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The question $\mathcal{L}$ is very ample, $\mathcal{U}$ is generated by global sections $\Rightarrow$ $\mathcal{L} \otimes \mathcal{U}$ is very ample asks for a proof of the following statement:

Let $X$ be Noetherian and of finite type over a Noetherian ring, $\mathcal L$ and $\mathcal M$ line bundles on $X$ with $\mathcal L$ very ample and $\mathcal M$ generated by global sections. Then $\mathcal L \otimes \mathcal M$ is very ample.

I followed the proof in the comments there but failed to find a step where the finite type assumption is necessary. On the other hand, this is exercise III.7.5 (d) in Hartshorne's Algebraic Geometry and there this assumption is stated explicitly for part (d).

So is this assumption necessary? (If no, does the proof in the linked thread use it somewhere nonetheless and I didn't realise?)

Jonathan Gruner
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    If not finite type and not Noetherian, what is your definition of very ample? – Mohan Apr 21 '16 at 01:47
  • A line bundle is very ample if it is the pullback of $\mathcal{O}(1)$ on $\mathbb{P}^n$ along an immersion. This should be the definition used by Hartshorne. – Jonathan Gruner Apr 21 '16 at 18:53
  • I assume you are working over a field. Can you think of an example of such a variety (immersed in $\mathbb{P}^n$) which is not finite type? – Mohan Apr 21 '16 at 20:06
  • I work over a Noetherian ring $A$. But now I see: $\mathbb{P}^n$ is of finite type over the ring and an immersion is locally of finite type and an immersion in a locally compact scheme is quasi-compact, so $X$ is of finite type. – Jonathan Gruner Apr 22 '16 at 12:04
  • But then I am really confused why Hartshorne asks for this property in III.7.5 (d). But maybe this is because he didn't want to give different assumptions for (d) and (e). – Jonathan Gruner Apr 22 '16 at 12:16

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