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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?


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In fact, you can ignore question ii). – Jonathan May 4 '12 at 21:43
up vote 1 down vote accepted

i) $\pi$ cannot be algebraic over $k$ because otherwise $\dim \mathcal{O}_{V,P} = 0$, and hence $V$ is not a curve.

ii) You have not used that $V$ is irreducible, but only that $P$ is a smooth point of $V$. Remember that a smooth point of a curve lies in a unique irreducible component, hence $V$ can be reducible but there is only one irreducible component of $V$ which contains $P$. If $V$ were reducible, you should replace $k(V)$ with the field of fractions of $\mathcal{O}_{V,P}$.

iii) Ok! :-)

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