Context: Deligne, Theorie de Hodge II, section 1.4.8.

Let $f:X\rightarrow Y$ be a map between spaces; $\mathcal{F}$ a sheaf of abelian groups on $X$, and $\mathcal{F} \rightarrow \mathcal{F}^{\bullet}$ a resolution of $\mathcal{F}$ by a complex of $f_{*}$-acyclic sheaves.

The canonical filtration of $f_{*}\mathcal{F}^{\bullet}$ is defined in the reference above as follows:

$$(\tau_{\leq p} f_{*}\mathcal{F}^{\bullet})^n = \begin{cases} f_{*}\mathcal{F}^n & \text{if }\ n<p \\ \text{Ker}(d) & \text{if }\ n = p \\ 0 & \text{if } n>p \end{cases}$$

Question 0: Should the right hand side of the preceding equation be changed by replacing "$p$" with "$-p$" everywhere?

Deligne writes the hypercohomology spectral sequence of $(f_{*}\mathcal{F}^{\bullet},\tau_{\leq p})$ as: $$E_1^{p,q}= H^{2p+q}(Y,R^{-p}f_{*}\mathcal{F}^{\bullet}) \Rightarrow H^{p+q}(X,\mathcal{F})$$

Question 1: Where did this latter spectral sequence come from? Specifically, how is it the hypercohomology spectral sequence of $(f_{*}\mathcal{F}^{\bullet},\tau_{\leq p})$ ? Assuming the correct answer to Question 0 is "Yes", I thought the hypercohomology spectral sequence should read:

$$R^{p+q}\Gamma(\text{Gr}^p f_{*}\mathcal{F}^{\bullet}) = H^{p+q}(Y,\mathcal{H}^{-p}(f_{*}\mathcal{F}^{\bullet})) = H^{p+q}(Y,R^{-p}f_{*}\mathcal{F}) \Rightarrow H^{p+q}(X,\mathcal{F})$$


1 Answer 1


This is really a comment, but too long to write as one. Deligne leaves some steps to the reader. The filtration $\tau$ is increasing. To write the standard spectral sequence, we need to switch signs (as you seem to surmise) to get a decreasing filtration. Let me write $T^p = \tau_{\le -p}$. Then we have $$E_1^{p+q} = H^{p+q}(Y, Gr_T^p \mathbb{R} f_*\mathcal{F})$$ converging to what you wrote. This simplifies to $$H^{p+q}(Y, R^{-p} f_*\mathcal{F}[p])=H^{2p+q}(Y, R^{-p} f_*\mathcal{F})$$ So all is well.


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