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It can be proven that if in a quasicompact scheme $X$ any finite subset is contained in an affine open subset then for any sheaf $\mathcal{F}$ on $X$ its Cech cohomology $\hat{H_{et}^{\bullet}}(X,\mat …

1)Yes, such decomposition follows from the fact that Frobenius on the de Rham-Witt differential forms acts in a way that slopes on $H^i(X, W\Omega^j)[1/p]$ are in the interval $[j,j+1)$. This forces t …

Let $k$ be a field of characteristics $p$ and $R$ be any ring where $p$ is not invertible. Asuume that $F:Var_{k}\to D(R-mod)$ is a cohomology theory of smooth algebraic varieties over the field $k$ …

For a smooth variety $X$ over a perfect field of characteristics $p$ the sheaf of differential operators is an Azumaya algebra(etale locally is isomorphic to endomorphisms of its center, which is bigg …

See Theorem 4.2.2 in https://www-fourier.ujf-grenoble.fr/~mbrion/lin.pdf In particular, in your example properness of $X=G/B$ simplifies the left part of the sequence, turning it into $$0\to \hat{G} …

Consider the class map $$cl:CH^i(X)\to H^{2i}_{cont}(X,\mathbb{Z}_l(i))$$ where the RHS is the continuous etale cohomology(defined by Jannsen in his paper "Continuous etale cohomology"). In this paper …

Let's first figure out why the definition given in Berthelot-Ogus coincides with the one from the Stacks project. Unraveling the definition 5.8.3 we see that for a sheaf $G$ on $(Y/W)_{cris}$ the in …

Here is an example when $\gamma $ is not injective. In general, if $A=A_0\times M$ is a constant abelian scheme, choose a presentation for $(A_0)_{an}$ as $\mathbb{C}^g/\mathbb{Z}^{2g}$. This induces …