments de mathématique: Algèbre commutative, Chapitres 5–6, Actualités Scientifiques et Industrielles 1308, Hermann, Paris, 1964.
  • W. Bruns and J. Herzog, Cohen–Macaulay rings, Cambridge Studies in Advanced Mathematics 39, Cambridge University Press, 1993.
  • H. E. A. Campbell, A. V. Geramita, I. P. Hughes, R. J. Shank, and D. L. Wehlau, “Non-Cohen–Macaulay vector invariants and a Noether bound for a Gorenstein ring of invariants”, Canad. Math. Bull. 42:2 (1999), 155–161.
  • E. Dufresne, J. Elmer, and M. Kohls, “The Cohen–Macaulay property of separating invariants of finite groups”, Transform. Groups 14:4 (2009), 771–785.
  • A. M. Duval, “Free resolutions of simplicial posets”, J. Algebra 188:1 (1997), 363–399.
  • A. Grothendieck, “Eléments de géométrie algébrique, I: Le langage des schémas”, Inst. Hautes Études Sci. Publ. Math. 4 (1960), 5–228.
  • D. Eisenbud, Commutative algebra with a view toward algebraic geometry, Graduate Texts in Mathematics 150, Springer, 1995.
  • G. Ellingsrud and T. Skjelbred, “Profondeur d'anneaux d'invariants en caractéristique $p$”, Compositio Math. 41:2 (1980), 233–244.
  • A. M. Garsia, “Combinatorial methods in the theory of Cohen–Macaulay rings”, Adv. in Math. 38:3 (1980), 229–266.
  • A. M. Garsia and D. Stanton, “Group actions of Stanley–Reisner rings and invariants of permutation groups”, Adv. in Math. 51:2 (1984), 107–201.
  • N. Gordeev and G. Kemper, “On the branch locus of quotients by finite groups and the depth of the algebra of invariants”, J. Algebra 268:1 (2003), 22–38.
  • T. D. Hamilton and T. Marley, “Non-Noetherian Cohen–Macaulay rings”, J. Algebra 307:1 (2007), 343–360.
  • P. Hersh, “Lexicographic shellability for balanced complexes”, J. Algebraic Combin. 17:3 (2003), 225–254.
  • P. Hersh, “A partitioning and related properties for the quotient complex $\Delta(B_{lm})/S_l\wr S_m$”, J. Pure Appl. Algebra 178:3 (2003), 255–272.
  • M. Hochster and J. A. Eagon, “Cohen–Macaulay rings, invariant theory, and the generic perfection of determinantal loci”, Amer. J. Math. 93 (1971), 1020–1058.
  • W. C. Huffman, “Imprimitive linear groups generated by elements containing an eigenspace of codimension two”, J. Algebra 63:2 (1980), 499–513.
  • V. Kac and K. Watanabe, “Finite linear groups whose ring of invariants is a complete intersection”, Bull. Amer. Math. Soc. $($N.S.$)$ 6:2 (1982), 221–223.
  • G. Kemper, “On the Cohen–Macaulay property of modular invariant rings”, J. Algebra 215:1 (1999), 330–351.
  • G. Kemper, “The depth of invariant rings and cohomology”, J. Algebra 245:2 (2001), 463–531.
  • G. Kemper, “The Cohen–Macaulay property and depth in invariant theory”, pp. 53–63 in Proceedings of the 33rd Symposium on Commutative Algebra (Japan), 2012.
  • C. Lange, “Characterization of finite groups generated by reflections and rotations”, J. Topol. 9:4 (2016), 1109–1129.
  • C. Lange and M. A. Mikhaîlova, “Classification of finite groups generated by reflections and rotations”, Transform. Groups 21:4 (2016), 1155–1201.
  • Q. Liu, Algebraic geometry and arithmetic curves, Oxford Graduate Texts in Mathematics 6, Oxford University Press, 2002.
  • M. Lorenz and J. Pathak, “On Cohen–Macaulay rings of invariants”, J. Algebra 245:1 (2001), 247–264.
  • J. R. Munkres, “Topological results in combinatorics”, Michigan Math. J. 31:1 (1984), 113–128.
  • M. D. Neusel and L. Smith, Invariant theory of finite groups, Mathematical Surveys and Monographs 94, American Mathematical Society, Providence, RI, 2002.
  • M. Raynaud, Anneaux locaux henséliens, Lecture Notes in Math. 169, Springer, 1970.
  • V. Reiner, “Quotients of Coxeter complexes and $P$-partitions”, pp. vi+134 Mem. Amer. Math. Soc. 460, Amer. Math. Soc., Providence, RI, 1992.
  • V. Reiner, Appendix to [hersh2?], pages 269–271 in J. Pure Appl. Algebra 178:3 (2003), 255–272.
  • G. A. Reisner, “Cohen–Macaulay quotients of polynomial rings”, Advances in Math. 21:1 (1976), 30–49.
  • C. P. Rourke and B. J. Sanderson, Introduction to piecewise-linear topology, Ergebnisse der Mathematik 69, Springer, Berlin, 1972. Reprinted in the series Springer Study Edition 69, Springer, Berlin, 1982.
  • L. Smith, Polynomial invariants of finite groups, Research Notes in Mathematics 6, A K Peters, Ltd., Wellesley, MA, 1995.
  • L. Smith, “Some rings of invariants that are Cohen–Macaulay”, Canad. Math. Bull. 39:2 (1996), 238–240.
  • R. P. Stanley, Enumerative combinatorics, I, Wadsworth & Brooks/Cole, Monterey, CA, 1986.
  • R. P. Stanley, “$f$-vectors and $h$-vectors of simplicial posets”, J. Pure Appl. Algebra 71:2-3 (1991), 319–331.