论文标题
Eilenberg-Moore光谱序列和Hodge共同体分类的堆栈
Eilenberg-Moore spectral sequence and Hodge cohomology of classifying stacks
论文作者
论文摘要
让$ g $成为一个平稳的连接还原组，而$ k $，$γ$是$ g $的中央子组。我们构建了Eilenberg-moore型光谱序列，这些频谱序列会融合到$ b（g/γ）$的Hodge和de Rham同胞。作为一个应用程序，基于Toda的作品并使用Totaro的不平等，我们表明，对于所有$ M \ geq 0 $ hodge and de rham共同体学代数，分类堆栈$ b \ mathrm {pgl} _ {4m+2} $ \ mathbb {f} _2 $是奇异$ \ mathbb {f} _2 $ - 相应谎言组的分类空间的体积学。从此，我们获得了$ h^{> 0}（\ Mathrm {gl} _ {4m+2}，\ propatorName {sym}^J（\ Mathfrak {pgl} _ {4m+2}^\ vee））$ h^{> 0}的完整说明\ operatoTorname {sym}^j（\ mathfrak {pso} _ {4m+2}^\ vee））$ of $ \ m athbb {f} _2 $。
Let $G$ be a smooth connected reductive group over a field $k$ and $Γ$ be a central subgroup of $G$. We construct Eilenberg-Moore-type spectral sequences converging to the Hodge and de Rham cohomology of $B(G/Γ)$. As an application, building upon work of Toda and using Totaro's inequality, we show that for all $m\geq 0$ the Hodge and de Rham cohomology algebras of the classifying stacks $B\mathrm{PGL}_{4m+2}$ and $B\mathrm{PSO}_{4m+2}$ over $\mathbb{F}_2$ are isomorphic to the singular $\mathbb{F}_2$-cohomology of the classifying space of the corresponding Lie group. From this we obtain a full description of $H^{>0}(\mathrm{GL}_{4m+2}, \operatorname{Sym}^j(\mathfrak{pgl}_{4m+2}^\vee))$ and $H^{>0}(\mathrm{SO}_{4m+2}, \operatorname{Sym}^j(\mathfrak{pso}_{4m+2}^\vee))$ over $\mathbb{F}_2$.
