sheaf at P, and the map returns the th Laurent coefficient, where . Arithmetic Riemann–Roch theorem A version of the arithmetic Riemann–Roch theorem states that if k is a global field, and f is a suitably admissible function of the adeles of k, then for every idele a, one has a Poisson summation formula: In the special case when k is the function field of an algebraic curve over a finite field and f is any character that is trivial on k, this recovers the geometric Riemann–Roch theorem. Other versions of the arithmetic Riemann–Roch theorem make use of Arakelov theory to resemble the traditional Riemann–Roch theorem more exactly. Generalizations of the Riemann–Roch theorem The Riemann–Roch theorem for curves was proved for Riemann surfaces by Riemann and Roch in the 1850s and for algebraic curves by Friedrich Karl Schmidt in 1931 as he was working on perfect fields of finite characteristic. As stated by Peter Roquette, The first main achievement of F. K. Schmidt is the discovery that the classical theorem of Riemann–Roch on compact Riemann surfaces can be transferred to function fields with finite base field. Actually, his proof of the Riemann–Roch theorem works for arbitrary perfect base fields, not necessarily finite. It is foundational in the sense that the subsequent theory for curves tries to refine the information it yields (for example in the Brill–Noether theory). There are versions in higher dimensions (for the appropriate notion of divisor, or line bundle). Their general formulation depends on splitting the theorem into two parts. One, which would now be called Serre duality, interprets the term as a dimension of a first sheaf cohomology group; with the dimension of a zeroth cohomology group, or space of sections, the left-hand side of the theorem becomes an Euler characteristic, and the right-hand side a computation of it as a degree corrected according to the topology of the Riemann surface. In algebraic geometry of dimension two such a formula was found by the geometers of the Italian school; a Riemann–Roch theorem for surfaces was proved (there are several versions, with the first possibly being due to Max Noether). An n-dimensional generalisation, the Hirzebruch–Riemann–Roch theorem, was found and proved by Friedrich Hirzebruch, as an application of characteristic classes in algebraic topology; he was much influenced by the work of Kunihiko Kodaira. At about the same time Jean-Pierre Serre was giving the general form of Serre duality,