Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum

Question

Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A_{q_n}$ as in this paper. Let $N$ be a closed submodule of $M$. I have some difficulty in understanding lemma $3.6$ of the above paper. That is I want to show that $N$ is co-admissible.

For that, consider $M=\varprojlim M_n$. The authors considers $N_n \subset M_n$, the $A_{q_n}$-submodule generated by the images of $N$ under the natural map $M \rightarrow M_n$. Then the authors say that $(N_n)_n$ is a coherent subsheaf of $(M_n)_n$, which is what I have some problem in understanding.

I am having difficulty in showing that $A_{q_{n}} \otimes_{A_{q_{n+1}}} N_{n+1}=N_n$. (I know that it should come from the equality $A_{q_{n}} \otimes_{A_{q_{n+1}}} M_{n+1}=M_n$ which is true by definition.) But somehow I cannot deduce it. Thank you for your help.

Answer 1

The map $A_{q_n} \otimes_{A_{q_{n+1}}} N_{n+1} \rightarrow N_n$ is surjective, by definition of $N_n$. To show that it is injective, it suffices to show that the composition $A_{q_n} \otimes_{A_{q_{n+1}}} N_{n+1} \rightarrow N_n \rightarrow M_n$ is injective. But this composition factors as $$ A_{q_n} \otimes_{A_{q_{n+1}}} N_{n+1} \rightarrow A_{q_n} \otimes_{A_{q_{n+1}}} M_{n+1} \rightarrow M_n. $$ The first arrow is injective because $A_{q_{n}}$ is a flat $A_{q_{n+1}}$-module, and the second arrow is an isomorphism.