Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Observability in invariant theory II: Divisors and rational invariants


Author: Lex E. Renner
Journal: Proc. Amer. Math. Soc. 143 (2015), 4113-4121
MSC (2010): Primary 13A50, 14L30
DOI: https://doi.org/10.1090/proc/12804
Published electronically: June 16, 2015
MathSciNet review: 3373912
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ G\times X\to X$ be an action of the connected algebraic group $ G$ on the irreducible, affine variety $ X$. We discuss the relationship between $ [k[X]^G]$ and $ k(X)^G$, where $ [k[X]^G]$ denotes the quotient field of $ k[X]^G$. We are particularly interested in the following three questions. (1) When is the inclusion $ [k[X]^G]\subseteq k(X)^G$ a finite extension of fields? (2) What is the role of $ G$-invariant divisors? (3) What is the exact characterization of ``observable in codimension one''?


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

  • [1] M. F. Atiyah and I. G. Macdonald, Introduction to commutative algebra, Addison-Wesley Publishing Co., Reading, Mass.-London-Don Mills, Ont., 1969. MR 0242802 (39 #4129)
  • [2] Walter Ferrer Santos and Alvaro Rittatore, Actions and invariants of algebraic groups, Pure and Applied Mathematics (Boca Raton), vol. 269, Chapman & Hall/CRC, Boca Raton, FL, 2005. MR 2138858 (2006c:14067)
  • [3] Frank Grosshans, Localization and invariant theory, Advances in Math. 21 (1976), no. 1, 50-60. MR 0407040 (53 #10823)
  • [4] Frank D. Grosshans, Algebraic homogeneous spaces and invariant theory, Lecture Notes in Mathematics, vol. 1673, Springer-Verlag, Berlin, 1997. MR 1489234 (99b:13005)
  • [5] Lex E. Renner, Observability in invariant theory, Proc. Amer. Math. Soc. 141 (2013), no. 1, 205-216. MR 2988723, https://doi.org/10.1090/S0002-9939-2012-11321-8
  • [6] Lex E. Renner, Orbits and invariants of visible group actions, Transform. Groups 17 (2012), no. 4, 1191-1208. MR 3000484, https://doi.org/10.1007/s00031-012-9198-1
  • [7] Lex Renner and Alvaro Rittatore, Observable actions of algebraic groups, Transform. Groups 14 (2009), no. 4, 985-999. MR 2577204 (2011b:14102), https://doi.org/10.1007/s00031-009-9073-x
  • [8] Maxwell Rosenlicht, Some basic theorems on algebraic groups, Amer. J. Math. 78 (1956), 401-443. MR 0082183 (18,514a)
  • [9] Maxwell Rosenlicht, A remark on quotient spaces, An. Acad. Brasil. Ci. 35 (1963), 487-489. MR 0171782 (30 #2009)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 13A50, 14L30

Retrieve articles in all journals with MSC (2010): 13A50, 14L30


Additional Information

Lex E. Renner
Affiliation: Department of Mathematics, Western University, London, Ontario, Canada N6A 5B7
Email: lex@uwo.ca

DOI: https://doi.org/10.1090/proc/12804
Received by editor(s): October 6, 2013
Published electronically: June 16, 2015
Communicated by: Harm Derksen
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society