Compact Riemann Surfaces
In the theory of compact Riemann surfaces it is possible to make particularly elegant applications of the finiteness theorem. For such considerations we will always let X denote a connected, compact Riemann surface with structure sheaf O. With script letters like S we will denote, as before, coherent analytic sheaves over X. If the support of such a sheaf is finite then T will usually be written. For such a sheaf it is easy to see that H 1(X, T) = (0). The symbols ℱ, G are reserved for locally free O-sheaves. The letter ℒ is usual exclusively for locally free sheaves of rank 1. All tensor products are formed over O.
KeywordsExact Sequence Duality Theorem Compact Riemann Surface Divisor Class Free Sheaf
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