Remote Access Journal of the American Mathematical Society
Green Open Access

Journal of the American Mathematical Society

ISSN 1088-6834(online) ISSN 0894-0347(print)

 
 

 

Trace rings of generic matrices are Cohen-Macaulay


Author: Michel Van den Bergh
Journal: J. Amer. Math. Soc. 2 (1989), 775-799
MSC: Primary 14L30; Secondary 14M05
DOI: https://doi.org/10.1090/S0894-0347-1989-1001850-7
MathSciNet review: 1001850
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: In this paper, we prove that trace rings of generic matrics are Cohen-Macaulay (Theorem 7.3.6). This is done by relating this problem to a conjecture of Stanley about modules of invariants under a reductive group.

We prove a slightly weakened version (Conjecture 3.4') of this conjecture in special cases (Theorem 6.1.8). In particular, we obtain that Conjecture 3.4' is true for $ S{L_2}$ (Remark 6.1.10).


References [Enhancements On Off] (What's this?)

  • [1] M. Artin, On Azumaya algebras and finite dimensional representations of rings, J. Algebra 11 (1969), 532-563. MR 0242890 (39:4217)
  • [2] M. Artin and W. Schelter, Integral ring homomorphisms, Adv. in Math. 39 (1981), 289-329. MR 614165 (83e:16015)
  • [3] R. Bott, Homogeneous vector bundles, Ann. of Math. (2) 65 (1957), 203-248. MR 0089473 (19:681d)
  • [4] E. Formanek, Functional equations for character series associated with $ n\times n$ matrices, Trans. Amer. Math. Soc. 294 (1986), 647-663. MR 825728 (87k:15035)
  • [5] E. Formanek, P. Halpin, and W. Li, The Poincaré series of 2 by 2 matrices, J. Algebra 69 (1981), 105-112. MR 613860 (82i:16020)
  • [6] R. Hartshorne, Residues and duality, Springer-Verlag, New York, 1966. MR 0222093 (36:5145)
  • [7] W. H. Hesselink, Desingularisations of varieties of null forms, Invent. Math. 55 (1979), 141-163. MR 553706 (81b:14025)
  • [8] M. Hochster and J. Roberts, Rings of invariants of reductive groups acting on regular rings are Cohen-Macaulay, Adv. in Math. 13 (1974), 313-373. MR 0347810 (50:311)
  • [9] F. C. Kirwan, Cohomology of quotients in symplectic and algebraic geometry, Math. Notes, no. 31, Princeton Univ. Press, Princeton, NJ, 1984. MR 766741 (86i:58050)
  • [10] L. le Bruyn, Trace rings of generic $ 2\times 2$ matrices, Mem. Amer. Math. Soc., no. 363, Amer. Math. Soc., Providence, RI, 1987. MR 878906 (88b:16032)
  • [11] L. LeBruyn and C. Procesi, Etale local structure of matrix invariants and concomittants, Lecture Notes in Math., no. 1271, Springer-Verlag, Berlin and New York, 1986, pp. 143-176. MR 911138 (89b:16042)
  • [12] L. LeBruyn and M. Van den Bergh, Regularity of trace rings of generic matrices, J. Algebra 117 (1988), 19-29.
  • [13] G. Kempf, Collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229-239. MR 0424841 (54:12799)
  • [14] D. Mumford, Geometric invariant theory, Springer-Verlag, New York, 1982. MR 719371 (86a:14006)
  • [15] V. L. Popov, Stability criteria for the actions of a semisimple group on a factorial manifold, Izv. Akad. Nauk SSSR Ser. Math. 4 (1970), 527-535. MR 0262416 (41:7024)
  • [16] C. Procesi, Invariant theory of $ n\times n$-matrices, Adv. in Math. 19 (1976), 306-381. MR 0419491 (54:7512)
  • [17] -, Trace identities and standard diagrams, Proc. 1978 Conf. on Ring Theory (F. Van Oystaeyen, ed.), Marcel Dekker, New York, 1979, pp. 191-218. MR 563295 (81m:15012)
  • [18] L. Small and T. Stafford, Homological properties of generic matrix rings, Israel J. Math. 51 (1985), 27-32. MR 804474 (87a:16031)
  • [19] T. A. Springer, Linear algebraic groups, Progr. Math., Vol. 9, Birkhäuser, Boston, 1981.
  • [20] R. Stanley, Combinatorics and invariant theory, Proc. Sympos. Pure Math., Vol. 34, Amer. Math. Soc., Providence, RI, 1979, pp. 345-355. MR 525334 (80e:15020)
  • [21] Y. Teranishi, The Hilbert series of rings of matrix concomittants, Nagoya Math. J. 111 (1988), 143-156. MR 961222 (90a:16017)
  • [22] M. Van den Bergh, Cohen-Macaulayness of modules of invariants for $ S{L_2}$ (to appear).

Similar Articles

Retrieve articles in Journal of the American Mathematical Society with MSC: 14L30, 14M05

Retrieve articles in all journals with MSC: 14L30, 14M05


Additional Information

DOI: https://doi.org/10.1090/S0894-0347-1989-1001850-7
Article copyright: © Copyright 1989 American Mathematical Society

American Mathematical Society