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Let G be a simple, simply connected algebraic group defined over analgebraically closed field k of positive characteristic p. Let \sigma:G->G be astrict endomorphism (i. e., the subgroup G(\sigma) of \sigma-fixed points isfinite). Also, let G_\sigma\ be the scheme-theoretic kernel of \sigma, aninfinitesimal subgroup of G. This paper shows that the degree m cohomologyH^m(G(\sigma),L) of any irreducible kG(\sigma)-module L is bounded by aconstant depending on the root system \Phi\ of G and the integer m. A similarresult holds for the degree m cohomology of G_\sigma. These bounds are actuallyestablished for the degree m extension groups Ext^m_{G(\sigma)}(L,L') betweenirreducible kG(\sigma)-modules L and L', with again a similar result holdingfor G_\sigma. In these Ext^m results, of interest in their own right, thebounds depend also on L, or, more precisely, on length of the p-adic expansionof the highest weight associated to L. All bounds are, nevertheless,independent of the characteristic p. These results extend earlier work ofParshall and Scott for rational representations of algebraic groups G.  We also show that one can find bounds independent of the prime for the Cartaninvariants of G(\sigma) and G_{\sigma}, and even for the lengths of theunderlying PIMs. These bounds, which depend only on the root system of G andthe "height" of \sigma, provide in a strong way an affirmative answer to aquestion of Hiss, for the special case of finite groups G(\sigma) of Lie typein the defining characteristic.

Bounding extensions for finite groups and Frobenius kernels