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Direct sum decompositions of infinitely generated modules
Author(s):
D.
J.
Benson;
Wayne
W.
Wheeler
Journal:
Trans. Amer. Math. Soc.
351
(1999),
3843-3855.
MSC (1991):
Primary 20C20
Posted:
May 21, 1999
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Abstract:
Almost all of the basic theorems in the representation theory of finite groups have proofs that depend upon the Krull-Schmidt Theorem. Because this theorem holds only for finite-dimensional modules, however, the recent interest in infinitely generated modules raises the question of which results may hold more generally. In this paper we present an example showing that Green's Indecomposability Theorem fails for infinitely generated modules. By developing and applying some general properties of idempotent modules, we are also able to construct explicit examples of modules for which the cancellation property fails.
References:
- 1.
- D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205. MR 96h:20095
- 2.
- D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107-129. MR 93i:20058
- 3.
- D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, I, Math. Proc. Cambridge Phil. Soc. 118 (1995), 223-243. MR 96j:20006
- 4.
- D. J. Benson, J. F. Carlson, and J. Rickard, Complexity and varieties for infinitely generated modules, II, Math. Proc. Cambridge Phil. Soc. 120 (1996), 597-615. MR 97f:20008
- 5.
- D. J. Benson, J. F. Carlson, and J. Rickard, Thick subcategories of the stable module category, Fundamenta Mathematicae 153 (1997), 59-80. MR 98g:20021
- 6.
- D. J. Benson and G. Ph. Gnacadja, Phantom maps and purity in modular representation theory, to appear in Fundamenta Mathematicae.
- 7.
- S. Brenner and C. Ringel, Pathological modules over tame rings, J. London Math. Soc. (2) 14 (1976), 207-215. MR 55:5697
- 8.
- J. F. Carlson, P. W. Donovan, and W. W. Wheeler, Complexity and quotient categories for group algebras, J. Pure Appl. Algebra 93 (1994), 147-167. MR 95a:20009
- 9.
- J. F. Carlson and W. W. Wheeler, Homomorphisms in higher complexity quotient categories, Seattle Conference on Representations of Finite Groups, Proc. Sympos. Pure Math., vol. 63, Amer. Math. Soc., Providence, RI, 1998, pp. 115-155. CMP 98:08
- 10.
- J. A. Green, On the indecomposable representations of a finite group, Math. Z. 70 (1959), 430-445. MR 24:A1304
- 11.
- D. Happel, Triangulated Categories in the Representation Theory of Finite Dimensional Algebras, Cambridge Univ. Press, Cambridge, 1998. MR 89e:16035
- 12.
- M. Raynaud, Modules projectifs universels, Invent. Math. 6 (1968), 1-26. MR 38:4462
- 13.
- J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. (2) 56 (1997), 149-170. MR 98d:20058
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Additional Information:
D.
J.
Benson
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Email:
djb@byrd.math.uga.edu
Wayne
W.
Wheeler
Affiliation:
Department of Mathematics, University of Georgia, Athens, Georgia 30602
Address at time of publication:
Department of Mathematics and Computer Science, University of Leicester, University Road, Leicester LE1 7RH, England
Email:
www@sloth.math.uga.edu
DOI:
10.1090/S0002-9947-99-02316-8
PII:
S 0002-9947(99)02316-8
Received by editor(s):
November 12, 1997
Posted:
May 21, 1999
Additional Notes:
Both authors are partially supported by the NSF
Copyright of article:
Copyright
1999,
American Mathematical Society
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