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Support varieties for rational representations
- Part of
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- Journal:
- Compositio Mathematica / Volume 151 / Issue 4 / April 2015
- Published online by Cambridge University Press:
- 13 November 2014, pp. 765-792
- Print publication:
- April 2015
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- Article
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We introduce support varieties for rational representations of a linear algebraic group
$G$ of exponential type over an algebraically closed field 
$k$ of characteristic 
$p>0$. These varieties are closed subspaces of the space 
$V(G)$ of all 1-parameter subgroups of 
$G$. The functor 
$M\mapsto V(G)_{M}$ satisfies many of the standard properties of support varieties satisfied by finite groups and other finite group schemes. Furthermore, there is a close relationship between 
$V(G)_{M}$ and the family of support varieties 
$V_{r}(G)_{M}$ obtained by restricting the 
$G$ action to Frobenius kernels 
$G_{(r)}\subset G$. These support varieties seem particularly appropriate for the investigation of infinite-dimensional rational 
$G$-modules.
