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I would like to know whether the differentials in a particular hypercohomology spectral sequence can each be interpreted, in some natural way, as Yoneda products between extension groups.

More specifically, let $R$ be a ring, let $M$ be a left $R$-module, and let $C: 0 \rightarrow C^0 \rightarrow C^1 \rightarrow C^2 \rightarrow \cdots$ be a cochain complex of left $R$-modules. Then there exists a spectral sequence whose $E_2$-page is given by $E_2^{i,j} = \text{Ext}_R^i(M,H^j(C))$, and whose limit is the hypercohomology group $\mathbb{Ext}_R^{i+j}(M,C)$. Explicitly, this spectral sequence can be constructed by first taking a Cartan-Eilenberg resolution $Q = Q^{i,j}$ of $C$. Then $Hom_R(M,Q)$ is a double complex, and the aforementioned hypercohomology spectral sequence is one of the two spectral sequences that naturally arises from this double complex. (I believe it is the spectral sequence that arises from the row-wise filtration of the double complex.)

On the $E_2$-page of this spectral sequence, the differential $d_2: E_2^{i,j} \rightarrow E_2^{i+2,j-1}$ identifies with a map $\text{Ext}_R^i(M,H^j(C)) \rightarrow \text{Ext}_R^{i+2}(M,H^{j-1}(C))$. More generally, on the $E_r$-page, the differential $d_r: E_r^{i,j} \rightarrow E_r^{i+r,j+1-r}$ identifies with a map from a subquotient of $\text{Ext}_R^i(M,H^j(C))$ to a subquotient of $\text{Ext}_R^{i+r}(M,H^{j+1-r}(C))$.

Think of the differential $d_r: E_r^{i,j} \rightarrow E_r^{i+r,j+1-r}$ as a map from a subquotient of $\text{Ext}_R^i(M,H^j(C))$ to a subquotient of $\text{Ext}_R^{i+r}(M,H^{j+1-r}(C))$. Is there necessarily an extension class $\eta$ in $\text{Ext}_R^r(H^j(C),H^{j+1-r}(C))$ such that the differential $d_r$ is induced by left Yoneda multiplication by $\eta$?

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Let me give an interpretation for $d_2$ along the lines that you want. Let $\tau_{\le p}C$ be the truncation operator which sets all terms $C^k=0$ for $k>p$, keeps $C^k$ for $k<p$ and replaces $C^p$ with $\ker d$. This operator passes to the derived category $D= D^+(R\text{-}mod)$. We have a distinguished triangle $$ H^{p-1}(C )[-p+1]\to \tau_{\le p}C/\tau_{\le p-2}C\to H^p(C )[-p]\stackrel{\delta}{\to} H^{p-1}(C ) [-p +2]$$ The last arrow $\delta$ is an element of $$Hom_D(H^{p}(C )[-p], H^{p-1}(C )[-p + 2]) = Ext^2_R(H^{p}(C ), H^{p-1}(C ))$$ Up to sign, $d_2$ is induced from $\delta$ by $$Hom_D(M, H^p(C )[-p])\to Hom_D(M, H^{p-1}(C ) [-p+1])$$

To see that this description of $d_2$ is correct, apply the remark on the top of p 21 of Deligne Theorie de Hodge II to $Dec(\tau)$, where $Dec$ is defined on p 15. I'm sure there is a better reference.

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Perhaps too late, but in this paper the author constructs a change-of-rings SS and shows $d_2^{*,q}$ is given by taking a Yoneda product with a particular class. Unfortunately, there is no description of higher differentials.

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