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:
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If it were, then $f_*\mathcal{O}_X$ would be a flat sheaf of $\mathcal{O}_Y$-module: ok cf here
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problem 1: It is coherent: why we don't have in general $f_*($coherent$)=$coherent
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Hence it is locally free: ok (finit type+local+flat+noetherian)
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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.
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Hence $f_*\mathcal{O}_X$ is an invertible sheaf: ok, by definition.
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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?
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Problem 4: Hence $(f_*\mathcal{O}_X)_Q$ has two generators as $\mathcal{O}_Y$-module: why?
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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.
Best Answer
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.