I am trying to calculate some relative de Rham cohomology, but I am not too skilled with hypercohomology or spectral sequences, and the situation becomes more complicated because (1) the base is not affine, so I need higher direct images instead of sheaf cohomology, and (2) the base is not a field, so the graded pieces of the spectral sequence do not recover the full cohomology.
I have a certain filtration on one of the cohomology groups, but I'm not sure where it fits in a bigger picture. I would be really grateful for some help with this. My primary focus is $H^1_{dR}$, but it would be great to understand more.
Let $\pi: X \to Y$ be a morphism of schemes, and $\Omega_{X/Y}$ the sheaf of relative differentials.
We compute the relative de Rham cohomology as the hypercohomology of the complex $\Omega_{X/Y}^\bullet$.
The hypercohomology can be computed as the cohomology of the total complex $C^\bullet$: $$C^k = \bigoplus\limits_{i+j=k}C^{i,j}$$ where $$C^{i,j} := \check{C}^j\left(\mathcal{U}, \Omega^i_{X/Y}\right)$$ is module of $\check{C}$ech $j$-cocycles for an affine open cover $\mathcal{U}$ of $X$.
In particular $$C^1 = C^{0,1}\oplus C^{1,0}$$ so $\alpha = (\beta, \gamma) \in C^1$ consists of a term $\beta$ from $\mathcal{O}_X$ of "$\check{C}$ech degree 1" and a term $\gamma$ from $\Omega^1_{X/Y}$ of "$\check{C}$ech degree 0".
The $\check{C}$ech 1-cochains $C^{0,1}$ have a filtration: $$\text{cochains } \supset \text{ cocycles } \supset \text{ coboundaries } \supset 0$$
and we can lift this filtration to $C^1$ via the projection $$C^1 = C^{0,1}\oplus C^{1,0} \to C^{0,1}$$ $$\alpha = (\beta, \gamma) \mapsto \beta$$
and this filtration is: $$\operatorname{Fil}^0C^1 = C^1 $$ $$\operatorname{Fil}^1C^1 = \left\{(\beta,\gamma) \,|\; \beta \in \check{Z}^j(\mathcal{U}, \Omega^i_{X/Y})\right\}$$ $$\operatorname{Fil}^2C^1 = \left\{(\beta,\gamma) \,|\; \beta \in \check{B}^j(\mathcal{U}, \Omega^i_{X/Y})\right\}$$ $$\operatorname{Fil}^3C^1 = 0 $$
I am unsure about this, but I think that, at least when $X \to Y$ is relative dimension 1, this filtration passes to the 1st de Rham cohomology, and we get a complex $$R^0\pi_*\Omega_{X/Y} \to H^1_{dR}(X/Y) \to R^1\pi_*\mathcal{O}_X$$
which resembles the short exact sequence one gets from the "filtration bête", but it is not exact, let alone short exact. The complex factors into the sequence $$R^0\pi_*\Omega^1_{X/Y} \to \operatorname{Fil}^2H^1(X/Y) \to H^1(X/S) \to \operatorname{Gr}^1H^1(X/Y) \to R^1\pi_*\mathcal{O}_X$$
I must be looking at a small part of a spectral sequence associated to a filtered complex, but I am having trouble matching it to what I see in the literature.
Questions:
- What is the natural way to extend this filtration to the whole complex, and not just $C^1$?
- Which filtration and spectral sequence am I dealing with here?
- What exact sequence do I get from the spectral sequence to describe $H^1_{dR}$ (not just the associated graded)?
- Is there a better way to approach the computation of $R^\bullet\pi_*\Omega^\bullet_{X/Y}$? I do intend on generalizing to other complexes, like twisted de Rham, etc.
I would really appreciate any help.