I am reading the wonderful article Spectral sequences : friend or foe ? by Ravi Vakil. However I do not understand the notation used when on p.4 he says :
More precisely, there is a filtration $E_\infty^{o,k} \overset{E_\infty^{1,k-1}}{\hookrightarrow} ? \overset{E_\infty^{2,k-2}}{\hookrightarrow} ... \hookrightarrow H^k(E^{.})$ where quotients are displayed above each inclusion.
What does this statement mean ? I do not know the meaning of 'filtration' and do not understand which quotients are being talked about.
My second question is the following. He mentions that in general there is no isomorphism between $_{>}E_\infty^{p,q}$ and $_{\wedge}E_\infty^{p,q}$. Why is it,then, that in the proof of snake lemma the vanishing of the first horizontal page implies that the vertical pages stabilize to the page with all entries 0 ?
I am a novice to this subject, so please excuse my silly questions. Thanks.
Let $H$ be some object in a nice (e.g., abelian) category. An (ascending) filtration of $H$ is a collection $(F_i)$ of subobjects $$F_0 \subset F_1 \subset \cdots \subset H.$$
In your example, it is asserted that $H = H^k(E^\bullet)$ has a filtration of length $k+1$ given by $(F_i)_{0 \leq i \leq k}$ with $F_0 = E_\infty^{0,k}$ and $F_k = H^k(E^\bullet)$. Moreover, the filtration quotients are known: there are short exact sequences \begin{gather*} 0 \to F_0 \to F_1 \to E_\infty^{1,k-1} \to 0 \\ 0 \to F_1 \to F_2 \to E_\infty^{2,k-2} \to 0 \\ \vdots \\ 0 \to F_{k-1} \to F_k \to E_\infty^{k,0} \to 0. \end{gather*}
In principle, knowledge of the associated graded will let you reconstruct the filtered object $H$, up to extension problems.
As for your second question, while there is no isomorphism between the $E_\infty$-pages of the horizontal and vertical filtrations, the two spectral sequences must converge (if they converge) to the same object, in this case the cohomology $H^\bullet(E^\bullet)$ of the total complex.
In the example of the proof of the snake lemma, we know from the horizontal filtration that $H^\bullet(E^\bullet) = 0$. This implies that we must have $E_\infty = 0$ for the vertical filtration as well (otherwise we would get $H^\bullet(E^\bullet) \neq 0$). Then you can use this to deduce the existence of certain differentials that may not have been evident otherwise.