Computations of de Rham cohomology rings of classifying stacks at torsion primes (1909.13413v1)

Abstract: For the split group $G_{2}$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B(G_{2}){\mathbb{F}{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}{2}$-coefficients of $B(G{2}){\mathbb{C}}.$ For the spin groups $\textrm{Spin}(n)$ defined over $\mathbb{Z},$ we show that the de Rham cohomology ring of $B\textrm{Spin}(n){\mathbb{F}{2}}$ is isomorphic to the singular cohomology ring with $\mathbb{F}{2}$-coefficients of $B\textrm{Spin}(n){\mathbb{C}}$ for $n \leq 10.$ For $n=11,$ we make a full computation of the de Rham cohomology ring of $B\textrm{Spin}(11){\mathbb{F}{2}},$ which is not isomorphic to the singular cohomology ring with $\mathbb{F}{2}$-coefficients of $B\textrm{Spin}(11){\mathbb{C}}.$ We also show that the Hodge spectral sequence for $BG{\mathbb{F}_{2}}$ degenerates for all of the groups $G$ mentioned above.