This is a beginner's question.
I don't understand the description after defining Weil divisor in Chapter 2 of Hartshorne's book 'Algebraic Geometry'.
Let $ X $ be a noetherian, integral, separated, regular in codimension one scheme.
Let $ Y $ be the prime divisor of $ X $ and $ \eta $ be the generic point of $ Y $.
It is written that the local ring $ \mathcal {O} _ {\eta, X} $ in $ \eta $ becomes a DVR with the function field $ K $ of $ X $ as the field of fractions.
However, in Ex.3.6 (I'm sorry if the problem number is different because it is the Japanese version), it is written that the local ring at the generic point of the integral scheme becomes the function field of $ X $.
If I follow this, $ \mathcal {O} _ {\eta, X} $ becomes a field, and I don't think it becomes the DVR.
What am I doing wrong?
Best Answer
Exercise 3.6 says that if $X$ is an integral scheme and $\eta$ is the generic point of $X$, then $\mathcal{O}_{\eta,X}$ is the function field of $X$. However, in the context you are asking about, $\eta$ is not the generic point of $X$ itself, but rather of the subscheme $Y$. So, Exercise 3.6 doesn't tell you anything about $\mathcal{O}_{\eta,X}$ (instead it would tell you about $\mathcal{O}_{\eta,Y}$).