Spectral Sequences
What are spectral sequences and what are they good for?
In short, spectral sequences are tools for computing cohomology of certain complexes that allows us to use the information about the approximation or pieces of the complex to get information about the cohomology of the whole complex.
Motivation
Consider a resolution of a sheaf \(\mathcal{F}\)
\[0\to \mathcal{F}\to \mathcal{G}^0 \to \mathcal{G}^1 \to \dots \to \mathcal{G}^n \to \dots\]
which can be thought as an approximation of \(\mathcal{F}\) using \(\mathcal{G}^\bullet\). Here the key point is that we are not necessarily using a \(\Gamma\)-acyclic resolution, just any resolution.
By breaking it into short exact sequences and repeatedly apply the process of resolution, we can resolve each of the \(\mathcal{G}^i\) and form a resolution of the sequence as a double complex
where the second line is an exact sequence of injective resolutions. By removing the first line and first column, we have a double complex \(I^{\bullet,\bullet}\) of injective modules and two differentials \(d_I, d_{II}\) (on which \(d_{II}\) comes with a sign) such that \(d_Id_{II}+d_{II}d_I=0\)
Form a total complex \(T^n = \bigoplus_{p+q=n} I^{p,q}\) and \(d = d_I + d_{II}\), we can define the cohomology of this total complex. Here the following properties hold
The vertical cohomologies are the cohomologies of the approximating objects \(\mathcal{G}^i\)
\[H^q(\Gamma I^{p,\bullet},d_{II}) = H^q(X,\mathcal{G}^p).\]
The horizontal cohomologies almost vanish
\[H^p(\Gamma I^{\bullet,q},d_I) = 0, \text{ for } p>0.\]
The inclusion \(\Gamma(I^\bullet)\subset \Gamma(T^\bullet)\) given by \(x_q\mapsto (x_q,0,\dots,0)\) is a quasi-isomorphism, i.e.
\[R^i\Gamma(\mathcal{F}) = H^i(\Gamma I^\bullet) = H^i(\Gamma T^\bullet)\]
which tells you that computing the derived functors of \(\mathcal{F}\) is the same as computing the cohomology of the total complex.