References:
Principles of Algebraic Geometry by Griffiths and Harris, Chapter 3
\(p\)-adic Cohomology by Kedlaya
Methods of Homological Algebra by Gelfand and Manin
Homological Algebra by Weibel
For any filtered complex \(K^\bullet\), there exists a spectral sequence \((E_r, d_r:E_r^{p,q}\to E_r^{d+r,d-r+1})\) such that
\[E_0^{p,q} = \mathrm{Gr}^p K^q[p] = \frac{F^p K^{p+q}}{F^{p+1} K^{p+q}}\]
\[E_1^{p,q} = H^q(E_0^{p,\bullet}) = H^{q}(\mathrm{Gr}^p K^\bullet[p]) = H^{p+q}(\mathrm{Gr}^p K^\bullet)\]
\[E_\infty^{p,q} = \mathrm{Gr}^p (H^{p+q}(K^\bullet))\]
i.e. the spectral sequence converges to the graded pieces of the total cohomology of the original complex \(K^\bullet\), some information is lost in the process.
If we think the higher index in the filtration \(F^r\), the smaller and closer to zero the element is, we can use this idea to talk about approximate cocycles. This will enable use to form the approximate cohomology groups. Define the approximate cocycles as
\[ A^{p,p+r} = A^p_r := \{ e \in F^pC : \mathrm{d}e \in F^{p+r}C \}. \]
Let \(\eta^p:F^pC\to \mathrm{Gr}^p C\xrightarrow{[p]} \mathrm{Gr}^p C[p] = E_0^p\) be the composition of projection and degree shift, then we define
\[ Z_r^p := \eta^p(A_r^p),\] note that \(\mathrm{d}A^{p-r+1}_{r-1}\subset A^p_r\), we can also define the boundary subgroup of \(Z_r^p\) as \[ B^p_r := \eta^p(\mathrm{d}A^{p-(r-1),p}) = \eta^p(\mathrm{d}A^{p-r+1}_{r-1}).\]
Let’s evaluate them in detail. We have
\[\begin{align*} Z_r^p &= \eta^p(A^{p,p+r}) \\ &= \frac{A^{p,p+r} + F^{p+1}C}{F^{p+1}C}[p] \\ &= \frac{A^{p,p+r}}{A^{p,p+r}\cap F^{p+1}C}[p] \\ &= \frac{A^{p,p+r}}{A^{p+1,p+r}}[p], \end{align*}\]
and
\[\begin{align*} B^p_r &= \eta^p(\mathrm{d}A^{p-(r-1),p}) \\ &= \frac{\mathrm{d}A^{p-(r-1),p} + F^{p+1}C}{F^{p+1}C}[p] \\ &= \frac{\mathrm{d}A^{p-(r-1),p}}{\mathrm{d}A^{p-(r-1),p}\cap F^{p+1}C}[p] \\ &= \frac{\mathrm{d}A^{p-(r-1),p}}{\mathrm{d}A^{p-(r-1),p+1}}[p]. \end{align*}\]
Using this, we can form the cohomology \(E_r^p = H^p(E_{r-1})\) as
\[\begin{align*} E_r^p &= \frac{Z_r^p}{B_r^p} = \frac {\ker\left(Z_{r-1}^p\xrightarrow{d_{r-1}^p} E_{r-1}^{p+r-1}\right)} {\mathrm{Im}\left( E_{r-1}^{p-(r-1)} \xrightarrow{d_{r-1}^{p-(r-1)}} Z_{r-1}^p \right) } \\ &= \frac { A^{p,p+r} } { A^{p+1,p+r} + dA^{p-(r-1),p} } [p]. \end{align*}\]
The differential of \(d_r:E^p_r\to E^{p+r}_r\) is induced from the original differential \(d:C\to C[1]\), which is of degree \(1\), so to match the degree on \(A^{p,p+r}\to A^{p+r,p+2r}[1]\), we finally have a map
\[ d_r : E^p_r \to E^{p+r}_r [1-r], \]
whose cohomology gives the next page \(E_{r+1}^p\).
When \(r\) is eventually large enough, we can see that
\[\begin{align*} E_\infty^p &= \frac { A^{p,\infty} } { A^{p+1,\infty} + dA^{-\infty,p} } [p] \\ &= \frac { \{c\in F^pC : dc = 0\} } { \{c\in F^{p+1}C: dc = 0\} + (\mathrm{d}C)\cap F^pC } [p] \\ &= \frac { \ker(d|_{F^pC}) } { \ker(d|_{F^{p+1}C}) + \mathrm{d}C\cap F^pC } [p] \\ &= \mathrm{Gr}^p H(C) [p]. \end{align*}\]
Some questions