Let $M$ be a co-admissible module over a Frechet Stein Algebra $A=\varprojlim A\_{q\_n}$ as in [this paper](https://arxiv.org/pdf/math/0206056.pdf). 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.



Coherent subsheaf of co-admissible modules of Schneider and Teitelbaum
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.


