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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Homological algebra for the representation Green functor for abelian groups


Author: Joana Ventura
Journal: Trans. Amer. Math. Soc. 357 (2005), 2253-2289
MSC (2000): Primary 55P91, 18G10
Published electronically: May 10, 2004
MathSciNet review: 2140440
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Abstract: In this paper we compute some derived functors ${Ext}$ of the internal homomorphism functor in the category of modules over the representation Green functor. This internal homomorphism functor is the left adjoint of the box product.

When the group is a cyclic $2$-group, we construct a projective resolution of the module fixed point functor, and that allows a direct computation of the graded Green functor ${Ext}$.

When the group is $G=\mathbb{Z} /2\times\mathbb{Z} /2$, we can still build a projective resolution, but we do not have explicit formulas for the differentials. The resolution is built from long exact sequences of projective modules over the representation functor for the subgroups of $G$ by using exact functors between these categories of modules. This induces a filtration which gives a spectral sequence which converges to the desired ${Ext}$ functors.


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Additional Information

Joana Ventura
Affiliation: Departamento de Matemática, Instituto Superior Técnico, Av. Rovisco Pais, 1049-001 Lisboa, Portugal
Email: jventura@math.ist.utl.pt

DOI: http://dx.doi.org/10.1090/S0002-9947-04-03566-4
PII: S 0002-9947(04)03566-4
Received by editor(s): August 22, 2003
Published electronically: May 10, 2004
Additional Notes: The author was partially supported by FCT grant Praxis XXI/BD/11357/97 and a one year research grant from Calouste Gulbenkian Foundation
Article copyright: © Copyright 2004 American Mathematical Society