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Before starting algebraic geometry, I got some understanding of Compact Riemann Surfaces which is more or less rigorous; and then I attended a couple of lectures on analytic geometry. I did not quite fully grasp the analytic geometry rigorously; but it stuck in my mind and this previous experiences helped me greatly when learning algebraic geometry. However I am not able to imagine what would be the precise analogue of Cartan's Theorem A for algebraic varieties over a field. There is the wikipedia article:


which tells me the precise analogue of Cartan Theorem B in the book of Hartshorne. And indeed it is really analogous. But it does not mention Theorem A. Note that here I am restricting the situation to varieties out of fear of any pathologies schemes might have.

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up vote 3 down vote accepted

Over an affine scheme any quasi-coherent sheaf is generated by its global sections, so I guess Theorem A is always true.

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Ah! Was rather stupid of me not to find it out by myself. Thank you. –  user977 Aug 9 '10 at 19:04
Heh, no problem, I just learned about it from Hartshorne! –  curious Aug 9 '10 at 19:05

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