# Phantom maps and purity in modular representation theory, I

Fundamenta Mathematicae (1999)

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

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## Abstract

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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

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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

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