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.
Categories: math.AG math.RT