Hochschild-Serre for ├ętale cohomology vs. Galois cohomology

Let X be an irreducible smooth projective variety of dimension d over a number field K. Let $ \bar X$ denote its base change to an algebraic closure of K. Then, if Understand it right, the Hochschil-Serre spectral sequence can be used to prove that for any $ i\geq 1$ and any integer $ m$ , there is an isomorphism

$ $ H^i_{et}(X,Z_p(m)) = H^1(G_K,H_{et}^{i-1}(\bar X,Z_p(m))) $ $

Since I know very little about spectral sequences, I am confused about some consequences one could derive from such isomorphism.

For instance take $ d=1$ , $ i=1$ , $ m=0$ , so $ X$ is a curve. Then

$ $ H^1_{et}(X,Z_p) = H^1(G_K,H_{et}^{0}(\bar X,Z_p)) $ $

The left-hand side is a very rich object, sitting in an exact sequence which shows that it is an extension of $ H_{et}^1(\bar X,Z_p)$ and $ G_K \otimes Z_p$ . The right-hand side looks rather trivial because $ H_{et}^{0}(\bar X,Z_p) = Z_p$ and hence $ H^1(G_K,H_{et}^{0}(\bar X,Z_p)) = Hom(G_K, Z_p) = (G_K \otimes Z_p)^*$ .

There is something that I’m obviously doing wrong, because the way I put it, the left-hand side is far from isomorphic to the right-hand side.

Quantum cup product and Dolbeault cohomology

Let $ X$ be a smooth projective variety over $ \mathbb{C}$ . We consider the small quantum cup product $ \star$ on the deRham cohomology ring $ \displaystyle H^*(X;\mathbb{C})=\bigoplus_{p,q}H^{p,q}(X)$ . Let $ H^{(i)}$ denote $ \displaystyle\bigoplus_{p-q=i}H^{p,q}(X)$ .

In ”Gamma Conjecture via Mirror Symmetry”, Galkin and Iritani claimed in the Appendix that $ H^{(i)}\star H^{(j)}\subseteq H^{(i+j)}$ in order to draw a conclusion about the spectrum of the operator $ c_1(X)\star$ acting on various subspaces of $ H^*(X)$ . They said this claim follows from the motivic axiom of the Gromov Witten theory. How should one argue following this line?

Thanks.

Remark: If $ X$ is convex, then $ \overline{\cal M}_{0,3}(X,\beta)$ is a smooth projective variety so the claim should be obvious. But in general we have to use virtual fundamental cycle to define $ \star$ . I don’t know what we should do then.

A cohomology associated to a vector field on a Riemannian manifold

Let $ X$ be a vector field on a Riemannian manifold $ (M,g)$ . So we have a $ 1$ -form $ \beta$ with $ \beta(Y)=<Y,X>_g$ .

We consider the following subcomplex of de Rham complex $ \Omega^*(M)$ : $ $ \{\alpha\in \Omega^*(M)| d\alpha=\alpha \wedge \beta\}$ $

This generates a cohomology with the standard exterior derivation. For zero vector field we get the standard de Rham cohomology.

Is this cohomology always a finite dimensional space? Does it depend on the Riemannian structure? Does it contain some dynamical information about the vector field $ X$ ?

Is there an appropriate analogy of this cohomology in algebraic topology when we replace the $ 1$ -form $ \beta$ with a $ 1$ -cochain and the exterior derivative with cup product?

Comparing cohomology using homotopy fibre

I have a question, which might be very basic, but I don’t know enough topology to answer.

Suppose you have a map of topological spaces (or homotopy types) $ f : X \to Y$ , with homotopy fibre given by $ F$ . We get an induced morphism on cohomology $ f^{*} : H^{*}(Y) \to H^{*}(X)$ . If $ F$ is $ n$ -connected (for all homotopy fibres), can we thereby conclude that $ f^{*}$ is an isomorphism up to degree $ n$ ?

I know that this holds in the case that $ f$ is locally a fibration between manifolds (for example a submersion).

Another case I have in mind is the truncation map $ p : X \to K(\pi_{1}(X),1)$ , which has homotopy fibre given by the universal cover $ \tilde{X}$ of $ X$ . In this case we see that if $ \tilde{X}$ is $ n$ -connected, then $ H^{k}(\pi_{1}(X)) \cong H^{k}(X)$ for $ k \leq n$ , where the left cohomology group refers to group cohomology.

(Co)homology of ind-schemes

I am trying to understand how $ \ell$ -adic homology and cohomology of ind-schemes are defined.

It seems that by definition one chooses a sequence of schemes which limits to the ind-scheme, and takes the limit of the cohomologies. However, I can’t prove that the limit is well-defined, i.e. independent of the sequence.

Also, does one take the limit of the $ \ell$ -adic cohomologies, or instead the limit of the finite cohomologies, and the pass to a limit in $ \ell^n$ ?

A canonical complex computing etale cohomology

Crystalline cohomology can be computed as the hypercohomology of the de Rham-Witt complex.

If we want to compute the etale cohomology of the constant sheaves $ \mathbb{Z}_l$ or $ \mathbb{Q}_l$ (well, to consider them as sheaves you need pro-etale site or something like that but you get the idea), is there some canonical choice of an injective resolution? There can be topological obstructions to this.

The question probably splits into 3 parts: schemes are defined over a field of zero characteristic, schemes are defined over a field of characteristic $ l=p$ , schemes are defined over a field of positive characteristic $ p\neq l$ .

Cohomology of $SO(p,q,\mathbb{Z})$ with p=3,q=19

I would like to understand the topology of the moduli space of Einstein orbifold metrics on the $ K3$ -surface. It is known that this space is given by the bi-quotient $ SO(3,19;\mathbb{Z})\setminus SO(3,19)/SO(3)\times SO(19)$ and I know that this has the same rational cohomology as $ SO(3,19;\mathbb{Z})$ . Is there anything known about these groups? Unfortunately my knowledge on arithmetic groups is very limited.

Weil cohomology theories “genuinely” of positive characteristic

One of the reasons why Weil cohomology theories are required to have coefficients in a field of characteristic 0 is that they are supposed to be robust enough to solve Weil conjectures, i.e. to count points (and if coefficients have $ \mathrm{char}\:p$ you could only count $ \mathrm{mod}\:p$ , barring some inventiveness).

The question is: if we take the Wikipedia definition of Weil cohomology theory and keep all of the requirements but allow characteristic to be positive, will we get any new theories?

I am not very experienced with etale cohomology to see whether it taken with $ F_p$ -coefficients satisfies all of the requirements. Assuming it does, taking it is a kind of a cheat because $ l$ -adic cohomology has a natural integral structure and $ F_p$ -coefficients are obtained by quotienting out the maximal ideal. The same objection applies to crystalline cohomology with truncated coefficients.

I wonder whether there are examples “genuinely” of $ \mathrm{char}\:p$ , i.e. such that there is no Weil cohomology theory with $ \mathrm{char}\:0$ coefficients that has a natural integral structure over a DVR such that the quotient by maximal ideal produces the cohomology theory under consideration. Possibly there are some other more-or-less trivial “cheats” I am missing, point them out in the comments. An answer to this question should not rely on such “cheats” (the distinction is not very precise admittedly but I hope the idea is clear).