Direct sum decompositions

of infinitely generated modules

Authors:
D. J. Benson and Wayne W. Wheeler

Journal:
Trans. Amer. Math. Soc. **351** (1999), 3843-3855

MSC (1991):
Primary 20C20

DOI:
https://doi.org/10.1090/S0002-9947-99-02316-8

Published electronically:
May 21, 1999

MathSciNet review:
1608277

Full-text PDF

Abstract | References | Similar Articles | Additional Information

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.

**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:
https://doi.org/10.1090/S0002-9947-99-02316-8

Received by editor(s):
November 12, 1997

Published electronically:
May 21, 1999

Additional Notes:
Both authors are partially supported by the NSF

Article copyright:
© Copyright 1999
American Mathematical Society