# Group Actions on Rings

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*Susan Montgomery*

Ring theorists and researchers in invariant theory and operator algebra met at Bowdoin for the 1984 AMS-IMS-SIAM Joint Summer Research Conference to exchange ideas about group actions on rings and to stimulate general interaction among their various fields. Collecting together the proceedings of that meeting, this interdisciplinary volume reveals many topics common to the three fields: \(K\)-theory, dual actions, semi-invariants, and crossed products. A high percentage of the papers here are expository; of particular significance are the papers by the four survey lecturers, Edward Formanek on noncommutative invariant theory; Melvin Hochster on invariant theory of commutative rings; D. S. Passman on algebraic crossed products; and Marc A. Reiffel on \(K\)-theory of crossed products of \(C*\)-algebras by discrete groups.

# Table of Contents

## Group Actions on Rings

- Table of Contents v6 free
- Introduction vii8 free
- Program of the Conference ix10 free
- Algebraic Group Actions on Affine Spaces 112 free
- 1. The linearization problem 213
- 2. The affine Cremona group GAn 314
- 3. Unipotent group actions 516
- 4. The affine cancellation problem 617
- 5. Fixed points 718
- 6. Some basic invariant theory 920
- 7. The affine quotient n: X → X/G 1324
- 8. Some linearization results 1526
- 9. The fiber spaces G vH W 1728
- 10. The Étale Slice of Luna 1829
- 11. Equivariant K-theory 1930

- Everybody Knows What a Hopf Algebra Is 2536
- Hopf Algebras Acting on Semiprime Algebras 4960
- Automorphfsms of the Free Algebra of Rank Two 6374
- Toward Multiplicative Invariant Theory 6980
- Invariants of Finite Linear Groups Acting on Relatively Free Algebras: A Survey 8192
- Noncommutative Invariant Theory 8798
- 1. Introduction. 8899
- 2. T-ideals and Relatively Free Rings. 89100
- 3. The Representation Theory of GL(n,K). 91102
- 4. Hilbert Series and Character Series of Relatively Free Algebras. 93104
- 5. Holien's Theorem for Relatively Free Algebras. 99110
- 6. Finite Generation of Rings of Invariants. 101112
- 7. Relative Freeness of Rings of Invariants. 103114
- 8. The Ring of Generic Matrices. 106117
- 9. Homogeneous C*-algebras and Azumaya Algebras. 114125

- Invariants and Formal Group Law Actions 121132
- Classification of Compact Group Actions on Locally 137148
- The Rohlin Property for Automorphisms of UHF C*-Algebras 155166
- Invariant Theory of Commutative Rings 161172
- Index for Subrings of Rings 181192
- Duality for Actions and Coactions of Groups 191202
- Algebraic Crossed Products 209220
- K-Theoretic Freeness of Finite Group Actions on C*-Algebras 227238
- Skew Group Rings in the Computation of Grothendieck Groups and in the Representation Theory of Artin Algebras 245256
- K-Theory of Crossed Products of C*-Algebras by Discrete Groups 253264
- Groups Acting on Fields: Noether's Problem 267278