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I'm reading Hartshorne Chapter 2.7.

Let $X$ be a scheme over $A$ and $f:X\rightarrow \mathbb P_A^n$ be an $A$ morphism. Define $\mathcal L = f^*(\mathcal O(1))$ and $s_i = f^*(x_i)$. Then $\mathcal L$ is an invertible sheaf and $s_i$ generate the sheaf $\mathcal L$.

I have 2 questions:

1.Why $\mathcal L$ is an invertible sheaf. In general, is that true that for any $f:X\rightarrow Y$ and locally free sheaf $\mathcal F$, we have $f^*\mathcal F$ is still locally free on $X$?

2.Why $s_i$ generate the sheaf $\mathcal L$? In general, let $f:X\rightarrow Y$ be any morphism and $\mathcal F$ be any sheaf on $Y$ and $s$ be a section of $\mathcal F$. Then, is there any relation between $f^*(s)_x$ and $s_x$?

Hydrogen
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    Question 1 has been addressed on this site before: pullback of locally free sheaves is locally free. For question 2, do you know the definition of the pullback and how it interacts with stalks? – KReiser Oct 06 '20 at 00:44
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    Pullback is symmetric monoidal with respect to the tensor product (we have a natural isomorphism $f^{\ast}(F \otimes G) \cong f^{\ast}(F) \otimes f^{\ast}(G)$) and symmetric monoidal functors send invertible objects to invertible objects. – Qiaochu Yuan Oct 06 '20 at 01:56
  • @KReiser I know the definition of pullback. But the construction is too complicated for me(involving pullback sheaves, tensor and sheafification). Hence I have no idea how it interacts with stalks. – Hydrogen Oct 06 '20 at 02:42
  • @QiaochuYuan Thank you. I'm still confused about question 2. Why $s_i$ actually generated $\mathcal L$? – Hydrogen Oct 06 '20 at 16:48
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    @Hydrogen: this is maybe a bit more abstract than necessary, like my first answer, but; to say that some sections $s_i, 0 \le i \le n$ generate a sheaf $L$ is to say that the $s_i$ define an epimorphism $\mathcal{O}^{n+1} \to L$. Pullback is the left adjoint of pushforward so it preserves cokernels and hence preserves epimorphisms, and it also sends structure sheaves to structure sheaves, so the pullback of such an epimorphism is another. It would be good to get some intuition for these statements by checking what they say for affine schemes and categories of modules. – Qiaochu Yuan Oct 06 '20 at 16:52

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