On the Castelnuovo-Mumford regularity of rings of polynomial invariants

Abstract

We show that when a group acts on a polynomial ring over a field the ring of invariants has Castelnuovo-Mumford regularity at most zero. As a consequence, we prove a well-known conjecture that the invariants are always generated in degrees at most $n(|G|-1)$, where $n >1$ is the number of polynomial generators and $|G|>1$ is the order of the group. We also prove some other related conjectures in invariant theory.

  • [bensonpoly] D. J. Benson, Polynomial Invariants of Finite Groups, Cambridge: Cambridge Univ. Press, 1993, vol. 190.
    @book {bensonpoly, MRKEY = {1249931},
      AUTHOR = {Benson, David J.},
      TITLE = {Polynomial Invariants of Finite Groups},
      SERIES = {London Math. Soc. Lecture Note Ser.},
      VOLUME = {190},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1993},
      PAGES = {x+118},
      ISBN = {0-521-45886-2},
      MRCLASS = {13A50 (13E15 20F29)},
      MRNUMBER = {1249931},
      MRREVIEWER = {Frank D. Grosshans},
      ZBLNUMBER = {0864.13001},
      }
  • [b1] Go to document D. J. Benson, "Dickson invariants, regularity and computation in group cohomology," Illinois J. Math., vol. 48, iss. 1, pp. 171-197, 2004.
    @article {b1, MRKEY = {2048221},
      AUTHOR = {Benson, David J.},
      TITLE = {Dickson invariants, regularity and computation in group cohomology},
      JOURNAL = {Illinois J. Math.},
      FJOURNAL = {Illinois Journal of Mathematics},
      VOLUME = {48},
      YEAR = {2004},
      NUMBER = {1},
      PAGES = {171--197},
      ISSN = {0019-2082},
      CODEN = {IJMTAW},
      MRCLASS = {20J06 (13A50 13D45)},
      MRNUMBER = {2048221},
      MRREVIEWER = {Ioannis Emmanouil},
      URL = {http://projecteuclid.org/getRecord?id=euclid.ijm/1258136180},
      ZBLNUMBER = {1041.20036},
      }
  • [b2] Go to document D. J. Benson, "On the regularity conjecture for the cohomology of finite groups," Proc. Edinb. Math. Soc., vol. 51, iss. 2, pp. 273-284, 2008.
    @article {b2, MRKEY = {2465907},
      AUTHOR = {Benson, David J.},
      TITLE = {On the regularity conjecture for the cohomology of finite groups},
      JOURNAL = {Proc. Edinb. Math. Soc.},
      FJOURNAL = {Proceedings of the Edinburgh Mathematical Society. Series II},
      VOLUME = {51},
      YEAR = {2008},
      NUMBER = {2},
      PAGES = {273--284},
      ISSN = {0013-0915},
      MRCLASS = {20J06 (13D45)},
      MRNUMBER = {2465907},
      MRREVIEWER = {Petter Andreas Bergh},
      DOI = {10.1017/S0013091505001203},
      ZBLNUMBER = {1202.20057},
      }
  • [bs] Go to document M. P. Brodmann and R. Y. Sharp, Local Cohomology: An Algebraic Introduction with Geometric Applications, Cambridge: Cambridge Univ. Press, 1998, vol. 60.
    @book {bs, MRKEY = {1613627},
      AUTHOR = {Brodmann, M. P. and Sharp, R. Y.},
      TITLE = {Local Cohomology: An Algebraic Introduction with Geometric Applications},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {60},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1998},
      PAGES = {xvi+416},
      ISBN = {0-521-37286-0},
      MRCLASS = {13D45 (14B15)},
      MRNUMBER = {1613627},
      MRREVIEWER = {L{ê} Tu{â}n Hoa},
      DOI = {10.1017/CBO9780511629204},
      ZBLNUMBER = {0903.13006},
      }
  • [broer] A. Broer, Remarks on invariant theory of finite groups, 1997.
    @misc{broer,
      author={Broer, A.},
      TITLE={Remarks on invariant theory of finite groups},
      NOTE={preprint, Université de Montréal},
      YEAR={1997},
      }
  • [bh] W. Bruns and J. Herzog, Cohen-Macaulay Rings, Cambridge: Cambridge Univ. Press, 1993, vol. 39.
    @book {bh, MRKEY = {1251956},
      AUTHOR = {Bruns, Winfried and Herzog, J{ü}rgen},
      TITLE = {Cohen-{M}acaulay Rings},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {39},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1993},
      PAGES = {xii+403},
      ISBN = {0-521-41068-1},
      MRCLASS = {13H10 (13-02)},
      MRNUMBER = {1251956},
      MRREVIEWER = {Matthew Miller},
      ZBLNUMBER = {0788.13005},
      }
  • [cghsw] Go to document 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., vol. 42, iss. 2, pp. 155-161, 1999.
    @article {cghsw, MRKEY = {1692004},
      AUTHOR = {Campbell, H. E. A. and Geramita, A. V. and Hughes, I. P. and Shank, R. J. and Wehlau, D. L.},
      TITLE = {Non-{C}ohen-{M}acaulay vector invariants and a {N}oether bound for a {G}orenstein ring of invariants},
      JOURNAL = {Canad. Math. Bull.},
      FJOURNAL = {Canadian Mathematical Bulletin. Bulletin Canadien de Mathématiques},
      VOLUME = {42},
      YEAR = {1999},
      NUMBER = {2},
      PAGES = {155--161},
      ISSN = {0008-4395},
      CODEN = {CMBUA3},
      MRCLASS = {13A50},
      MRNUMBER = {1692004},
      MRREVIEWER = {Dmitri I. Panyushev},
      DOI = {10.4153/CMB-1999-018-4},
      ZBLNUMBER = {0942.13007},
      }
  • [dk] H. Derksen and G. Kemper, "Computational invariant theory," in Invariant Theory and Algebraic Transformation Groups, I, New York: Springer-Verlag, 2002, vol. 130, p. x.
    @incollection {dk, MRKEY = {1918599},
      AUTHOR = {Derksen, Harm and Kemper, Gregor},
      TITLE = {Computational invariant theory},
      BOOKTITLE = {Invariant Theory and Algebraic Transformation Groups, {\rm I}},
      SERIES={Encyclopaedia Math. Sci.},
      VOLUME={130},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2002},
      PAGES = {x+268},
      ISBN = {3-540-43476-3},
      MRCLASS = {13A50 (13P10)},
      MRNUMBER = {1918599},
      MRREVIEWER = {Dmitri I. Panyushev},
      ZBLNUMBER = {1011.13003},
      }
  • [em] S. Eilenberg and J. C. Moore, "Foundations of relative homological algebra," Mem. Amer. Math. Soc. No., vol. 55, p. 39, 1965.
    @article {em, MRKEY = {0178036},
      AUTHOR = {Eilenberg, Samuel and Moore, J. C.},
      TITLE = {Foundations of relative homological algebra},
      JOURNAL = {Mem. Amer. Math. Soc. No.},
      FJOURNAL = {Memoirs of the American Mathematical Society},
      VOLUME = {55},
      YEAR = {1965},
      PAGES = {39},
      ISSN = {0065-9266},
      MRCLASS = {18.20},
      MRNUMBER = {0178036},
      MRREVIEWER = {A. Heller},
      ZBLNUMBER = {0129.01101},
      }
  • [eisenbud] D. Eisenbud, Commutative Algebra with a View Toward Algebraic Geometry, New York: Springer-Verlag, 1995, vol. 150.
    @book {eisenbud, MRKEY = {1322960},
      AUTHOR = {Eisenbud, David},
      TITLE = {Commutative Algebra with a View Toward Algebraic Geometry},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {150},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1995},
      PAGES = {xvi+785},
      ISBN = {0-387-94268-8; 0-387-94269-6},
      MRCLASS = {13-01 (14A05)},
      MRNUMBER = {1322960},
      MRREVIEWER = {Matthew Miller},
      ZBLNUMBER = {0819.13001},
      }
  • [eisenbud2] D. Eisenbud, The Geometry of Syzygies, New York: Springer-Verlag, 2005, vol. 229.
    @book {eisenbud2, MRKEY = {2103875},
      AUTHOR = {Eisenbud, David},
      TITLE = {The Geometry of Syzygies},
      SERIES = {Grad. Texts in Math.},
      VOLUME = {229},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {2005},
      PAGES = {xvi+243},
      ISBN = {0-387-22215-4},
      MRCLASS = {13D02 (13-02 13D40 14H45 14H51)},
      MRNUMBER = {2103875},
      MRREVIEWER = {Juan C. Migliore},
      ZBLNUMBER = {1086.14044},
      }
  • [ej] E. E. Enochs and O. M. G. Jenda, Relative Homological Algebra, Berlin: Walter de Gruyter & Co., 2000, vol. 30.
    @book {ej, MRKEY = {1753146},
      AUTHOR = {Enochs, Edgar E. and Jenda, Overtoun M. G.},
      TITLE = {Relative Homological Algebra},
      SERIES = {de Gruyter Exp. Math.},
      VOLUME = {30},
      PUBLISHER = {Walter de Gruyter \& Co.},
      ADDRESS = {Berlin},
      YEAR = {2000},
      PAGES = {xii+339},
      ISBN = {3-11-016633-X},
      MRCLASS = {16E65 (13H10 16-02 16E10 18G25)},
      MRNUMBER = {1753146},
      MRREVIEWER = {J. Kuzmanovich},
      ZBLNUMBER = {0952.13001},
      }
  • [fleischmann] Go to document P. Fleischmann, "The Noether bound in invariant theory of finite groups," Adv. Math., vol. 156, iss. 1, pp. 23-32, 2000.
    @article {fleischmann, MRKEY = {1800251},
      AUTHOR = {Fleischmann, Peter},
      TITLE = {The {N}oether bound in invariant theory of finite groups},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {156},
      YEAR = {2000},
      NUMBER = {1},
      PAGES = {23--32},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {13A50},
      MRNUMBER = {1800251},
      MRREVIEWER = {Thomas Garrity},
      DOI = {10.1006/aima.2000.1952},
      ZBLNUMBER = {0973.13003},
      }
  • [fogarty] Go to document J. Fogarty, "On Noether’s bound for polynomial invariants of a finite group," Electron. Res. Announc. Amer. Math. Soc., vol. 7, pp. 5-7, 2001.
    @article {fogarty, MRKEY = {1826990},
      AUTHOR = {Fogarty, John},
      TITLE = {On {N}oether's bound for polynomial invariants of a finite group},
      JOURNAL = {Electron. Res. Announc. Amer. Math. Soc.},
      FJOURNAL = {Electronic Research Announcements of the American Mathematical Society},
      VOLUME = {7},
      YEAR = {2001},
      PAGES = {5--7},
      ISSN = {1079-6762},
      MRCLASS = {13A50},
      MRNUMBER = {1826990},
      DOI = {10.1090/S1079-6762-01-00088-9},
      ZBLNUMBER = {0980.13003},
      }
  • [goebel] Go to document M. Göbel, "Computing bases for rings of permutation-invariant polynomials," J. Symbolic Comput., vol. 19, iss. 4, pp. 285-291, 1995.
    @article {goebel, MRKEY = {1339909},
      AUTHOR = {G{ö}bel, Manfred},
      TITLE = {Computing bases for rings of permutation-invariant polynomials},
      JOURNAL = {J. Symbolic Comput.},
      FJOURNAL = {Journal of Symbolic Computation},
      VOLUME = {19},
      YEAR = {1995},
      NUMBER = {4},
      PAGES = {285--291},
      ISSN = {0747-7171},
      MRCLASS = {13A50 (13P99 68Q40)},
      MRNUMBER = {1339909},
      MRREVIEWER = {Frank D. Grosshans},
      DOI = {10.1006/jsco.1995.1017},
      ZBLNUMBER = {0832.13006},
      }
  • [hartshorne] Go to document R. Hartshorne, Local Cohomology. A Seminar Given by A. Grothendieck, Harvard University, Fall, New York: Springer-Verlag, 1967, vol. 1961.
    @book {hartshorne, MRKEY = {0224620},
      AUTHOR = {Hartshorne, Robin},
      TITLE = {Local Cohomology. A Seminar Given by A. Grothendieck, Harvard University, Fall},
      SERIES={Lecture Notes in Math.},
      VOLUME = {1961},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1967},
      PAGES = {vi+106},
      MRCLASS = {14.55 (18.00)},
      MRNUMBER = {0224620},
      MRREVIEWER = {F. Oort},
      ZBLNUMBER = {0185.49202},
      DOI={10.1007/BFb0073971},
      }
  • [24] S. B. Iyengar, G. J. Leuschke, A. Leykin, C. Miller, E. Miller, A. K. Singh, and U. Walther, Twenty-Four Hours of Local Cohomology, Providence, RI: Amer. Math. Soc., 2007, vol. 87.
    @book {24, MRKEY = {2355715},
      AUTHOR = {Iyengar, Srikanth B. and Leuschke, Graham J. and Leykin, Anton and Miller, Claudia and Miller, Ezra and Singh, Anurag K. and Walther, Uli},
      TITLE = {Twenty-Four Hours of Local Cohomology},
      SERIES = {Graduate Stud. Math.},
      VOLUME = {87},
      PUBLISHER = {Amer. Math. Soc.},
      ADDRESS = {Providence, RI},
      YEAR = {2007},
      PAGES = {xviii+282},
      ISBN = {978-0-8218-4126-6},
      MRCLASS = {13D45 (14B15 55N30)},
      MRNUMBER = {2355715},
      MRREVIEWER = {Ngo Viet Trung},
      ZBLNUMBER = {1129.13001},
      }
  • [ks] Go to document D. B. Karagueuzian and P. Symonds, "The module structure of a group action on a polynomial ring: a finiteness theorem," J. Amer. Math. Soc., vol. 20, iss. 4, pp. 931-967, 2007.
    @article {ks, MRKEY = {2328711},
      AUTHOR = {Karagueuzian, Dikran B. and Symonds, Peter},
      TITLE = {The module structure of a group action on a polynomial ring: a finiteness theorem},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {20},
      YEAR = {2007},
      NUMBER = {4},
      PAGES = {931--967},
      ISSN = {0894-0347},
      MRCLASS = {20C20 (13A50)},
      MRNUMBER = {2328711},
      MRREVIEWER = {U{\u{g}}ur Madran},
      DOI = {10.1090/S0894-0347-07-00563-2},
      ZBLNUMBER = {1138.13004},
      }
  • [kemper] G. Kemper, "Hilbert series and degree bounds in invariant theory," in Algorithmic Algebra and Number Theory, New York, 1999, pp. 249-263.
    @inproceedings{kemper, MRKEY = {1672054},
      AUTHOR = {Kemper, Gregor},
      TITLE = {Hilbert series and degree bounds in invariant theory},
      BOOKTITLE = {Algorithmic Algebra and Number Theory},
      VENUE={{H}eidelberg, 1997},
      PAGES = {249--263},
      PUBLISHER = {Springer-Verlag},
      ADDRESS = {New York},
      YEAR = {1999},
      MRCLASS = {13A50 (13D40)},
      MRNUMBER = {1672054},
      MRREVIEWER = {Neil L. White},
      ZBLNUMBER = {0972.13005},
      }
  • [knop] Go to document F. Knop, "Der kanonische Modul eines Invariantenrings," J. Algebra, vol. 127, iss. 1, pp. 40-54, 1989.
    @article {knop, MRKEY = {1029400},
      AUTHOR = {Knop, Friedrich},
      TITLE = {Der kanonische {M}odul eines {I}nvariantenrings},
      JOURNAL = {J. Algebra},
      FJOURNAL = {Journal of Algebra},
      VOLUME = {127},
      YEAR = {1989},
      NUMBER = {1},
      PAGES = {40--54},
      ISSN = {0021-8693},
      CODEN = {JALGA4},
      MRCLASS = {14L30 (14D25 32M05)},
      MRNUMBER = {1029400},
      MRREVIEWER = {Dennis Snow},
      DOI = {10.1016/0021-8693(89)90271-8},
      ZBLNUMBER = {0716.20021},
      }
  • [richman1] Go to document D. R. Richman, "On vector invariants over finite fields," Adv. Math., vol. 81, iss. 1, pp. 30-65, 1990.
    @article {richman1, MRKEY = {1051222},
      AUTHOR = {Richman, David R.},
      TITLE = {On vector invariants over finite fields},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {81},
      YEAR = {1990},
      NUMBER = {1},
      PAGES = {30--65},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {15A72 (13E15 20G40)},
      MRNUMBER = {1051222},
      MRREVIEWER = {Joseph Kung},
      DOI = {10.1016/0001-8708(90)90003-6},
      ZBLNUMBER = {0715.13002},
      }
  • [richman2] Go to document D. R. Richman, "Invariants of finite groups over fields of characteristic $p$," Adv. Math., vol. 124, iss. 1, pp. 25-48, 1996.
    @article {richman2, MRKEY = {1423197},
      AUTHOR = {Richman, David R.},
      TITLE = {Invariants of finite groups over fields of characteristic {$p$}},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {124},
      YEAR = {1996},
      NUMBER = {1},
      PAGES = {25--48},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {13A50 (15A72)},
      MRNUMBER = {1423197},
      MRREVIEWER = {Frank D. Grosshans},
      DOI = {10.1006/aima.1996.0076},
      ZBLNUMBER = {0879.13004},
      }
  • [stanley] Go to document R. P. Stanley, "Invariants of finite groups and their applications to combinatorics," Bull. Amer. Math. Soc., vol. 1, iss. 3, pp. 475-511, 1979.
    @article {stanley, MRKEY = {0526968},
      AUTHOR = {Stanley, Richard P.},
      TITLE = {Invariants of finite groups and their applications to combinatorics},
      JOURNAL = {Bull. Amer. Math. Soc.},
      FJOURNAL = {American Mathematical Society. Bulletin. New Series},
      VOLUME = {1},
      YEAR = {1979},
      NUMBER = {3},
      PAGES = {475--511},
      ISSN = {0273-0979},
      CODEN = {BAMOAD},
      MRCLASS = {20C15 (05A15 13H10 14H10 15A72 51F15)},
      MRNUMBER = {0526968},
      MRREVIEWER = {Ralph Strebel},
      DOI = {10.1090/S0273-0979-1979-14597-X},
      ZBLNUMBER = {0497.20002},
      }
  • [s] Go to document P. Symonds, "Structure theorems over polynomial rings," Adv. Math., vol. 208, iss. 1, pp. 408-421, 2007.
    @article {s, MRKEY = {2304322},
      AUTHOR = {Symonds, Peter},
      TITLE = {Structure theorems over polynomial rings},
      JOURNAL = {Adv. Math.},
      FJOURNAL = {Advances in Mathematics},
      VOLUME = {208},
      YEAR = {2007},
      NUMBER = {1},
      PAGES = {408--421},
      ISSN = {0001-8708},
      CODEN = {ADMTA4},
      MRCLASS = {20C05},
      MRNUMBER = {2304322},
      MRREVIEWER = {Mark L. Lewis},
      DOI = {10.1016/j.aim.2006.02.012},
      ZBLNUMBER = {1108.13007},
      }
  • [s1] Go to document P. Symonds, "On the Castelnuovo-Mumford regularity of the cohomology ring of a group," J. Amer. Math. Soc., vol. 23, iss. 4, pp. 1159-1173, 2010.
    @article {s1, MRKEY = {2669710},
      AUTHOR = {Symonds, Peter},
      TITLE = {On the {C}astelnuovo-{M}umford regularity of the cohomology ring of a group},
      JOURNAL = {J. Amer. Math. Soc.},
      FJOURNAL = {Journal of the American Mathematical Society},
      VOLUME = {23},
      YEAR = {2010},
      NUMBER = {4},
      PAGES = {1159--1173},
      ISSN = {0894-0347},
      MRCLASS = {20J06 (57S15)},
      MRNUMBER = {2669710},
      DOI = {10.1090/S0894-0347-10-00666-1},
      ZBLNUMBER = {pre05797892},
      }
  • [wehlau] D. L. Wehlau, "The Noether number in invariant theory," C. R. Math. Acad. Sci. Soc. R. Can., vol. 28, iss. 2, pp. 39-62, 2006.
    @article {wehlau, MRKEY = {2257602},
      AUTHOR = {Wehlau, David L.},
      TITLE = {The {N}oether number in invariant theory},
      JOURNAL = {C. R. Math. Acad. Sci. Soc. R. Can.},
      FJOURNAL = {Comptes Rendus Mathématiques de l'Académie des Sciences. La Société Royale du Canada. Mathematical Reports of the Academy of Science. The Royal Society of Canada},
      VOLUME = {28},
      YEAR = {2006},
      NUMBER = {2},
      PAGES = {39--62},
      ISSN = {0706-1994},
      MRCLASS = {13A50 (20G15)},
      MRNUMBER = {2257602},
      MRREVIEWER = {Anne Moreau},
      ZBLNUMBER = {1108.13008},
      }
  • [weibel] C. A. Weibel, An Introduction to Homological Algebra, Cambridge: Cambridge Univ. Press, 1994, vol. 38.
    @book {weibel, MRKEY = {1269324},
      AUTHOR = {Weibel, Charles A.},
      TITLE = {An Introduction to Homological Algebra},
      SERIES = {Cambridge Stud. Adv. Math.},
      VOLUME = {38},
      PUBLISHER = {Cambridge Univ. Press},
      ADDRESS = {Cambridge},
      YEAR = {1994},
      PAGES = {xiv+450},
      ISBN = {0-521-43500-5; 0-521-55987-1},
      MRCLASS = {18-01 (16-01 17-01 20-01 55Uxx)},
      MRNUMBER = {1269324},
      MRREVIEWER = {Kenneth A. Brown},
      ZBLNUMBER = {0797.18001},
      }

Authors

Peter Symonds

School of Mathematics
University of Manchester
Oxford Road
Manchester M13 9PL
United Kingdom