This booklet and the subsequent moment quantity is an creation into smooth algebraic geometry. within the first quantity the tools of homological algebra, idea of sheaves, and sheaf cohomology are constructed. those equipment are integral for contemporary algebraic geometry, yet also they are basic for different branches of arithmetic and of serious curiosity of their own.
within the final bankruptcy of quantity I those options are utilized to the idea of compact Riemann surfaces. during this bankruptcy the writer makes transparent how influential the tips of Abel, Riemann and Jacobi have been and that some of the glossy tools were expected via them.
For this moment variation the textual content was once thoroughly revised and corrected. the writer additionally additional a quick part on moduli of elliptic curves with N-level buildings. This new paragraph anticipates many of the thoughts of quantity II.

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