Geodesic
Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 155-161

Voir la notice de l'article provenant de la source Cambridge University Press

This paper contains two essentially independent results in the invariant theory of finite groups. First we prove that, for any faithful representation of a non-trivial $p$ -group over a field of characteristic $p$ , the ring of vector invariants of $m$ copies of that representation is not Cohen-Macaulay for $m\,\ge \,3$ . In the second section of the paper we use Poincaré series methods to produce upper bounds for the degrees of the generators for the ring of invariants as long as that ring is Gorenstein. We prove that, for a finite non-trivial group $G$ and a faithful representation of dimension $n$ with $n\,>\,1$ , if the ring of invariants is Gorenstein then the ring is generated in degrees less than or equal to $n(\left| G \right|\,-\,1)$ . If the ring of invariants is a hypersurface, the upper bound can be improved to $\left| G \right|$ .
DOI : 10.4153/CMB-1999-018-4
Mots-clés : 13A50 
Campbell, H. E. A.; Geramita, A. V.; Hughes, I. P.; Shank, R. J.; Wehlau, D. L. Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants. Canadian mathematical bulletin, Tome 42 (1999) no. 2, pp. 155-161. doi: 10.4153/CMB-1999-018-4
@article{10_4153_CMB_1999_018_4,
     author = {Campbell, H. E. A. and Geramita, A. V. and Hughes, I. P. and Shank, R. J. and Wehlau, D. L.},
     title = {Non {Cohen-Macaulay} {Vector} {Invariants} and a {Noether} {Bound} for a {Gorenstein} {Ring} of {Invariants}},
     journal = {Canadian mathematical bulletin},
     pages = {155--161},
     year = {1999},
     volume = {42},
     number = {2},
     doi = {10.4153/CMB-1999-018-4},
     url = {http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-018-4/}
}
TY  - JOUR
AU  - Campbell, H. E. A.
AU  - Geramita, A. V.
AU  - Hughes, I. P.
AU  - Shank, R. J.
AU  - Wehlau, D. L.
TI  - Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
JO  - Canadian mathematical bulletin
PY  - 1999
SP  - 155
EP  - 161
VL  - 42
IS  - 2
UR  - http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-018-4/
DO  - 10.4153/CMB-1999-018-4
ID  - 10_4153_CMB_1999_018_4
ER  - 
%0 Journal Article
%A Campbell, H. E. A.
%A Geramita, A. V.
%A Hughes, I. P.
%A Shank, R. J.
%A Wehlau, D. L.
%T Non Cohen-Macaulay Vector Invariants and a Noether Bound for a Gorenstein Ring of Invariants
%J Canadian mathematical bulletin
%D 1999
%P 155-161
%V 42
%N 2
%U http://geodesic.mathdoc.fr/articles/10.4153/CMB-1999-018-4/
%R 10.4153/CMB-1999-018-4
%F 10_4153_CMB_1999_018_4

[1] [1] Benson, D. J., Polynomial invariants of finite groups. London Math. Soc. Lecture Note Ser. 190, Cambridge University Press, Cambridge, 1993. Google Scholar

[2] [2] Bruns, W. and Herzog, J., Cohen-Macaulay rings. Cambridge Stud. Adv. Math. 39, Cambridge University Press, Cambridge, 1993. Google Scholar

[3] [3] Eagon, J. A. and Hochster, M., Cohen-Macaulay rings, invariant theory, and the generic perfection of determinantal loci. Amer. J. Math. 93 (1971), 1020–1058. Google Scholar

[4] [4] Noether, E., Der endlichkeitssatz der invarianten endlicher gruppen. Math. Ann. 77 (1916), 89–92. Google Scholar

[5] [5] Smith, L., Polynomial invariants of finite groups. A. K. Peters, Wellesley, MA, 1995. Google Scholar

[6] [6] Smith, L., Polynomial invariants of finite groups—a survey of recent developments. Bull. Amer. Math. Soc. (3) 34 (1997), 211–248. Google Scholar

[7] [7] Springer, T. A., Invariant Theory. Lecture Notes in Math. 585, Springer-Verlag, Berlin, 1977. Google Scholar

[8] [8] Stanley, R. P., Invariants of finite groups and their applications to combinatorics. Bull. Amer. Math. Soc. (3) 1 (1979), 475–511. Google Scholar

[9] [9] Weyl, H., The classical groups. Princeton University Press, Princeton, NJ, 1946. Google Scholar

Cité par Sources :