I have a question on the proof of Lemma 53.2.3. the claim is:
Lemma 52.2.3. Let $k$ be a field. Let $f:X\to Y$ be a nonconstant morphism of curves over $k$. If $Y$ is normal, then $f$ is flat.
the curves are assumed to be smooth and therefore all stalks $\mathcal{O}_{Y,y}$ are either fields or a discrete valuation rings (Lemma 32.43.16). $f$ is non constant and therefore dominant, since $X,Y$ are irreducible of dimension $1$. 'This implies that $\mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$ is injective. $\mathcal{O}_{X,x}$ is torsion-free as a $\mathcal{O}_{Y,y}$-module ...'
The last argument I not understand. Why injectivity of $\mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$ implies that $\mathcal{O}_{X,x}$ is torsion-free as a $\mathcal{O}_{Y,y}$-module?
The theory of finitely generated modules over pid (dvr are pid) gives decomposition of $\mathcal{O}_{X,x}$ as $\mathcal{O}_{X,x}= \mathcal{O}_{Y,y}^m \bigoplus \oplus_{j=1} ^k \mathcal{O}_{Y,y}/\mathfrak{m}_y$ with torsion part $\oplus_{j=1} ^k \mathcal{O}_{Y,y}/\mathfrak{m}_y$. recall $\mathcal{O}_{Y,y}$ has only one non trivial prime $\mathfrak{m}_y$.
Question: Why injectivity of $\mathcal{O}_{Y,y} \to \mathcal{O}_{X,x}$ and this decomption implies that $\mathcal{O}_{X,x}$ is torsion-free as a $\mathcal{O}_{Y,y}$-module?
That’s actually general. Assume that $f: A \rightarrow B$ is an injective ring morphism and assume $B$ is an integral domain (eg, a DVR). Let $x \in B$ be nonzero of torsion, let $a \in A$ be such that $0= a \cdot x=f(a)x$. Thus since $B$ is integral, $f(a)=0$, so since $f$ injective $a=0$.