Duke Mathematical Journal

Π-supports for modules for finite group schemes

Eric M. Friedlander and Julia Pevtsova

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We introduce the space Π(G) of equivalence classes of π-points of a finite group scheme G and associate a subspace Π(G)M to any G-module M. Our results extend to arbitrary finite group schemes G over arbitrary fields k of positive characteristic and to arbitrarily large G-modules, the basic results about “cohomological support varieties” and their interpretation in terms of representation theory. In particular, we prove that the projectivity of any (possibly infinite-dimensional) G-module can be detected by its restriction along π-points of G. Unlike the cohomological support variety of a G-module M, the invariant MΠ(G)M satisfies good properties for all modules, thereby enabling us to determine the thick, tensor-ideal subcategories of the stable module category of finite-dimensional G-modules. Finally, using the stable module category of G, we provide Π(G) with the structure of a ringed space which we show to be isomorphic to the scheme ProjH(G,k)

Article information

Duke Math. J., Volume 139, Number 2 (2007), 317-368.

First available in Project Euclid: 31 July 2007

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 16G10: Representations of Artinian rings
Secondary: 20C20: Modular representations and characters 20G10: Cohomology theory


Friedlander, Eric M.; Pevtsova, Julia. $\Pi$ -supports for modules for finite group schemes. Duke Math. J. 139 (2007), no. 2, 317--368. doi:10.1215/S0012-7094-07-13923-1. https://projecteuclid.org/euclid.dmj/1185891825

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