Cohomology for Frobenius kernels of $SL_2$

TL;DR

This paper provides a complete description of the cohomology groups for Frobenius kernels of $SL_2$, proves the Cohen-Macaulay property of their cohomology rings, and relates their spectra to geometric fiber products, also extending results to quantum groups.

Contribution

It offers the first comprehensive computation of cohomology for $(SL_2)_r$ and establishes geometric and algebraic properties, extending to quantum groups.

Findings

Cohomology groups for $(SL_2)_r$ are explicitly described.

The cohomology ring $ ext{H}^ullet((SL_2)_r,k)_{ ext{red}}$ is Cohen-Macaulay.

The spectrum of the cohomology ring is homeomorphic to a fiber product $G imes_B raku^r$.

Abstract

Let $(SL_2)_r$ be the $r$-th Frobenius kernels of the group scheme $SL_2$ 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 $(SL_2)_r$. 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 $(SL_2)_r$ 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 $SL_2$.