Some basic questions on crystalline cohomology

Let $ X_0$ be a smooth projective variety over $ \mathbf{F}_q$ and $ {X}$ its base change to an algebraic closure $ k$ of $ \mathbf{F}_q$ .

Crystalline cohomology $ H^*_{\rm cris}(X) := H^*((X/W(k))_{\rm cris},\mathcal{O}_{(X/W)_{\rm cris}})[1/p]$ is known to be a Weil cohomology.

It is also known to be computed as the Zariski hypercohomology of the de Rham-Witt complex $ W\Omega^*_X$ , and the Hodge to de Rham spectral sequence:

$ $ E_2^{i,j} := H^i(X_{\rm Zar},W\Omega^j_X)[1/p]\Rightarrow H^{i+j}_{\rm cris}(X)$ $

degenerates at the second page.

  • Do we therefore have a “Hodge decomposition” $ $ \bigoplus_{i+j=n}H^i(X_{\rm Zar},W\Omega^j_X)[1/p]=H^n_{\rm cris}(X)\ \ ?$ $
  • When $ n$ is even and $ i=j=n/2$ , does the geometric Frobenius act on $ H^i(X_{\rm Zar},W\Omega^i_X)$ by multiplication by $ q^i$ ?
  • Are the Frobenius eigenvalues on $ H^i(X_{\rm Zar},W\Omega^j_X)[1/p]$ (base changed to an algebraic closure of $ \text{Frac}(W)$ ) not integers when $ i\neq j$ ?
  • And on a more naive level, where do we crucially use that $ k$ is algebraically closed? We know there is no Weil cohomology with coefficients in $ \mathbf{Q}_p$ , so when $ p = q$ we know crystalline cohomology with $ W(\mathbf{F}_p)[1/p]$ coefficients can’t work. Where do the proofs of the axioms break down when $ k$ is not algebraically closed?

How to compute Galois representations from etale cohomology groups of a generalized flag variety?

Let $ G$ be a connected reductive group over a number field $ K$ , $ P$ be a parabolic subgroup of $ G$ defined over $ K$ , $ X=G/P$ be the generalized flag variety which is a smooth projective variety over $ K$ and $ p$ be a prime number. For a positive integer $ i>0$ , consider the etale cohomology $ V=H^i(X_{K^{alg}},\mathbb Q_p)$ as a Galois representation of $ G_K$ .

How to compute such Galois representation? This may be done somewhere but I can’t find a reference. Firstly, the dimension may be computed by using Betti numbers and some combination datas from the Lie algebra (the odd dimension shall vanish).

Secondly, is the representation semi-simple? For the projective space it’s obviously true as the dimension is no bigger than $ 1$ . If that’s true, then must the direct summand be some $ \mathbb Q_p(-i/2)$ ? Some density theorem may reduce this to the finite field case.

In a short word, how to completely decide the Galois representation? As the flag variety has a stratification by affine spaces, this seems reachable.

(Singular) metric associated to the higher cohomology

Suppose $ X$ is a smooth complex variety and $ L$ is a line bundle with a metric $ h_L$ , then a section $ s \in H^0(X, L)$ gives another metric $ \tilde h_L:= e^{-\phi}h_L$ where $ \phi=\log \|s\|^2_{h_L}$ .

If $ u \in H^q(X,L)$ is a section (or just $ u\in H^q(X,K_X)$ ), is there a way to construct a metric on $ L$ with some relation to $ u$ ?

Serre spectral sequence for de Rham cohomology

Suppose we a given a fibration of manifolds $ p\colon E\to M$ with a path connected fiber $ F$ and simply connected $ M$ , then we have the Serre spectral sequence with

$ $ E_2^{p,q} = H^p(M,\underline{H^q(F)}) $ $

The standard proof of its convergence to $ H^n(E)$ is purely topological and goes for singular or cellular cohomology. Can one give the proof in terms of de Rham cohomology?

In fact, I’m even more interested in de Rham cohomology with compact support. Now we have some difficulties in defining the local system as cohomology with compact support are no longer homotopy invariant but I hope these problems can be overcome.

The sheafification of taking cohomology is trivial?

Consider the Nisnevich site of a noetherian scheme $ S$ of finite Krull dimension (the objects are schemes $ U$ smooth and of finite type over $ S$ ), let $ A$ be a sheaf of abelian groups on this site. I want to know:

Is the Nisnevich sheafification of the presheaf $ $ U\mapsto\mathrm{H}_{\mathrm{Nis}}^{n}(U, A)$ $ trivial for $ n>0$ (or does this presheaf have trivial stalks)?

One can also assume $ S=\mathrm{Spec}(k)$ for $ k$ a field if needed.

The case of $ n=1$ is trivial since $ \mathrm{H}_{\mathrm{Nis}}^{1}$ classifies torsors, which are Nisnevich locally trivial.

Compute the cohomology of $\mathrm{Hom} (\Omega^*(M),\Omega^*(M))$

Let $ M$ be a compact smooth manifold. And particularly I am interested in the case the torus $ M=T^n$ .

Consider the de Rham complex $ (\Omega^*(M), d)$ and the cochain complex $ $ C:=\mathrm{Hom} (\Omega^*(M),\Omega^*(M)) $ $ with the differential map $ \delta$ given by $ $ \delta(f)=d \circ f – f\circ d$ $

Question: what is the cohomology $ H^*(C,\delta)$ ? And, what should be the right tool to do the computation? Is this cohomology related to the usual de Rham cohomology $ H^*_{dR}(M)$ ?

Independence of $\ell$ in $\ell$-adic cohomology

Let $ X$ be the base change of a smooth projective variety over a finite field, to a separable closure $ k$ of the ground field.

Do we expect for any endomorphism $ f$ of $ X$ to have an effect $ f^* : H^n(X,\mathbf{Q}_{\ell})\to H^n(X,\mathbf{Q}_{\ell})$ whose characteristic polynomial has coefficients in $ \mathbf{Q}$ ? or at least independent of $ \ell$ in some suitable sense?

Is there an example of an $ f$ such that its characteristic polynomial on $ \ell$ -adic cohomology does depend on $ \ell$ ?

Differentials in Weil model for equivariant cohomology

Why should we define the differential in Weil model as follows? I could understand $ \sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k$ plays a role in the formula because it is the dual of the structure map of $ \mathfrak{g}$ . The rest of formula looks mysterious to me.

Cartan-Weil model for Equivariant Cohomology

We define its Weil algebra by $ W^*(\mathfrak{g}^*)=S^*(\mathfrak{g}^*) \otimes \wedge^*(\mathfrak{g}^*)$ there is also a natural differential operator $ d_W$ which makes $ W*(\mathfrak{g}^*)$ into a complex. We define $ d_W$ as follows:

Choose a basis $ e_1,…,e_n$ for $ \mathfrak{g}$ and let $ e^*_1,…e^*_n$ its dual basis in $ \mathfrak{g}^*$ . Let $ \theta_1,…,\theta_n$ be the image of $ e^*_1,…e^*_n$ in $ \wedge(\mathfrak{g}^*)$ and let $ \Omega_1,…,\Omega_n$ be the image of $ e^*_1,…e^*_n$ in $ S(\mathfrak{g}^*)$ . Let $ c_{jk}^i$ be the structure constants of $ \mathfrak{g}$ , that is $ [e_j,e_k]=\sum_{i=1}^nc_{jk}^ie_i$ . Define $ d_W$ by \begin{eqnarray} d_W\theta_i=\Omega_i- \frac{1}{2}\sum_{j,k} c_{jk}^i \theta_j \wedge \theta_k \end{eqnarray} and \begin{eqnarray} d_W\Omega_i=\sum_{j,k}c_{jk}^i\theta_j \Omega_k \end{eqnarray} and extending $ d_W$ to $ W(\mathfrak{g})$ as a derivation.

Group cohomology of “twisted” projective SU(N) with various coefficients

Given a group $ $ G= PSU(N) \rtimes \mathbb{Z}_2, $ $ where $ PSU(N)$ is a projective special unitary group. Say $ a \in PSU(N)$ , $ c \in \mathbb{Z}_2$ , then $ $ c a c= a^*, $ $ which $ c$ flips $ a$ to its complex conjugation written as $ a^*$ .

My question is what can we say about the group cohomology of $ G= PSU(N) \rtimes \mathbb{Z}_2$ with integer coefficient or finite abelian group coefficient? Say $ $ H^k(G, \mathbb{Z})=? $ $ $ $ H^k(G, \mathbb{Z}_2)=? $ $ $ $ H^k(G, \mathbb{Z}_n)=? $ $ Here we can either regard the group cohomology $ H^k(G,*)$ , or regard it as the topological cohomology of the classifying space of the group $ H^k(BG,*)$ .

I am ONLY interested in $ k=1,2,3,4$ . Here $ n=2$ or $ n=4$ is enough. And when $ N$ as an even integer is enough.

P.S: For your assitance — When $ N=2$ , I already know that $ $ H^3(SO(3), \mathbb{Z})=H^2(SO(3), \mathbb{R}/\mathbb{Z})=H^2(SO(3), \mathbb{Z}_2)=\mathbb{Z}_2. $ $ $ $ H^4(SO(3), \mathbb{Z})=H^3(SO(3), \mathbb{R}/\mathbb{Z})=\mathbb{Z}, $ $ $ $ H^2(PSU(N), \mathbb{Z}_N)=\mathbb{Z}_N, $ $

See a related post: Group cohomology of orthogonal groups with integer coefficient