Download PDF - Phantom maps and purity in modular representation theory, IAll rights reserved. Copyright exceptions and limitations apply

ArticleOriginal scientific text

Title

Phantom maps and purity in modular representation theory, I

Authors 1, 1

Affiliations

  1. Department of Mathematics, University of Georgia, Athens, GA 30602, U.S.A.

Abstract

Let k be a field and G a finite group. By analogy with the theory of phantom maps in topology, a map f : M → ℕ between kG-modules is said to be phantom if its restriction to every finitely generated submodule of M factors through a projective module. We investigate the relationships between the theory of phantom maps, the algebraic theory of purity, and Rickard's idempotent modules. In general, adding one to the pure global dimension of kG gives an upper bound for the number of phantoms we need to compose to get a map which factors through a projective module. However, this bound is not sharp. For example, for the group ℤ/4×ℤ/2 in characteristic two, the composite of 6 phantom maps always factors through a projective module, whereas the pure global dimension of the group algebra can be arbitrarily large.

Bibliography

  1. F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math. 13, Springer, Berlin, 1974.
  2. M. Auslander, Representation theory of Artin algebras II, Comm. Algebra 1 (1974), 269-310.
  3. D. Baer, H. Brune and H. Lenzing, A homological approach to representations of algebras II: tame hereditary algebras, J. Pure Appl. Algebra 26 (1982), 141-153.
  4. D. Baer and H. Lenzing, A homological approach to representations of algebras I: the wild case, ibid. 24 (1982), 227-233.
  5. D. J. Benson, Representations and Cohomology I: Basic representation theory of finite groups and associative algebras, Cambridge Stud. Adv. Math. 30, Cambridge Univ. Press, 1991.
  6. D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Stud. Adv. Math. 31, Cambridge Univ. Press, 1991.
  7. D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205.
  8. D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107-129.
  9. D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, I, Math. Proc. Cambridge Philos. Soc. 118 (1995), 223-243.
  10. D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, II, ibid. 120 (1996), 597-615.
  11. D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), 59-80.
  12. 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.
  13. J. F. Carlson and W. W. Wheeler, Homomorphisms in higher complexity quotient categories, to appear.
  14. J. D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), 284-339.
  15. J. D. Christensen and N. P. Strickland, Phantom maps and homology theories, Topology 37 (1998), 339-364.
  16. G. Ph. Gnacadja, Phantom maps in the stable module category, J. Algebra 201 (1998), 686-702.
  17. B. Gray, Spaces of the same n-type, for all n, Topology 5 (1966), 241-243.
  18. L. Gruson et C. U. Jensen, Modules algébriquement compacts et foncteurs varprojlimi, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1651-A1653.
  19. L. Gruson et C. U. Jensen, Dimensions cohomologiques reliées aux foncteurs varprojlimi, in: Séminaire d'Algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math. 867, Springer, Berlin, 1981, 234-294.
  20. A. Heller, The loop-space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382-394.
  21. M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 610 (1997).
  22. C. U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach, 1989.
  23. H. Krause, Generic modules over Artin algebras, Proc. London Math. Soc. (3) 76 (1998), 276-306.
  24. J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, MA, 1966.
  25. C. A. McGibbon, Phantom maps, in: Handbook of Algebraic Topology, I. M. James (ed.), North-Holland, Amsterdam, 1995, 1209-1257.
  26. A. Neeman, On a theorem of Brown and Adams, Topology 36 (1997), 619-645.
  27. B. L. Osofsky, Homological dimension and the continuum hypothesis, Trans. Amer. Math. Soc. 132 (1968), 217-230.
  28. B. L. Osofsky, Homological Dimensions of Modules, CBMS Regional Conf. Series Math. 12, Amer. Math. Soc., 1973.
  29. M. Prest, Model Theory and Modules, London Math. Soc. Lecture Note Ser. 130, Cambridge Univ. Press, 1988.
  30. J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149-170.
  31. J. E. Roos, Sur les foncteurs dérivés de varprojlim. Applications, C. R. Acad. Sci. Paris 252 (1961), 3702-3704.
  32. R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699-719.
  33. A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. 58 (1987), 129-143.
  34. M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149-213.
Fundamenta Mathematicae
1999/161/1-2
Export to PBN, Opens in new tab
Pages:
37-91
Main language of publication
English
Received
1997-11-17
Published
1999
Exact and natural sciences