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.