Questions about example Hartshorne III.9.7.1 In Hartshorne III.9.7.1 page 258 the author consider a curve $Y$ with a node and $f:X\to Y$ its normalization. He wants to prove that this map is not flat. He reasons as follow:

*

*If it were, then $f_*\mathcal{O}_X$ would be a flat sheaf of $\mathcal{O}_Y$-module: ok cf here


*problem 1: It is coherent: why we don't have in general $f_*($coherent$)=$coherent


*Hence it is locally free: ok (finit type+local+flat+noetherian)


*Problem 2: $f_*\mathcal{O}_X$ has rank 1: why? in ring I guess that is to say $B\otimes_A A_\mathfrak{p}$ has rank 1 as $A_\mathfrak{p}$-module that is (isn't?) that $B_\mathfrak{q}\otimes k(\mathfrak{p})$ is a dimension 1 $k(\mathfrak{p})$-vectorial space: it is always unclear to me.


*Hence $f_*\mathcal{O}_X$ is an invertible sheaf: ok, by definition.


*Problem 3: There are 2 points $P_1$, $P_2$ going to the node $Q$ of $Y$: why? In exercise I.5.6 it's proven but when $X$ is the blowing-up. Maybe for curves one has blowing-up=normalization?


*Problem 4: Hence $(f_*\mathcal{O}_X)_Q$ has two generators as $\mathcal{O}_Y$-module: why?


*Problem 5: Hence $(f_*\mathcal{O}_X)_Q$ is not locally free: why? Does the author make a mistake and he means that it's not an invertible sheaf?
Thanks for all your answers, even if there are partial.
 A: *

*Coherent has a finiteness assumption which can be fail very easily. Consider the natural projection $\pi:\Bbb A^1_k\to \operatorname{Spec} k$. Then $\pi_*(\mathcal{O}_{\Bbb A^1_k})$ is an infinite-dimensional vector space which is not coherent.

*The rank of $\mathcal{F}$ at a point $x\in X$ is $\dim_{k(x)} \mathcal{F}_x\otimes k(x)$, and when we talk about rank of a sheaf, we usually mean at the generic point. Since there is a neighborhood of the generic point where normalization is an isomorphism, we see that the rank of $f_*\mathcal{O}_X$ is 1.

*A normal variety is singular in codimension two, which means that a normal curve is smooth. The same method you used to show that the (proper, birational) blowdown map from a smooth curve to your node has two points in the fiber over the node should work here.

*Consider the indicator functions of the distinct points in the fiber (the function which is $1$ on that point and $0$ on the other point of the fiber). There's no $\mathcal{O}_Y$ relation between them (no function on $Y$ can tell the two points apart since they both map to the same point of $Y$), so they have to be independent.

*In a line bundle, any stalk of a nonvanishing section should be a generator of the stalk, so there should be an $\mathcal{O}_Y$ relation between the two sections. This contradicts 4.
