论文标题
$ \ text {psl} _2（q）$的同谋
Cohomology of $\text{PSL}_2(q)$
论文作者
论文摘要
在2011年，Guralnick和Tiep证明，如果$ g $是一个带有borel子组$ b $的雪佛兰集团，而$ v $ a nordreducible $ g $ -mmodule具有交叉特征，则$ v^b = 0 $，则$ h^1（g，v）$的尺寸由$ b $ b $ b $ b $ osets ocets ocets ocets ocets ocets ocets ocets ocets ocets ocets ocets ocets of b $ b $ b $ b $。 We generalise this theorem to higher cohomology and an arbitrary finite group, so that if $H \leq G$ such that $O_{r'}(H) = O^r(H)$ and $V^H = 0$ for $V$ a $G$-module in characteristic $r$ then $\dim H^1(G,V)$ is determined by the structure of the permutation module on cosets of $H$, and $ h^n（g，v）$ by $ \ text {ext} _g^{n-1}（n-1}（v^*，m）$，依赖于$ h $的$ kg $ -module $ m $。我们还确定$ \ text {ext} _g^n（v，w）$ for $ $ kg $ -modules $ v $，$ w $ for $ g \ in \ in \ in \ {\ text {psl} _2 _2 _2（q），\ text {pgl} _2（q）
In 2011, Guralnick and Tiep proved that if $G$ was a Chevalley group with Borel subgroup $B$ and $V$ an irreducible $G$-module in cross characteristic with $V^B = 0$, then the the dimension of $H^1(G,V)$ is determined by the structure of the permutation module on the cosets of $B$. We generalise this theorem to higher cohomology and an arbitrary finite group, so that if $H \leq G$ such that $O_{r'}(H) = O^r(H)$ and $V^H = 0$ for $V$ a $G$-module in characteristic $r$ then $\dim H^1(G,V)$ is determined by the structure of the permutation module on cosets of $H$, and $H^n(G,V)$ by $\text{Ext}_G^{n-1}(V^*,M)$ for some $kG$-module $M$ dependent on $H$. We also determine $\text{Ext}_G^n(V,W)$ for all irreducible $kG$-modules $V$, $W$ for $G \in \{\text{PSL}_2(q), \text{PGL}_2(q), \text{SL}_2(q)\}$ in cross characteristic.
