Let $V$ be an irreducible curve with $P \in V$ a smooth point. Let $\pi_P$ be a local parameter of $P$. I'd like to show that $\mathrm{dim}_k \mathcal{O}_{V,P} / (\pi_P^n) = n$ for $n \in \mathbb N$.
At first glance, I'd expect $1, \pi, \pi^2 , \ldots , \pi^{n-1}$ to be a basis. In fact, it's pretty obvious they span the space (since $\mathcal{O}_{V,P} = \{ f \ | \ \nu_P(f) \geq 0 \}$). Suppose that $a_0 + a_1 \pi + \ldots + a_{n-1} \pi^{n-1} \in (\pi^n)$ (*) for $a_i \in k$ not all $0$. Then observe the following Lemma:
Let $P$ be a smooth point of the irreducible curve $V$. If $f,g \in k(V)$, then $\nu_P(f + g) \geq \mathrm{min}(\nu_P(f), \nu_P(g))$, with equality if $\nu_P(f) \neq \nu_P(g)$.
This tells me that $\nu_P(a_0 + \ldots + a_{n-1}\pi^{n-1}) = i $ where $i$ is the smallest such that $a_i$ is not $0$. In particular, $\nu_P(a_0 + \ldots + a_{n-1}\pi^{n-1}) \leq n-1$. The valuation at $P$ of the RHS of (*) however is at least $n$ (provided it isn't $0 \in \mathcal{O}_{V,P}$). I have the following queries:
i) What happens if it is $0$, though? Is it possible to have $a_0 + \ldots + a_{n-1} \pi^{n-1} = 0 \in \mathcal{O}_{V,P}$?
ii) Where have I used irreducibility of $V$? Does the Lemma hold for $V$ non-irreducible?
iii) Is my reasoning OK?
Thanks!