Phantom maps and purity in modular representation theory, I

D. Benson; G. Gnacadja

Fundamenta Mathematicae (1999)

  • Volume: 161, Issue: 1-2, page 37-91
  • ISSN: 0016-2736

Abstract

top
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.

How to cite

top

Benson, D., and Gnacadja, G.. "Phantom maps and purity in modular representation theory, I." Fundamenta Mathematicae 161.1-2 (1999): 37-91. <http://eudml.org/doc/212404>.

@article{Benson1999,
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.},
author = {Benson, D., Gnacadja, G.},
journal = {Fundamenta Mathematicae},
keywords = {pure global dimensions; finite groups; phantom maps; pure projective modules; pure short exact sequences; direct sums of finitely generated modules; pure injective modules; idempotent modules; stable module categories; cohomology varieties; stable endomorphism rings},
language = {eng},
number = {1-2},
pages = {37-91},
title = {Phantom maps and purity in modular representation theory, I},
url = {http://eudml.org/doc/212404},
volume = {161},
year = {1999},
}

TY - JOUR
AU - Benson, D.
AU - Gnacadja, G.
TI - Phantom maps and purity in modular representation theory, I
JO - Fundamenta Mathematicae
PY - 1999
VL - 161
IS - 1-2
SP - 37
EP - 91
AB - 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.
LA - eng
KW - pure global dimensions; finite groups; phantom maps; pure projective modules; pure short exact sequences; direct sums of finitely generated modules; pure injective modules; idempotent modules; stable module categories; cohomology varieties; stable endomorphism rings
UR - http://eudml.org/doc/212404
ER -

References

top
  1. [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math. 13, Springer, Berlin, 1974. 
  2. [2] M. Auslander, Representation theory of Artin algebras II, Comm. Algebra 1 (1974), 269-310. Zbl0285.16029
  3. [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. Zbl0504.16021
  4. [4] D. Baer and H. Lenzing, A homological approach to representations of algebras I: the wild case, ibid. 24 (1982), 227-233. Zbl0504.16020
  5. [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. Zbl0718.20001
  6. [6] D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Stud. Adv. Math. 31, Cambridge Univ. Press, 1991. Zbl0731.20001
  7. [7] D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205. Zbl0879.20004
  8. [8] D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107-129. Zbl0807.20044
  9. [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. Zbl0848.20003
  10. [10] D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, II, ibid. 120 (1996), 597-615. Zbl0888.20003
  11. [11] D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), 59-80. Zbl0886.20007
  12. [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. Zbl0811.20002
  13. [13] J. F. Carlson and W. W. Wheeler, Homomorphisms in higher complexity quotient categories, to appear. Zbl0899.20001
  14. [14] J. D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), 284-339. Zbl0928.55010
  15. [15] J. D. Christensen and N. P. Strickland, Phantom maps and homology theories, Topology 37 (1998), 339-364. Zbl1001.55009
  16. [16] G. Ph. Gnacadja, Phantom maps in the stable module category, J. Algebra 201 (1998), 686-702. Zbl0910.20001
  17. [17] B. Gray, Spaces of the same n-type, for all n, Topology 5 (1966), 241-243. Zbl0149.20102
  18. [18] L. Gruson et C. U. Jensen, Modules algébriquement compacts et foncteurs lim ← ( i ) , C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1651-A1653. Zbl0259.18015
  19. [19] L. Gruson et C. U. Jensen, Dimensions cohomologiques reliées aux foncteurs lim ← ( i ) , in: Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math. 867, Springer, Berlin, 1981, 234-294. 
  20. [20] A. Heller, The loop-space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382-394. Zbl0096.25502
  21. [21] M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 610 (1997). Zbl0881.55001
  22. [22] C. U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach, 1989. 
  23. [23] H. Krause, Generic modules over Artin algebras, Proc. London Math. Soc. (3) 76 (1998), 276-306. Zbl0908.16016
  24. [24] J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, MA, 1966. 
  25. [25] C. A. McGibbon, Phantom maps, in: Handbook of Algebraic Topology, I. M. James (ed.), North-Holland, Amsterdam, 1995, 1209-1257. Zbl0867.55013
  26. [26] A. Neeman, On a theorem of Brown and Adams, Topology 36 (1997), 619-645. 
  27. [27] B. L. Osofsky, Homological dimension and the continuum hypothesis, Trans. Amer. Math. Soc. 132 (1968), 217-230. Zbl0157.08201
  28. [28] B. L. Osofsky, Homological Dimensions of Modules, CBMS Regional Conf. Series Math. 12, Amer. Math. Soc., 1973. Zbl0254.13015
  29. [29] M. Prest, Model Theory and Modules, London Math. Soc. Lecture Note Ser. 130, Cambridge Univ. Press, 1988. 
  30. [30] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149-170. Zbl0910.20034
  31. [31] J. E. Roos, Sur les foncteurs dérivés de lim ← . Applications, C. R. Acad. Sci. Paris 252 (1961), 3702-3704. Zbl0102.02501
  32. [32] R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699-719. Zbl0172.04801
  33. [33] A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. 58 (1987), 129-143. Zbl0638.55020
  34. [34] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149-213. Zbl0593.16019

NotesEmbed ?

top

You must be logged in to post comments.

To embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.

Only the controls for the widget will be shown in your chosen language. Notes will be shown in their authored language.

Tells the widget how many notes to show per page. You can cycle through additional notes using the next and previous controls.

    
                

                
Note: Best practice suggests putting the JavaScript code just before the closing </body> tag.