Spectral Sequences Of Filtered Complex

Author: Eiko

Tags: algebraic geometry, hodge filtration, homological algebra, spectral sequences

Time: 2024-12-22 12:19:55 - 2024-12-22 12:19:55 (UTC)

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
Stacks Project Chapter 13.21, 50

Filtered Complex Induces A Spectral Sequence

For any filtered complex $K^{∙}$ , there exists a spectral sequence $(E_{r}, d_{r} : E_{r}^{p, q} \to E_{r}^{d + r, d - r + 1})$ such that

The $0$ -th page is given by all graded pieces of the complex

$E_{0}^{p, ∙} = {Gr}^{p} K^{∙} [p] = \frac{F^{p} K^{p + q}}{F^{p + 1} K^{p + q}} .$

Think of a first quadrant lattice where you put all pieces of the filtered complex on each dot, but with a downward shifting. The $0$ -th column is the ${Gr}^{0} K^{∙}$ , the $1$ -st column is ${Gr}^{1} K^{∙} [1]$ , and so on.
The first page is given by the cohomologies of graded piece of our complex,

$E_{1}^{p, q} = H^{q} (E_{0}^{p, ∙}) = H^{q} ({Gr}^{p} K^{∙} [p]) = H^{p + q} ({Gr}^{p} K^{∙}) .$

Think about taking all vertical cohomology of the vertical complexes in the $0$ -th page. Interestingly, you obtain horizontal differentials in this page.
The eventual goal of convergence is the graded piece of the total cohomology of the original complex,

$E_{\infty}^{p, q} = {Gr}^{p} (H^{p + q} (K^{∙})) .$

i.e. the spectral sequence converges to the graded pieces of the total cohomology of the original complex $K^{∙}$ , some information is lost in the process.

Approximate Cocycles

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

$A^{p, p + r} = A_{r}^{p} := {e \in F^{p} C : d e \in F^{p + r} C} .$

Let $η^{p} : F^{p} C \to {Gr}^{p} C \overset{[p]}{\to} {Gr}^{p} C [p] = E_{0}^{p}$ be the composition of projection and degree shift, then we define

$Z_{r}^{p} := η^{p} (A_{r}^{p}),$ note that $d A_{r - 1}^{p - r + 1} \subset A_{r}^{p}$ , we can also define the boundary subgroup of $Z_{r}^{p}$ as $B_{r}^{p} := η^{p} (d A^{p - (r - 1), p}) = η^{p} (d A_{r - 1}^{p - r + 1}) .$

Let’s evaluate them in detail. We have

$\begin{aligned} Z_{r}^{p} & = η^{p} (A^{p, p + r}) \\ = \frac{A^{p, p + r} + F^{p + 1} C}{F^{p + 1} C} [p] \\ = \frac{A^{p, p + r}}{A^{p, p + r} \cap F^{p + 1} C} [p] \\ = \frac{A^{p, p + r}}{A^{p + 1, p + r}} [p], \end{aligned}$

and

$\begin{aligned} B_{r}^{p} & = η^{p} (d A^{p - (r - 1), p}) \\ = \frac{d A^{p - (r - 1), p} + F^{p + 1} C}{F^{p + 1} C} [p] \\ = \frac{d A^{p - (r - 1), p}}{d A^{p - (r - 1), p} \cap F^{p + 1} C} [p] \\ = \frac{d A^{p - (r - 1), p}}{d A^{p - (r - 1), p + 1}} [p] . \end{aligned}$

Using this, we can form the cohomology $E_{r}^{p} = H^{p} (E_{r - 1})$ as

$\begin{aligned} E_{r}^{p} & = \frac{Z_{r}^{p}}{B_{r}^{p}} = \frac{\ker (Z_{r - 1}^{p} \overset{d_{r - 1}^{p}}{\to} E_{r - 1}^{p + r - 1})}{Im (E_{r - 1}^{p - (r - 1)} \overset{d_{r - 1}^{p - (r - 1)}}{\to} Z_{r - 1}^{p})} \\ = \frac{A^{p, p + r}}{A^{p + 1, p + r} + d A^{p - (r - 1), p}} [p] . \end{aligned}$

The differential of $d_{r} : E_{r}^{p} \to E_{r}^{p + r}$ is induced from the original differential $d : C \to C [1]$ , which is of degree $1$ , so to match the degree on $A^{p, p + r} \to A^{p + r, p + 2 r} [1]$ , we finally have a map

$d_{r} : E_{r}^{p} \to E_{r}^{p + r} [1 - r],$

whose cohomology gives the next page $E_{r + 1}^{p}$ .

When $r$ is eventually large enough, we can see that

$\begin{aligned} E_{\infty}^{p} & = \frac{A^{p, \infty}}{A^{p + 1, \infty} + d A^{- \infty, p}} [p] \\ = \frac{{c \in F^{p} C : d c = 0}}{{c \in F^{p + 1} C : d c = 0} + (d C) \cap F^{p} C} [p] \\ = \frac{\ker (d |_{F^{p} C})}{\ker (d |_{F^{p + 1} C}) + d C \cap F^{p} C} [p] \\ = {Gr}^{p} H (C) [p] . \end{aligned}$

Some questions

Is every page of spectral sequence computing the same thing? i.e. Is $H (Tot (E_{r})) = H (C)$ for all $r$ ?

Spectral Sequence Of Double Complex Is Equivalent To Spectral Sequence of Filtered Complex

Hodge-to-de Rham Spectral Sequence

Let $X \to S$ be a morphism of schemes, we know that any abelian category with enough injectives have a Cartan-Eilenberg resolution for every bounded below complex $K^{∙}$ .

The Hodge Filtration

Let $X \to S$ be a morphism of schemes, the Hodge Filtration on $H_{d R}^{n} (X / S)$ is the filtration induced by the Hodge-to-de Rham spectral sequence.