as we now know it. Alexander Grothendieck proved a far-reaching generalization in 1957, now known as the Grothendieck–Riemann–Roch theorem. His work reinterprets Riemann–Roch not as a theorem about a variety, but about a morphism between two varieties. The details of the proofs were published by Armand Borel and Jean-Pierre Serre in 1958. Later, Grothendieck and his collaborators simplified and generalized the proof. Finally a general version was found in algebraic topology, too. These developments were essentially all carried out between 1950 and 1960. After that the Atiyah–Singer index theorem opened another route to generalization. Consequently, the Euler characteristic of a coherent sheaf is reasonably computable. For just one summand within the alternating sum, further arguments such as vanishing theorems must be used. See also Arakelov theory Grothendieck–Riemann–Roch theorem Hirzebruch–Riemann–Roch theorem Kawasaki's Riemann–Roch formula Hilbert polynomial Moduli of algebraic curves Notes References Grothendieck, Alexander, et al. (1966/67), Théorie des Intersections et Théorème de Riemann–Roch (SGA 6), LNM 225, Springer-Verlag, 1971. See pages 208–219 for the proof in the complex situation. Note that Jost uses slightly different notation. , contains the statement for curves over an algebraically closed field. See section IV.1. . Vector bundles on Compact Riemann Surfaces, M. S. Narasimhan, pp. 5–6. Misha Kapovich, The Riemann–Roch Theorem (lecture note) an elementary introduction J. Gray, The Riemann–Roch theorem and Geometry, 1854–1914. Is there a Riemann–Roch for smooth projective curves over an arbitrary field? on MathOverflow Theorems in algebraic geometry Geometry of divisors Topological methods of algebraic geometry Theorems in complex analysis Bernhard Riemann