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This answer is due to Jon Pridham. While we might not expect $H^i(G_k,V)=H^i(G_k^\mathrm{alg},V)$ for every finite dimensional, continuous $G_k$-representation $V$, there are certain results from the …

To answer your second question, for any nilpotent neutral Tannakian category $\mathcal{C}$, (i.e. one in which every object is an iterated extension of the unit object $\underline{1}$), with fibre fun …

I've been thinking about Bertrand Toen's approach to studying the homotopy theory of schemes, and I've come across an inconsistency in my understanding of the subject that I was hoping somebody might …

Let $k$ be a complete, non-archimedean field, and $X$ a Berkovich space over $k$ (as nice as you like, for arguments sake let's say strictly $k$-analytic, good, and geometrically connected). As discus …

Suppose that $k$ is a perfect field of characteristic $p>0$, $\mathcal{V}$ is a complete discrete valuation ring with residue field $k$ and quotient field $K$, of characteristic $0$. Then when one d …

This question is about 6 operations formalism for 'crystalline' cohomology theories - more specifically the infinitesimal cohomology of smooth $\mathbb{C}$-varieties, and the overconvergent cohomology …

So maybe everything I'm about to say you already know, so apologies if I'm teaching my grandmother to suck eggs. This is discussed a bit at the end of a paper of Fontaine "Representations $\ell$-adiq …

According to Wikipedia, a Weil cohomology theory is a functor from the category of smooth projective varieties over a field $k$, to graded algebras over a field $K$ of characteristic zero, together wi …

Al ulrich says, this is not true in general. If $X$ is non-compact (i.e. affine), then the Lie algebra of U is isomorphic to the free Lie algebra on $H^1_\mathrm{et}(X,\mathbb{Q}_p)^\vee$. Hence in th …

This is a question about the category $F\text{-}D^b_\mathrm{coh}(\mathscr{D}^\dagger_{\mathscr{X},\mathbb{Q}})$ of complexes of arithmetic $\mathscr{D}$-modules with Frobenius structure on a smooth fo …

Suppose that $X/\mathbb{Q}_p$ is a smooth, projective variety, and choose a prime $\ell\neq p$. Then the weight-monodromy conjecture says that the graded pieces $\mathrm{Gr}_k^M$ of the monodromy filt …