Restriction of $(\varphi, N)$-modules

Question

For any $p$-dic field $K$, we have an equivalence of categories $$D_{st}:Rep_{\mathbb{Q}_p}^{st}(G_K)\rightarrow MF_K^{ad}(\varphi,N),\quad V\mapsto (B_{st}\otimes_{\mathbb{Q}_p} V)^{G_K}$$ with quasi-inverse $V_st$ determined by $$V_{st}(D)=(B_{st}\otimes_{K_0} D)^{\varphi=1,N=0}\cap Fil^0(B_{dR}\otimes_{K_0} D).$$ If $L/K$ is a finite Galois extension, I'd like to understand the natural restriction functor $$ \mathcal{R}:MF_K^{ad}(\varphi,N)\overset{V_{st}}{\longrightarrow}Rep_{\mathbb{Q}_p}^{st}(G_K) \overset{Res^L_K}{\longrightarrow} Rep_{\mathbb{Q}_p}^{st}(G_L)\overset{D_{st}}{\longrightarrow} MF_L^{ad}(\varphi,N).$$ If we take $D\in MF_K^{ad}(\varphi,N)$, then $\mathcal{R}(D)=\left( B_{st}\otimes_{\mathbb{Q}_p} V_{st}(D)\right)^{G_L}.$ By definition $\left( B_{st}\otimes_{\mathbb{Q}_p} V_{st}(D)\right)^{G_K}\cong D$, and thus we have $\mathcal{R}(D)^{\text{Gal}(L/K)}\cong D$. So, maybe a bit naively, one might expect that $$\mathcal{R}(D)= D\otimes_{K_0} L_0 .$$ Is this true? The main reason why I think it might be wrong is this paper by E. De Shalit and G.Porat, where they suggest a variation of the definition of $\varphi$-modules which "allows [them] to compute the functors of induction and restriction for $(\varphi,\Gamma)$-modules, when the ground field changes". So that would imply that the variation is necessary to understand restriction. The reason why I'd prefer to understand the restriction on the "original" definition is that their variation classifies $Rep^{st}_{\mathcal{O}_L}(G_L)$ whereas the "original" definition classifies $Rep^{st}_{\mathbb{Q}_p}(G_L)$.

Answer 1

Don't confuse $(\phi, N)$-modules (which are finite-dimensional vector spaces over $\mathbf{Q}_p$ with various extra structures) with $(\phi, \Gamma)$-modules (which are modules over a much bigger and more complicated ring, but see all Galois reps, not just semistable ones). De Shalit and Porat are working on the $(\phi, \Gamma)$-theory.

In your case, what you write is almost correct. You have only written "half" of the base-extension functor because you are not specifying the filtration; the half you have written (specifying the $K_0$-vector space with its $\phi$ and $N$ actions) is correct, and the filtration is what you would naively expect: $$\mathrm{Fil}^i(\mathcal{R}(D) \otimes_{L_0} L) = \mathrm{Fil}^i D \otimes_{K} L.$$