Let $M$ be a a left module over the algebra of formal power series, $K=\mathbb{C}[[h]]$

Question

let $M$ be a a left module over the algebra of formal power series, $K=\mathbb{C}[[h]]$. Consider the family $(h^nM)_{n>0}$ of submodules and let $M_n = \frac{M}{h^nM}$. Consider the family of canonical $K$-linear projections, $$p_n: M_n \rightarrow M_{n-1}$$

This confused me because it seems to me that $h^{n-1}M \subset h^nM$ and thus $M_{n} \subset M_{n-1}$, so it seems like these arn't projections but rather embeddings.... I must be making a mistake somewhere, can somebody help me out?

Answer 1

See, $h^n M \subseteq h^{n-1}M$ because a torsion element of order $h^{n-1}$ is not a torsion element of order $h^n$ and so doesn't belong to $h^nM$ while a torsion element of order $h^{n}$ belongs to $h^{n-1} M$.

For example, $2^2 \mathbb{Z} \subset 2 \mathbb{Z}$.

Thus, $M_n \supseteq M_{n-1}$. So $ M_n \to M_{n-1}$ is not an embedding.

You can think the map $M_n \to M_{n-1}$ , the multiplication by $h$.

It is a morphism which is a projection also.

This is in more details, a multiplication by $h$ map.

Answer 2

$h^n(M)=h^{n-1}(h(M))\subseteq h^{n-1}(M)$ since $h(M)\subseteq M$.

So whatever it may “seem” to you, this is the case, and not necessarily the other way around.