Book Review
The AMS does not provide abstracts of book reviews.
You may download the entire review from the links below.
Full text of review:
PDF
This review is available free of charge.
Book Information:
Author:
Martin Lorenz
Title:
Multiplicative invariant theory
Additional book information:
Encyclopaedia of Mathematical Sciences, vol. 135, Invariant Theory and Algebraic Transformation Groups, VI,
Springer-Verlag, Berlin,
2005,
xii+177 pp.,
ISBN 3-540-24323-2,
US$109.00$
George M. Bergman, The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc. 157 (1971), 459–469. MR 280489, DOI 10.1090/S0002-9947-1971-0280489-8
Christine Bessenrodt and Lieven Le Bruyn, Stable rationality of certain $\textrm {PGL}_n$-quotients, Invent. Math. 104 (1991), no. 1, 179–199. MR 1094051, DOI 10.1007/BF01245071
N. Bourbaki, Éléments de mathématique. Fasc. XXXIV. Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles [Current Scientific and Industrial Topics], No. 1337, Hermann, Paris, 1968 (French). MR 0240238
Frank DeMeyer and Thomas McKenzie, On generic polynomials, J. Algebra 261 (2003), no. 2, 327–333. MR 1966633, DOI 10.1016/S0021-8693(02)00678-6
Daniel R. Farkas, Multiplicative invariants, Enseign. Math. (2) 30 (1984), no. 1-2, 141–157. MR 743674
Daniel R. Farkas, Toward multiplicative invariant theory, Group actions on rings (Brunswick, Maine, 1984) Contemp. Math., vol. 43, Amer. Math. Soc., Providence, RI, 1985, pp. 69–80. MR 810644, DOI 10.1090/conm/043/810644
Daniel R. Farkas, Reflection groups and multiplicative invariants, Rocky Mountain J. Math. 16 (1986), no. 2, 215–222. MR 843049, DOI 10.1216/RMJ-1986-16-2-215
Edward Formanek, The center of the ring of $3\times 3$ generic matrices, Linear and Multilinear Algebra 7 (1979), no. 3, 203–212. MR 540954, DOI 10.1080/03081087908817278
Edward Formanek, The center of the ring of $4\times 4$ generic matrices, J. Algebra 62 (1980), no. 2, 304–319. MR 563230, DOI 10.1016/0021-8693(80)90184-2
[Jo] Camille Jordan, Mémoire sur l'équivalence des formes, J. École Polytech. 48 (1880), 112-150.
Ming Chang Kang, Picard groups of some rings of invariants, J. Algebra 58 (1979), no. 2, 455–461. MR 540650, DOI 10.1016/0021-8693(79)90172-8
Nicole Marie Anne Lemire, Reduction in the rationality problem for multiplicative invariant fields, J. Algebra 238 (2001), no. 1, 51–81. MR 1822183, DOI 10.1006/jabr.2000.8652
H. W. Lenstra Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299–325. MR 347788, DOI 10.1007/BF01389732
Martin Lorenz, Class groups of multiplicative invariants, J. Algebra 177 (1995), no. 1, 242–254. MR 1356370, DOI 10.1006/jabr.1995.1296
Martin Lorenz, Picard groups of multiplicative invariants, Comment. Math. Helv. 72 (1997), no. 3, 389–399. MR 1476055, DOI 10.1007/s000140050023
Martin Lorenz, On the Cohen-Macaulay property of multiplicative invariants, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1605–1617. MR 2186988, DOI 10.1090/S0002-9947-05-03764-5
Claudio Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. Ia (8) 8 (1967), 237–255 (English, with Italian summary). MR 224657
C. Procesi, The invariant theory of $n\times n$ matrices, Advances in Math. 19 (1976), no. 3, 306–381. MR 419491, DOI 10.1016/0001-8708(76)90027-X
J. E. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), no. 3, 385–447. MR 491797, DOI 10.1112/plms/s3-36.3.385
David J. Saltman, Generic Galois extensions and problems in field theory, Adv. in Math. 43 (1982), no. 3, 250–283. MR 648801, DOI 10.1016/0001-8708(82)90036-6
David J. Saltman, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), no. 2-3, 165–215. MR 738167, DOI 10.1007/BF02760515
G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canad. J. Math. 6 (1954), 274–304. MR 59914, DOI 10.4153/cjm-1954-028-3
Robert Steinberg, On a theorem of Pittie, Topology 14 (1975), 173–177. MR 372897, DOI 10.1016/0040-9383(75)90025-7
[Sy] J. J. Sylvester, On the involution of two matrices of the second order, British Assoc. Report (Southport) (1883), 430-432. Reprinted in: ``Collected Mathematical Papers'', Vol. 4, Chelsea, 1973, pp. 115-117.
- [Be]
- George M. Bergman, The logarithmic limit-set of an algebraic variety, Trans. Amer. Math. Soc. 157 (1971), 459-469. MR 0280489
- [BL]
- Christine Bessenrodt and Lieven Le Bruyn, Stable rationality of certain -quotients, Invent. Math. 104 (1991), no. 1, 179-199. MR 1094051
- [Bo]
- N. Bourbaki, Groupes et algèbres de Lie. Chapitre IV: Groupes de Coxeter et systèmes de Tits. Chapitre V: Groupes engendrés par des réflexions. Chapitre VI: systèmes de racines, Actualités Scientifiques et Industrielles, No. 1337, Hermann, Paris, 1968. MR 0240238
- [DM]
- Frank DeMeyer and Thomas McKenzie, On generic polynomials, J. Algebra 261 (2003), no. 2, 327-333. MR 1966633
- [Fa1]
- Daniel R. Farkas, Multiplicative invariants, Enseign. Math. (2) 30 (1984), nos. 1-2, 141-157. MR 0743674
- [Fa2]
- -, Toward multiplicative invariant theory, in ``Group Actions on Rings'' (Brunswick, Maine, 1984), 69-80, Contemp. Math. 43, Amer. Math. Soc., Providence, RI, 1985. MR 0810644
- [Fa3]
- -, Reflection groups and multiplicative invariants, Rocky Mountain J. Math. 16 (1986), no. 2, 215-222. MR 0843049
- [Fo1]
- Edward Formanek, The center of the ring of generic matrices, Linear and Multilinear Algebra 7 (1979), no. 3, 203-212. MR 0540954
- [Fo2]
- -, The center of the ring of generic matrices, J. Algebra 62 (1980), no. 2, 304-319. MR 0563230
- [Jo]
- Camille Jordan, Mémoire sur l'équivalence des formes, J. École Polytech. 48 (1880), 112-150.
- [Ka]
- Ming-Chang Kang, Picard groups of some rings of invariants, J. Algebra 58 (1979), no. 2, 455-461. MR 0540650
- [Lm]
- Nicole Lemire, Reduction in the rationality problem for multiplicative invariant fields, J. Algebra 238 (2001), no. 1, 51-81. MR 1822183
- [Ln]
- H. W. Lenstra Jr., Rational functions invariant under a finite abelian group, Invent. Math. 25 (1974), 299-325. MR 0347788
- [Lo1]
- Martin Lorenz, Class groups of multiplicative invariants, J. Algebra 177 (1995), no. 1, 242-254. MR 1356370
- [Lo2]
- -, Picard groups of multiplicative invariants, Comment. Math. Helv. 72 (1997), no. 3, 389-399. MR 1476055
- [Lo3]
- -, On the Cohen-Macaulay property of multiplicative invariants, Trans. Amer. Math. Soc. 358 (2006), no. 4, 1605-1617. MR 2186988
- [Pr1]
- Claudio Procesi, Non-commutative affine rings, Atti Accad. Naz. Lincei Mem. Cl. Sci. Fis. Mat. Natur. Sez. I (8) 8 (1967), 237-255. MR 0224657
- [Pr2]
- -, The invariant theory of matrices, Advances in Math. 19 (1976), no. 3, 306-381. MR 0419491
- [Ro]
- J. E. Roseblade, Prime ideals in group rings of polycyclic groups, Proc. London Math. Soc. (3) 36 (1978), no. 3, 385-447. MR 0491797
- [Sa1]
- David J. Saltman, Generic Galois extensions and problems in field theory, Adv. in Math. 43 (1982), no. 3, 250-283. MR 0648801
- [Sa2]
- -, Retract rational fields and cyclic Galois extensions, Israel J. Math. 47 (1984), nos. 2-3, 165-215. MR 0738167
- [ST]
- G. C. Shephard and J. A. Todd, Finite unitary reflection groups, Canadian J. Math. 6 (1954), 274-304. MR 0059914
- [St]
- Robert Steinberg, On a theorem of Pittie, Topology 14 (1975), 173-177. MR 0372897
- [Sy]
- J. J. Sylvester, On the involution of two matrices of the second order, British Assoc. Report (Southport) (1883), 430-432. Reprinted in: ``Collected Mathematical Papers'', Vol. 4, Chelsea, 1973, pp. 115-117.
Review Information:
Reviewer:
D. S. Passman
Affiliation:
University of Wisconsin, Madison
Email:
passman@math.wisc.edu
Journal:
Bull. Amer. Math. Soc.
44 (2007), 157-162
Published electronically:
August 28, 2006
Review copyright:
© Copyright 2006
American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.