Let $X$ be a variety over a field $k$ with separable closure $k_s$. Let $A$, $B$ be étale sheaves on $X$. Consider now the Hochschild–Serre spectral sequences.
\begin{align*}
E_2^{pq}: H^p(k, H^q(X_{k_s}, A)) & {}\Rightarrow H^{p + q}(X, A) \\
E_2^{pq}: H^p(k, H^q(X_{k_s}, B)) & {}\Rightarrow H^{p + q}(X, B) \\
E_2^{pq}: H^p(k, H^q(X_{k_s}, A \otimes B)) & {}\Rightarrow H^{p + q}(X, A \otimes B).
\end{align*}
The cup product for $H^{\bullet}(X_{k_s}, \cdot)$ gives a bilinear pairing $H^q(X_{k_s}, A) \times H^{q'}(X_{k_s}, B) \to H^{q + q'}(X_{k_s}, A \otimes B)$. This induces via the cup product on $H^{\bullet}(k, \cdot)$ a bilinear pairing $E^{pq}_2 \times E^{p' q'}_2 \to E^{(p + p')(q + q')}_2$. This should be the same bilinear pairing (up to a sign) as the one coming from the usual cup product pairing $H^{p + q}(X, A) \times H^{p' + q'}(X, B) \to H^{p + p' + q +q'}(X, A \otimes C)$ (part of the statement being that the cup product is compatible with the filtration).
For the Serre spectral sequence in algebraic topology this is described in Hatcher – Algebraic topology – Spectral sequences page 543 or, Hutchings – Introduction to spectral sequences. Does anyone know of a good reference where this is explained for the Hochschild–Serre spectral sequence? The Leray spectral sequence would also be fine, since Hochschild–Serre is a special case. Especially the signs seems like they might be subtle.
Best Answer
The original reference is
On page 118 they introduce a filtration $\{A_j\}_j$ of the cochain complex $A = C^*(G; M)$, which is compatible with cup products pairings and gives the desired pairing of Hochschild-Serre spectral sequences, as stated on page 119. The same kind of filtration was used by Serre in the context of cubical cohomology to define the multiplicative structure on the Serre spectral sequence of a fibration. Perhaps Chapters 5 and 6 of
https://www.uio.no/studier/emner/matnat/math/MAT9580/v21/dokumenter/spseq.pdf
are helpful.