# Leray spectral sequence from hypercohomology

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 \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:

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

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.