Cohomology for Frobenius kernels of $S{L}_2$

Abstract

Let $(S{L}_2{)}_r$ be the $r$-th Frobenius kernels of the group scheme $S{L}_2$ defined over an algebraically field of characteristic $p>2$. In this paper we give for $r\ge 1$ a complete description of the cohomology groups for $(S{L}_2{)}_r$. We also prove that the reduced cohomology ring $\opH^\bullet((SL_2)_r,k)_{\red}$ is Cohen-Macaulay. Geometrically, we show for each $r\ge 1$ that the maximal ideal spectrum of the cohomology ring for $(S{L}_2{)}_r$ is homeomorphic to the fiber product $G\times_B\fraku^r$. Finally, we adapt our calculations to obtain analogous results for the cohomology of higher Frobenius-Luzstig kernels of quantized enveloping algebras of type $S{L}_2$.

published version; a section for the case p=2 is added

References in corpus (3)

• Cohomology of finite dimensional pointed Hopf algebras
• Representations and cohomology for Frobenius-Lusztig kernels
• Commuting varieties of $r$-tuples over Lie algebras

Cited by in corpus (3)

• Commuting varieties of $r$-tuples over Lie algebras
• Cohomology of $S{L}_2$ and related structures
• Extensions for Finite Chevalley Groups III: Rational and Generic Cohomology