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Cohomology of infinitesimal and discrete groups. (English) Zbl 0586.20021
An infinitesimal algebraic group over a field k is a connected algebraic group scheme whose coordinate ring is a finite dimensional k-algebra. For a reductive algebraic group G defined over $$F_ p$$, there is a close relation between the representation theory of G and its infinitesimal subgroups $$G_ r$$ $$(=\ker nels$$ of the Frobenius endomorphism $$\sigma^ r)$$ and its finite Chevalley subgroups $$G(F_ q)$$ of $$F_ q$$-rational points. In the case of $$r=1$$, earlier works of the authors and coworkers led to explicit cohomology calculations for $$G(F_ q)$$ with certain nontrivial coefficients.
This paper continues the study of infinitesimal subgroups of simple algebraic groups by proving various qualitative results on the cohomology and by extending the computational knowledge. The precise results are too numerous and too complicated to summarize. There are various applications as well as ample references to related works.
Reviewer: C.H.Sah

##### MSC:
 20G10 Cohomology theory for linear algebraic groups 14L15 Group schemes 20G40 Linear algebraic groups over finite fields
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