The following is from BGV paper (https://eprint.iacr.org/2011/277.pdf) p. 12.
$\text{FHE.Add}(pk,\textbf{c}_1,\textbf{c}_2)$: Takes two ciphertexts encrypted under the same $\textbf{s}_j$ (If they are not initially, use $\text{FHE.Refresh}$ (below) to make it so.) Set $\textbf{c}_3\leftarrow \textbf{c}_1+\textbf{c}_2 \mod q_j$. Interpret $\textbf{c}_3$ as a ciphertext under $\textbf{s}_j'$ ($\textbf{s}_j'$'s coefficients include all of $\textbf{s}_j$'s since $\textbf{s}_j'=\textbf{s}_j\otimes\textbf{s}_j$ and $\textbf{s}_j$'s first coefficientn is $1$) and output:
\begin{equation}
\textbf{c}_4\leftarrow\text{FHE.Refresh}(\textbf{c}_3,\tau_{\textbf{s}_j''\to\textbf{s}_{j-1}},q_j,q_{j-1})
\end{equation}
If $\textbf{c}_3$ is a ciphertext under $\textbf{s}_j'$, we should be able to define an inner product $\langle \textbf{c}_3,\textbf{s}_j'\rangle$. But $\textbf{c}_3$ is in $R^n$ where $R$ is some unital commutative ring (specified in the paper but probably not important for this question), while $\textbf{s}_j'$ is in $R^{n^2}$. Shouldn't $\textbf{c}_3$ and $\textbf{s}_j'$ be in the same space in order for an inner product to be defined?
