Phantom maps and purity in modular representation theory, I
Fundamenta Mathematicae, Tome 161 (1999) no. 1, pp. 37-91
Cet article a éte moissonné depuis la source Institute of Mathematics Polish Academy of Sciences
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.
@article{10_4064_fm_161_1_2_37_91,
author = {D. J. Benson and G. Ph. Gnacadja},
title = {Phantom maps and purity in modular representation theory, {I}},
journal = {Fundamenta Mathematicae},
pages = {37--91},
year = {1999},
volume = {161},
number = {1},
doi = {10.4064/fm-161-1-2-37-91},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.4064/fm-161-1-2-37-91/}
}
TY - JOUR AU - D. J. Benson AU - G. Ph. Gnacadja TI - Phantom maps and purity in modular representation theory, I JO - Fundamenta Mathematicae PY - 1999 SP - 37 EP - 91 VL - 161 IS - 1 UR - http://geodesic.mathdoc.fr/articles/10.4064/fm-161-1-2-37-91/ DO - 10.4064/fm-161-1-2-37-91 LA - en ID - 10_4064_fm_161_1_2_37_91 ER -
D. J. Benson; G. Ph. Gnacadja. Phantom maps and purity in modular representation theory, I. Fundamenta Mathematicae, Tome 161 (1999) no. 1, pp. 37-91. doi: 10.4064/fm-161-1-2-37-91
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