Riemann-Roch theorem

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In algebraic geometry the Riemann-Roch theorem states that if is a smooth algebraic curve, and is an invertible sheaf on Failed to parse (SVG (MathML can be enabled via browser plugin): Invalid response ("Math extension cannot connect to Restbase.") from server "https://wikimedia.org/api/rest_v1/":): {\displaystyle C} then the the following properties hold:

  • The Euler characteristic of is given by
  • There is a canonical isomorphism

Generalizations

Proofs

Using modern tools, the theorem is an immediate consequence of Serre's duality.