Phantom maps and purity in modular representation theory, I

Benson, D. ; Gnacadja, G.

Fundamenta Mathematicae, Tome 159 (1999), p. 37-91 / Harvested from The Polish Digital Mathematics Library

Résumé

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.

Détail

Publié le : 1999-01-01

Zbl 0944.20004

EUDML-ID : urn:eudml:doc:212404

Bibliographie

[00000] [1] F. W. Anderson and K. R. Fuller, Rings and Categories of Modules, Grad. Texts in Math. 13, Springer, Berlin, 1974.

[00001] [2] M. Auslander, Representation theory of Artin algebras II, Comm. Algebra 1 (1974), 269-310. | Zbl 0285.16029

[00002] [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. | Zbl 0504.16021

[00003] [4] D. Baer and H. Lenzing, A homological approach to representations of algebras I: the wild case, ibid. 24 (1982), 227-233. | Zbl 0504.16020

[00004] [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. | Zbl 0718.20001

[00005] [6] D. J. Benson, Representations and Cohomology II: Cohomology of groups and modules, Cambridge Stud. Adv. Math. 31, Cambridge Univ. Press, 1991. | Zbl 0731.20001

[00006] [7] D. J. Benson, Cohomology of modules in the principal block of a finite group, New York J. Math. 1 (1995), 196-205. | Zbl 0879.20004

[00007] [8] D. J. Benson and J. F. Carlson, Products in negative cohomology, J. Pure Appl. Algebra 82 (1992), 107-129. | Zbl 0807.20044

[00008] [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. | Zbl 0848.20003

[00009] [10] D. J. Benson, J. F. Carlson and J. Rickard, Complexity and varieties for infinitely generated modules, II, ibid. 120 (1996), 597-615. | Zbl 0888.20003

[00010] [11] D. J. Benson, J. F. Carlson and J. Rickard, Thick subcategories of the stable module category, Fund. Math. 153 (1997), 59-80. | Zbl 0886.20007

[00011] [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. | Zbl 0811.20002

[00012] [13] J. F. Carlson and W. W. Wheeler, Homomorphisms in higher complexity quotient categories, to appear. | Zbl 0899.20001

[00013] [14] J. D. Christensen, Ideals in triangulated categories: phantoms, ghosts and skeleta, Adv. Math. 136 (1998), 284-339. | Zbl 0928.55010

[00014] [15] J. D. Christensen and N. P. Strickland, Phantom maps and homology theories, Topology 37 (1998), 339-364. | Zbl 1001.55009

[00015] [16] G. Ph. Gnacadja, Phantom maps in the stable module category, J. Algebra 201 (1998), 686-702. | Zbl 0910.20001

[00016] [17] B. Gray, Spaces of the same n-type, for all n, Topology 5 (1966), 241-243. | Zbl 0149.20102

[00017] [18] L. Gruson et C. U. Jensen, Modules algébriquement compacts et foncteurs $\varprojlim ^(i)$, C. R. Acad. Sci. Paris Sér. A-B 276 (1973), A1651-A1653. | Zbl 0259.18015

[00018] [19] L. Gruson et C. U. Jensen, Dimensions cohomologiques reliées aux foncteurs $\varprojlim ^(i)$, in: Séminaire d’Algèbre Paul Dubreil et Marie-Paule Malliavin, Lecture Notes in Math. 867, Springer, Berlin, 1981, 234-294.

[00019] [20] A. Heller, The loop-space functor in homological algebra, Trans. Amer. Math. Soc. 96 (1960), 382-394. | Zbl 0096.25502

[00020] [21] M. Hovey, J. H. Palmieri and N. P. Strickland, Axiomatic stable homotopy theory, Mem. Amer. Math. Soc. 610 (1997). | Zbl 0881.55001

[00021] [22] C. U. Jensen and H. Lenzing, Model Theoretic Algebra, Gordon and Breach, 1989.

[00022] [23] H. Krause, Generic modules over Artin algebras, Proc. London Math. Soc. (3) 76 (1998), 276-306. | Zbl 0908.16016

[00023] [24] J. Lambek, Lectures on Rings and Modules, Blaisdell, Waltham, MA, 1966.

[00024] [25] C. A. McGibbon, Phantom maps, in: Handbook of Algebraic Topology, I. M. James (ed.), North-Holland, Amsterdam, 1995, 1209-1257. | Zbl 0867.55013

[00025] [26] A. Neeman, On a theorem of Brown and Adams, Topology 36 (1997), 619-645.

[00026] [27] B. L. Osofsky, Homological dimension and the continuum hypothesis, Trans. Amer. Math. Soc. 132 (1968), 217-230. | Zbl 0157.08201

[00027] [28] B. L. Osofsky, Homological Dimensions of Modules, CBMS Regional Conf. Series Math. 12, Amer. Math. Soc., 1973. | Zbl 0254.13015

[00028] [29] M. Prest, Model Theory and Modules, London Math. Soc. Lecture Note Ser. 130, Cambridge Univ. Press, 1988.

[00029] [30] J. Rickard, Idempotent modules in the stable category, J. London Math. Soc. 178 (1997), 149-170. | Zbl 0910.20034

[00030] [31] J. E. Roos, Sur les foncteurs dérivés de $\varprojlim $. Applications, C. R. Acad. Sci. Paris 252 (1961), 3702-3704. | Zbl 0102.02501

[00031] [32] R. B. Warfield, Purity and algebraic compactness for modules, Pacific J. Math. 28 (1969), 699-719. | Zbl 0172.04801

[00032] [33] A. Zabrodsky, On phantom maps and a theorem of H. Miller, Israel J. Math. 58 (1987), 129-143. | Zbl 0638.55020

[00033] [34] M. Ziegler, Model theory of modules, Ann. Pure Appl. Logic 26 (1984), 149-213. | Zbl 0593.16019