# Analogue of Cartan theorem A in algebraic geometry

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:

http://en.wikipedia.org/wiki/Cartan's_theorems_A_and_B

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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## 1 Answer

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