## Duke Mathematical Journal

### $\Pi$-supports for modules for finite group schemes

#### Abstract

We introduce the space $\Pi(G)$ of equivalence classes of $\pi$-points of a finite group scheme $G$ and associate a subspace $\Pi(G)_M$ 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 $\pi$-points of $G$. Unlike the cohomological support variety of a $G$-module $M$, the invariant $M \mapsto \Pi(G)_M$ 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 $\Pi(G)$ with the structure of a ringed space which we show to be isomorphic to the scheme $\rm{Proj} \rm{H}^{\bullet}(G,k)$

#### Article information

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

Dates
First available in Project Euclid: 31 July 2007

https://projecteuclid.org/euclid.dmj/1185891825

Digital Object Identifier
doi:10.1215/S0012-7094-07-13923-1

Mathematical Reviews number (MathSciNet)
MR2352134

Zentralblatt MATH identifier
1128.20031

#### Citation

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