Hochschild-Serre filtration and etale cohomology

Question

I encountered the Hochschild-Serre spectral sequence in étale cohomology

$$H^i(\text{Gal}(\overline{k}/k), H^j_{et}(X_{\overline{k}}, F))\Rightarrow H^{i+j}_{et}(X_{{k}}, F)$$

How is the filtration on $H^*_{et}(X_{{k}}, F)$ "associated to the Hochschild-Serre spectral sequence" defined? Several authors quote it without ever defining it.

Is $F^aH^*$ the image, or a union of images, of differentials in the spectral sequence?

Could you give a reference where the induced filtration on the Hochschild -Serre spectral sequence is explained?

Example What is $F^1H^n(X_k, F)\subset H^1(\text{Gal}(\overline{k}/k), H^{n-1}(X_{\overline{k}}, F))$ ?

Answer 1

Every time there is a spectral sequence, it is usually coming from a filtration. In this case it is coming from a filtration of the chain complex $C^*(X_k;F):=R\Gamma(X_k;F)$. The idea behind it, is to write the functor $R\Gamma(X_k;-)$ as $R\Gamma(k;-)\circ Rp_*$ where $p:X_k\to\mathrm{Spec}\,k$ is the structure map.

Now $Rp_*F=C^*(X_{\bar k};F)$ as a chain complex with $G_k=\mathrm{Gal}(\bar k/k)$-action. We can filter it by connectivity and get a filtered complex $$C^*(X_{\bar k};F)\leftarrow \cdots t_{\ge -2}C^*(X_{\bar k};F) \leftarrow t_{\ge -1}C^*(X_{\bar k};F) \leftarrow t_{\ge 0}C^*(X_{\bar k};F) \leftarrow 0$$ The associated graded of this filtered complex is precisley $\bigoplus_i H^i(X_{\bar k};F)[-i]$. Applying $R\Gamma(k;-)$ we obtain another filtered complex $$C^*(X_k;F)=R\Gamma(k;C^*(X_{\bar k};F))\leftarrow \cdots R\Gamma(k;t_{\ge -2}C^*(X_{\bar k};F)) \leftarrow R\Gamma(k;t_{\ge -1}C^*(X_{\bar k};F)) \leftarrow R\Gamma(k;t_{\ge 0}C^*(X_{\bar k};F)) \leftarrow 0$$ The associated graded of this filtration is $$\bigoplus_i R\Gamma(k;H^i(X_{\bar k};F))[-i]$$ and we obtain a spectral sequence with $E_2$-page the homotopy of the associated graded, converging to the associated graded of the induced filtration on homotopy groups $$F^iH^n(X_k;F)=\mathrm{Im}\left(\mathbb{H}^n(G_k;t_{\ge -i}C^*(X_{\bar k};F))\to H^n(X_k;F)\right)$$ Now, you might complain that this is not very explicit, and the answer is that unfortunately it is not. But in low degrees we are able to describe this filtered pieces in terms of edge homomorphisms in the spectral sequence. For example $F^1H^n(X_k;F)$ is given by the kernel of $$H^n(X_k;F)\to H^n(X_{\bar k};F)^{G_k}$$ (you should recognize the target of the map as the 0-line of the spectral sequence).

[Disclaimer: this answer is using homological indexing. Yes this means that cohomology will be in negative degree. Sorry, I cannot seem to get the indexing right with any other convention]