Projective representations and relative semisimplicity

E. Aljadeff; U. Onn; Y. Ginosar

doi:10.1006/jabr.1998.7783

Projective representations and relative semisimplicity

E. Aljadeff^*, U. Onn, Y. Ginosar

^*Corresponding author for this work

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Let R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element α∈H²(G,R*) one can form the twisted group ring R^αG. The "twisting problem" asks whether there exists a class α s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra94 (1994), 1-15]. Here we answer this question for rings of the form Z_p^s, s≥2. The main tools are the classification of modular representation of the Klein 4 group over Z₂ and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599-606] for twisted group rings.

Original language	English
Pages (from-to)	249-274
Number of pages	26
Journal	Journal of Algebra
Volume	217
Issue number	1
DOIs	https://doi.org/10.1006/jabr.1998.7783
State	Published - 1 Jul 1999
Externally published	Yes

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title = "Projective representations and relative semisimplicity",

abstract = "Let R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element α∈H2(G,R*) one can form the twisted group ring RαG. The {"}twisting problem{"} asks whether there exists a class α s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra94 (1994), 1-15]. Here we answer this question for rings of the form Zps, s≥2. The main tools are the classification of modular representation of the Klein 4 group over Z2 and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599-606] for twisted group rings.",

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T1 - Projective representations and relative semisimplicity

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AU - Onn, U.

AU - Ginosar, Y.

PY - 1999/7/1

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N2 - Let R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element α∈H2(G,R*) one can form the twisted group ring RαG. The "twisting problem" asks whether there exists a class α s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra94 (1994), 1-15]. Here we answer this question for rings of the form Zps, s≥2. The main tools are the classification of modular representation of the Klein 4 group over Z2 and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599-606] for twisted group rings.

AB - Let R be a local (commutative) ring and let p be a prime not invertible in R. Let G be a finite group of order divisible by p. It is well known that the group ring RG admits nonprojective lattices (e.g., R itself with the trivial action). For any element α∈H2(G,R*) one can form the twisted group ring RαG. The "twisting problem" asks whether there exists a class α s.t. the corresponding twisted group ring admits only projective lattices. For fields of characteristic p, the answer is in E. Aljadeff and D. J. S. Robinson [J. Pure Appl. Algebra94 (1994), 1-15]. Here we answer this question for rings of the form Zps, s≥2. The main tools are the classification of modular representation of the Klein 4 group over Z2 and a Chouinard-like theorem [E. Aljadeff and Y. Ginosar, J. Algebra179 (1996), 599-606] for twisted group rings.

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Projective representations and relative semisimplicity

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