Finiteness of analytic cohomology of Lubin-Tate \(( \varphi_L,\Gamma_L)\)-modules (Q6585657)

Let $p>2$ be a prime number and let $L/\mathbb Q_{p}$ be a finite extension and let $\varphi_{L}$ be a Frobenius power series for a uniformiser $\pi_{L}$. Suppose $L_\infty$ is the Lubin-Tate extension of $L$ attached to $\varphi_{L}$ and $\Gamma_{L}=\mathrm{Gal}(L_\infty/L)$. When $L=\mathbb Q_{p}$ and $\varphi(T)=(T+1)^{p}-1$ there is an equivalence of categories between $p$-adic Galois representations and étale $(\varphi,\Gamma)$-modules. For a simple descriptions of the Galois and Iwasawa cohomology of a representation $V$ in terms of the $(\varphi,\Gamma)$-module attached to $V$ we can use $(\varphi,\Gamma)$-modules theory [\textit{F. Cherbonnier} and \textit{P. Colmez}, J. Am. Math. Soc. 12, No. 1, 241--268 (1999; Zbl 0933.11056); \textit{L. Herr}, Bull. Soc. Math. Fr. 126, No. 4, 563--600 (1998; Zbl 0967.11050)]. So, similarly it can be used the Lubin-Tate $(\varphi_{L},\Gamma_{L})$-modules attached to \(L\)-linear representations of $G_{L}$ to obtain similar results [\textit{B. Kupferer} and \textit{O. Venjakob}, Mathematika 68, No. 1, 74--147 (2022; Zbl 1523.11210); \textit{P. Schneider} and \textit{O. Venjakob}, Springer Proc. Math. Stat. 188, 401--468 (2016; Zbl 1409.11107)]. To do so, we need to work with $(\varphi_{L},\Gamma_{L})$-modules over the Robba ring $R_{L}$. It is known that the category of étale $(\varphi_{L},\Gamma_{L})$-modules over $R_{L}$ is equivalent to the category of overconvergent representations. In the case of $L=\mathbb Q_{p}$ every Galois representation is overconvergent [\textit{F. Cherbonnier} and \textit{P. Colmez}, Invent. Math. 133, No. 3, 581--611 (1998; Zbl 0928.11051)], but the theory breaks down if $L\neq \mathbb Q_{p}$. The author of the paper under review motivated by the work of \textit{K. S. Kedlaya} et al. [J. Am. Math. Soc. 27, No. 4, 1043--1115 (2014; Zbl 1314.11028)] and specialises to their situation in the case of $L=\mathbb Q_{p}$ to prove finiteness and base change properties for analytic cohomology of families of $L$-analytic $(\varphi_{L},\Gamma_{L})$-modules parametrised by affinoid algebras in the sense of Tate.