*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:


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



Leray spectral sequence from hypercohomology
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.


